Physics Lab Blogs

Containing the blogs I make while I'm taking my undergrad degree. Currently with 101.1 (Mechanics) and 103.1 (Optics and Thermodynamics) stuff.

[matophysics-blog]

Quantifying Heat Exchange

We previously talked about how heat and other factors interact with gasses. Now on our last topic for this lab series, and the last topic for thermodynamics, we will placing numbers and equations on heat exchange.

The amount of heat required to change the temperature of an object depends on a lot of different factors. The following equation shows this relationship

where Q is the amount of heat needed, m is the mass of the object in question, c is the specific heat capacity (or the amount of heat needed to change one gram of an object by one degree C/K), and (delta)T is the change in temperature of the material. The specific heat of an object depends on the type of substance being treated. Water, one of the most used substances in thermodynamic experiments in the classroom, is known as having a high specific heat capacity at 4.184 J/(g * C). Materials with high specific heat capacities take a lot of energy to change temperature; thus, they are great for cooling down hot objects.

In today’s experiment, we will determine the specific heat capacities of brass and copper using a method known as calorimetry.

A calorimeter is an insulated vessel, which prevents heat exchange between the interactions within the calorimeter and the surroundings. These calorimeters allow us to determine the heat capacities of an object by having it in thermal contact with another substance of known specific heat capacity (usually water) within its body. In this experiment, the calorimeter is a coffee-cup calorimeter that consists of stacked coffee cups. Styrofoam coffee-cups are usually used as they are good insulators that reduce the amount of heat that escapes to the surroundings.

They work on the basic thermodynamic law, where the sum of all total heat exchange between system and surrounds is zero, expressed in the equations:

In this experiment, we will identify the specific heat capacities of brass and copper samples by having hot samples interact with cold water within the calorimeter. When the hot sample enters the water, heat exchange occurs and energy is transferred into both the cold water and the calorimeter. The hot metal drops in temperature, losing heat, while both the water and calorimeter gain heat.

You may be wondering how we know the specific heat capacity of our calorimeter if its unique to all experimentation. That is true, there is no fixed formula for identifying the components of the calorimeter and directly computing for the specific heat capacity. Instead, we take the calorimeter’s heat capacity (the amount of energy needed to change the calorimeter’s temperature by 1 degree C/K) through calibration. 

Calibration

Calibration of coffee-cup calorimeters involves having a reaction of known heat exchange occur within the calorimeter to find out how much heat is transferred to the calorimeter. In this activity, hot water was added to tap water, with the heat lost by the hot water equaling the sum of the heat gained by the tap water and the heat gained by the calorimeter.

Temperatures in this experiment were measured using a digital thermocouple with smallest reading 0.1 degree Celsius.

50 mL of tap water, its temperature recorded, was placed in the calorimeter. 50 mL of hot water was then taken from a boiling heat bath, with water temperature also measured, and made to mix with the tap water.

The temperature of the water in the calorimeter was then measured at thirty (30) second intervals until a total time of five (5) minutes had elapsed. Data was then graphed twice: first a graph of T (temperature) versus t (time) was made; then ln(T) vs t graphs were also done. A linear fit was placed on the ln(T) vs t graphs, and the y-intecept was taken. This y-intercept was used to take the temperature of the mixture at initial contact T0, by using EXP(y-intercept). Using equations (1), (2), and (3), the heat capacity of the calorimeter Ccal could be obtained.

This was done several times to get the best Ccal value for our calorimeter. The figures below present our data.

Figure 1. Trial 1 for calibration

Figure 2. Trial 2 for calibration

Figure 3. Trial 3 for calibration

Figure 4. Trial 4 for calibration

It was observed that during our experimentation, Ccal values would always be negative. We know this was not supposed to happen, as Ccal values are known to be negative and it requires an input of energy to increase temperature. We identified the main source of error to be an erratic thermocouple, which had difficulty reading temperatures above 40 degrees Celsius. This could have led directly to the negative values obtained as measurements would be important in this computation. We also considered limitations of the calorimeter in fully trapping heat and the presence of sediments in the water used making it having an increase in its specific heat capacity.

However, under the constraints of time and resources, we had to proceed. We continued the experiment under the assumption that this inconsistency would be a constant systematic error in the succeeding parts of the experiment. We decided to use the Trial 1 data for C at -109.53735 J/C as our Ccal.

Finding Specific Heat Capacity

We then attempted to find the specific heat of a 144.1 g brass sample and a 128.2 g copper sample (measurements taken using a digital scale of lowest reading 0.1 g) using calorimetry. The samples were placed in boiling water and had their temperatures rise to the temperature of the water (90 C). Individually, a sample was placed in 75 g of water, with temperature measured, in the calorimeter. The temperature of the water over time was measured similarly to what was done in calibration. Using the different equations again, the specific heats of the the samples were computed.

The following graphs were our ln(T) vs t graphs.

Figure 5. Specific heat capacity of brass

By determining T0 again, it was found that the specific heat capacity of brass was found to be -0.391 J/(g*C). Once again, a negative value was obtained. This could have been due to similar errors, as well as utilizing the negative Ccal value. Compared to a theoretical value of 0.380 J/(g*C) for brass, this came out at a 202.86% error. However, if we take the absolute value of the experimental specific heat capacity, we have only 2.86% error.

Figure 6. Specific heat capacity of copper

Similarly, it was found that the specific heat capacity of copper was -0.299 J/(g*C). We had the same problem of negative specific heat capacities and attribute it to the same errors. Compared to a theoretical value of 0.386 J/(g*C) for copper, the computed value had a 177.46% error, and an absolute specific heat capacity error of 22.54%.

Although it did not appear to be effective in our experimentation, calorimetry has been used in both physics and chemistry to explore thermodynamics and thermodynamic properties. We could improve this experiment by using better measurement devices and isolating our system more by increasing the quality of our calorimetry. There are other such calorimeters such as the bomb calorimeter that allow for different types of thermodynamic experimentation.

We are still discovering more and more things about thermodynamics and how heat interacts with all objects. As scientists, we aid in this discovery through our experimentations, both in our successes and failures. I hope these blog entries have inspired you to pursue the sciences, especially physics, to discover more about our world and how it works.

Until next time, thank you for reading.

REFERENCES

1.     Physics 103.1 Lab Manual, T5 Calorimetry, National Institute of Physics, Philippines, 2013. 2.     R.A Freedman, H.D. Young, University Physics with Modern Physics 13th Edition, Chapter 17, Pearson Education, San Francisco, 2012. 3. Specific heats taken from http://hyperphysics.phy-astr.gsu.edu/hbase/tables/sphtt.html

[matophysics-blog]

Not Just Hot Air: The Ideal Gas Laws

For a while now, we’ve been talking about temperature and how it influences the world around us. This time we will talk about how temperature, among other factors, influences the gases around us. 

Let us observe gas in a fixed container and what happens when we change its temperature; a balloon on a hot summer’s day for instance. Like we described before, temperature is the measurement of the average kinetic energy of the molecules of a substance. The higher the temperature, the faster the molecules move. So as the balloon heats up while it’s under the sun, the air molecules inside it move faster, hitting the walls of the balloon more frequently. This increases the balloon’s pressure, as pressure is the amount of force exerted per given unit of area. This also causes changes in the balloon’s volume, the balloon being forced to expand to accommodate the increase in pressure. Pressure and volume are at a balance though, because as one increases, the other would decrease. However, we all know that if the pressure inside the balloon gets too much, the balloon being only able to expand to a certain extent, it would finally *POP*. 

These three factors, temperature, pressure, and volume, are interrelated to one another through what we know as the Ideal Gas Laws. An ideal gas is a theoretical gas that is composed of many randomly moving point particles that do not interact with one another except when they collide elastically (kinetic energy is conserved). Although it is a theoretical model, it is used to describe dilute gases at low pressures and high temperatures, as gases in these conditions have properties very similar with the ideal gas model. Many common gases can be considered to be ideal gases at room temperatures and pressures close to 1 atm. 

There are three main laws composing the Ideal Gas Laws, each relating a different pair of factors among temperature, pressure, and volume, and each having been discovered experimentally. Boyle’s Law relates volume to pressure and states that they are inversely proportional to one another. Charles’s Law shows how volume is directly proportional to temperature. The last of the laws, Gay-Lussac’s Law, states that pressure is directly proportional to temperature. These three laws can be summarized in the ideal gas equation:

where P is the pressure in Pascals, V is the volume in m^3, N is the number of gas particles, k is the Boltzmann’s constant (k = 1.3806488 × 10−23 m2 · kg · s−2K−1), and T is the temperature measured in Kelvin.

In this experiment, we will be experimentally verifying the former two laws, namely Boyle’s Law and Charles’s Law, using our own set-up.

METHODOLOGY

We tackle verifying the two laws separately but each uses the same instruments and measuring devices. 

Our container of gas in this experiment was the mass lifter apparatus attached to an air chamber via rubber tubing. The mass lifter apparatus had a diameter of 0.0325 m as written on its dimensions as well as gradations of 1 mm measured from 0 to 100 mm. The air chamber (in grey) would be in direct contact with our heat/ice bath, varying the temperature of gas through this contact. 

The mass lifter apparatus was also attached to a Vernier LabQuest with a gas pressure sensor, measuring the pressure within the apparatus in kPa to a smallest reading of 0.01 kPa.

The hot bath consisted of a beaker ¾ filled with water, in which the air chamber was placed, within a pot ¾ filled with water over a hot plate. The water was made to boil and kept boiling throughout the entire experiment.

In the succeeding part of the experiment, the hot bath was removed from the hot plate and chunks of ice were placed directly in the water of the beaker, to cool the gas within the air chamber. Temperature was measured through a digital thermometer with lowest reading of 0.1 degrees Celsius.

Boyle’s Law

To verify Boyle’s Law, we constructed a set-up such that temperature could be kept constant while we varied pressure and volume. The air chamber, connected to the mass lifter apparatus through rubber tubing was, placed in the hot bath. The piston of the mass lifter apparatus was lifted to maximum height. Temperature was measured and kept constant at 353.65 K. 

A 50 g standard mass was placed on the platform of the mass lifter apparatus. After 5 seconds, the height h of the piston and the pressure P measured were recorded. This step was repeated for masses of 100 g, 150 g, 200 g, and 250 g.

Using the heights recorded, the volume of the cylinder, Vcyl, for each height was calculated. The reciprocal of the pressure readings, P^-1, in Pa were also computed. A plot of Vcyl vs P^-1 was graphed with a linear fit. It should be noted that the total volume of air in this experiment was the volume of the cylinder added to the volume of the chamber. Thus, the y-intercept would be the negative of the volume of the chamber. The number of particles N and the volume of the air chamber were calculated using the Ideal Gas Equation using the slope and y-intercept of the graph respectively.

Charles’s Law

To verify Charles’s Law, we constructed a set-up such that pressure could be kept constant while we varied temperature and volume. We removed the hot bath from atop the hot plate from the previous set-up. The piston was lifted to its maximum height. Pressure here was monitored to be a constant 101.61 kPa.

The temperature of the hot bath was measuring using a digital thermometer and the initial height h of the piston was recorded. Ice chunks were slowly added on the hot bath and h and T measurements at equal time intervals of 10 seconds until the set-up had cooled to close to room temperature.

A plot of V vs T was made with a linear fit with the equation and R^2 values present. The number of particles N and the volume of the air chamber were calculated using the Ideal Gas Equation using the slope and y-intercept of the graph respectively. 

We also note that if we increase the pressure in this experiment, the corresponding effect to its V vs T graph would be a decrease in slope. This could be seen by rearranging variables in the Ideal Gas Equations.

Gay-Lussac’s Law (optional)

Although not accomplished in this experimentation, an additional set-up to verify Gay-Lussac’s Law could be made using the materials used in this experiment by keeping the volume constant while pressure and temperature would be made to vary.

The air chamber would once again be placed in the hot bath and allowed to heat for a while. The hot bath would then be removed from atop the hot plate. The piston would be lifted to maximum height throughout the entire experiment. The initial temperature T and pressure P reading would be taken. Ice chunks would then be slowly added to the water of the beaker, and T and P readings would be taken at equal intervals of 10 seconds, until the set-up reaches near room temperature.

RESULTS AND DISCUSSION

The following graphs show the results of our experimentation.

This graph shows results of our verification of Boyle’s Law. The R^2 value describes the precision of the experiment, showing a moderately high precision. Using the slope of the experiment and our known measurement for the temperature in the Ideal Gas Equation, we could compute an N of 6.02E+21 particles of gas. The volume of the chamber can also be gathered from the y-intercept of the graph, and it is seen to be 0.000205557 m^3.

This graph shows results of our verification of Charles’s Law. The R^2 value is smaller, but still maintains to be moderately high. Once again, using the slope of the experiment and our known measurement for the pressure during the experiment in the Ideal Gas Equation, we find an N of 4.08E+21 particles of gas. The volume of the chamber can also be gathered from the y-intercept of the graph, and it is seen to be 0.000105002 m^3.

It is seen that despite the precision of the experiments as indicated by the R^2 values, some discrepancies occurred as the N and Vcham values are not the same. This could be attributed to some limitations with our experiment and what we could keep constant throughout the experiment. However, the results do remain in the same order of magnitude, describing some accuracy in experimentation.

CONCLUSIONS AND RECOMMENDATIONS

We conclude that the experiment were effective in verifying Boyle’s and Charles’s Laws, showing how volume is inversely proportional to pressure and how volume is directly proportional to temperature respectively. This was evident with the high precision of the linear fit of the graphs made as seen in the high R^2 values, which would only happen if these laws were valid. The experiment was not free of limitations, however. Discrepancies between the measured values of N and Vcham in Boyle’s and Charles Law (of 6.02E+21 particles and 0.000205557 m^3 in Boyle’s Law verification and of 4.08E+21 particles and 0.000105002 m^3 in Charles’s Law verification) show the limitations of our experiment. 

These discrepancies could be attributed to systematic faults in our experiment. It was observed that gas was leaking out of the air chamber and mass lifter apparatus, as the piston’s height would decrease over time even if no force was acting on the system. Although we attempted to overcome this by taking readings at fixed time intervals, this could have still led to the decrease in accuracy of the experiment as observed with the different N and Vcham readings observed. Another limitation would be that the fixed variables in either experiment (temperature for Boyle’s Law and pressure in Charles’s Law) were measured at single instances. The reality was that these factors still varied throughout the experiment. Since these were used in computation, this could have also had an effect on the precision accuracy of the experiment.

To resolve these, I would suggest that better equipment be used and all joints be properly sealed so as to limit the amount of gas exchange. It would also benefit the experiment if the measurements for the supposedly fixed constants would be taken over a longer period of time, and the value used would instead be the average of values over that time frame. With these suggestions, similar experiments could allow more accurate and precise manipulation of factors utilizing the gas laws. 

As gas remains one of the most chaotic substances in terms of movement, it is amazing that we were able to grasp the effect of temperature, pressure, and volume on them. Physics finds new ways to make even the most complex of ideas within our grasps. This experiment specifically shows how experimental physics can be on par with theoretical physics in discovering more and more about the world around us.

REFERENCES

1.     Physics 103.1 Lab Manual, T3 Gas Laws, National Institute of Physics, Philippines, 2013.

2.     R.A Freedman, H.D. Young, University Physics with Modern Physics 13th Edition, Chapter 18, Pearson Education, San Francisco, 2012.

3.     Image retrieved on April 10, 2016, from http://www.sarahmichellerupert.com/wp-content/uploads/2007/06/theBalloonPop-webdetail-640.jpg

[matophysics-blog]

Cool Things and Hot Stuff: Measuring Temperature

On a cold rainy day, it’s nice to stay in bed, cuddled up with your blankets and pillows, and take naps with the warmth around you. Later on in the day, too lazy to cook your own food, you check your refrigerator for the leftover pizza you had the night before. It’s cold, and from experience, you would rather skip on chowing down on cold pizza when you have the opportunity to make it taste better. You pop the pizza into the microwave, and after around thirty seconds of anticipation, out comes a “fresh” slice, with its cheesy aroma reminding you why you love pizza so much. A bit too hasty, you took a bite, burning your tongue in the process. You quickly grab a glass of ice water to soothe your tongue, and take your food back to bed for the rest of your lazy day.

Hot and cold are words we use to describe the temperature of an object. Temperature comes in a balance. Neither hot nor cold are inherently bad. We use objects of both degrees to assist us in everyday life, from cold refrigerators keeping our food preserved to hot microwaves to heat up our food when it’s time to eat. When we talk about temperature though, we’re actually describing another characteristic of an object – its internal kinetic energy. In fact, temperature is known as the measurement of the average internal kinetic energy of molecules within an object. A high temperature or something that is hot has molecules moving quickly due to their higher average in kinetic energy. On the other hand, an object of low temperature or something that is cold has its molecules moving relatively slower due to their lesser average kinetic energy.

The heat that we feel coming from a warm slice of pizza is actually a result of the difference between the temperature of the pizza and its surrounding air. We can then define heat as the energy transferred between objects as a result of a difference in temperature. When two objects transfer heat, they reach a point where their temperatures are equal and therefore, no further energy can be transferred. This is called thermal equilibrium. 

Hot and cold though are rather qualitative terms. Something that is hot to one person might just be moderate to another. A Filipino whose natural environment is in the tropics would find winter much colder than someone who is already accustomed to the chilly weather. This is why we find ways to standardize our ways we measure temperature.

We actually have three major degree scales we use to measure temperature. These would be the Celsius (C), Fahrenheit (F), and Kelvin (K) scales. Celsius uses water as its basis, with 0 degrees as the freezing point of water and 100 degrees as the boiling point of water. Fahrenheit also follows this, having 32 and 212 degrees marking the two phase changes respectively. The Kelvin scale has 0 K being absolute zero or the temperature at which there is no kinetic energy of molecules within the object and all molecules are at rest. Celsius and Fahrenheit are used in everyday life, depending on a person’s location in the world; however, we in the science community use Kelvin as the SI unit for temperature as it closely relates to our definition of what temperature actually is.

Measuring temperature requires the use of a thermometer. When a thermometer is placed in contact with a substance, heat transfer occurs until the thermometer and the substance reach thermal equilibrium. The thermometer actually reads its own temperature in this case. It should be noted that placing the thermometer in contact with the observed substance changes the substance’s temperature. It is then essential that the transfer of heat to or from the thermometer remain minimal.

Heat does not transfer linearly, however. The greater the difference between the temperatures of the two objects, the faster heat is transferred. Once the difference decreases, the rate at which heat is transferred also decreases. This is visualized in the following graph:

Figure 1. Graph of temperature versus time in a heating process. 

We let T(t) denote the temperature at a given time t. The shape of this graph or how fast a thermometer reaches thermal equilibrium depends on the thermal time constant (tau). Using our knowledge of exponential functions, we can solve for T(t) through the equation

When time is equal to different multiples of tau, we get the following

We can then observe that accurate measurements for the final temperature of an object are measured at t values between 3 tau and 5 tau. Shorter than that and you would have an imprecise reading while longer than that would risk heat transfer between the system and its surroundings.

Typical thermometers used would include alcohol and mercury thermometers, where a thin glass cylinder is filled with the substance and scaled to provide its readings. How exactly can we get readings from these thermometers? Well when these measuring devices increase in temperature, the substances within them expand as a result of the molecules, with increase in energy, moving further away from one another. For thermometers with measurements in Celsius, the usual way of marking would involve allowing the thermometer to reach temperatures of 0 and 100 degrees, using ice and boiling water, marking these points, and creating smaller, proportional divisions. Since the length of the substance within the thermometers expands proportional to the temperature, this method of creating divisions would remain accurate.

In this experiment, we used the concept of thermal equilibrium to obtain the thermal time constants for an alcohol thermometer and a mercury thermometer.

 Methodology

Two setups were used for experimentation.

Figure 2. Hot setup of boiling water in a 3/4 filled pot.

The first was our hot setup. A pot was filled with water up to 3/4 of its capacity and left to boil. This water would be boiling throughout the entire activity so as to allow the water to remain at temperatures near the boiling point.

Figure 3. Cold setup of ice and ice water

The second was our cold setup. A beaker was filled with ice water and left to stand. The ice would allow the water to remain at temperatures close to the freezing point of water.

To determine the thermal time constants for the given thermometers, two procedures were conducted: the heating and cooling procedures. For each of the procedures, three trials were conducted for each of the alcohol and mercury thermometers.

For the heating procedure, a thermometer was first placed in the hot water, and the water’s temperature was recorded as Tf. The thermometer was then dipped in the cold water. The temperature in the cold setup was recorded as Ti. T(tau) was then computed using the above equation. With initial temperature Ti from the cold setup, the thermometer was placed into the hot setup and the time taken for it to reach T(tau) was recorded as the thermal time constant for the thermometer. It should be noted that time measurements were taken using a digital stopwatch of lowest increment of 0.01 s. It was also ensured that the thermometer did not touch the sides of any of the containers while any measurement was being taken.

For the cooling procedure, a similar process was conducted; however, Tf was initially taken as the temperature of the cold setup while Ti was taken as the temperature of the hot setup. T(tau) was computed for once again. With initial temperature Tf from the hot setup, the thermometer was placed into the cold setup and the time taken for it to reach T(tau) was recorded as the thermal time constant for the thermometer.

Results and Discussion

After conducting the trials for experimentation, the data was tabulated as seen below.

Tables 2 and 3 show the data for the alcohol thermometer. It can be observed that the time constant for cooling is significantly larger than the time constant for heating. We look towards our data for the mercury thermometer for any similar trends. At a range of 3 tau to 5 tau, it would then take 17.46 s to 29.1 s to get an accurate measurement for heating and 21.15 to 35.35 s to get an accurate measurement for cooling, both using an alcohol thermometer.

Tables 4 and 5 show the data for the mercury thermometer. It is noted that the thermal time constant is much smaller than that of the alcohol thermometer. At a range of 3 tau to 5 tau, it would take 5.16 s to 8.10 s to obtain an accurate measurement for heating and 6.57 s and 10.95 s to get an accurate measurement for cooling, both for a mercury thermometer. This coincides with known concepts, as mercury thermometers are used to measure much higher temperatures and thus, expand to greater increments much faster than alcohol thermometers.

It is also of note that once again, the thermal time constants for cooling are significantly greater than the thermal time constants for heating for both alcohol and mercury thermometers. Intuitively, it would be proposed that they would have equal time constants as they are of the same material. However, it can be concluded that the time constants for cooling would indeed be greater than the time constants for heating, and thus, it would take a longer time for an object to decrease its temperature than to increase its temperature by the same difference. This could be attributed to the cooling pattern resembling a logarithmic function with a floor value as opposed to the function given above.

Conclusion and Recommendations

It was discovered that the thermal time constants for the used alcohol thermometer were 5.82 s and 7.05 s for heating and cooling respectively while the time constants for the used mercury thermometer were 1.72 s and 2.19 s for heating and cooling respectively. The thermal time constants were greater for the alcohol thermometer, meaning it would take a longer time for the thermometers to get an accurate reading for the alcohol thermometer than the mercury thermometer. It would also take a longer time to get an accurate reading for a cooling procedure than a heating one.

The deviation between results was significant enough that the results gathered may not have been as precise as we would want. These would be results of reaction time, which for small times would have a greater impact, and heat transfer to the surrounding. To reduce the amount of errors and ensure that a greater precision is achieved, additional trials for each procedure and each thermometer should be conducted.

It is important that we understand the thermal time constants of our thermometers. This is so we can ensure accurate and precise temperature measurements in our experimentation. It is also important that we understand the limitations of our measuring instruments so we would choose the most appropriate ones given our procedures. With this knowledge, we can head into more complex thermodynamic experimentations with confidence in our basics.

References

1.     Physics 103.1 Lab Manual, T1 Temperature Measurement, National Institute of Physics, Philippines, 2013.

2.     R.A Freedman, H.D. Young, University Physics with Modern Physics 13th Edition, Chapter 17, Pearson Education, San Francisco, 2012.

[matophysics-blog]

Ride the Light Wave with Slit Diffraction

Light is really something special. We know that through vast amounts of research that light is known to travel in quantized packets called photons. These photons are particles that represent light in its movement, transmitting energy but containing no mass. However, light also travels with wave-like properties IN ADDITION to exhibiting this particle-like behavior. That is the dual nature of light.

Light is commonly observed as a wave when it travels through a small opening. Take, for instance, light shining through an open door into a dark room. If light were merely a particle, we would expect it to travel in a straight line from the door, shining only a small band against the wall across with the exact same dimensions as the door. Instead of doing that, however, light fills up the entire room, shining light on the opposite wall wider than the width of the door. In a smaller scale, we shine light through a very small slit, in this case a slit of 0.02 mm. It is seen as such

This is called single-slit diffraction (SSD). When light passes through a single slit, it produces a diffraction pattern of alternating bright and dark fringes. This would only be possible if light were a wave, with bright fringes (called the intensity peak) representing constructive interference and dark fringes (called the intensity minima) representing destructive interference. This pattern that we observe would be symmetric across the center, directly in front of the slit, with the central bright fringe called the central maximum being the widest and the brightest. Bright fringes then decrease in brightest as you move away from the central maximum.

The diffraction pattern observed, consisting of these bright and dark fringes, depends on the slit width. At distances much greater in magnitude than the slit width away from the slit, the fringes observed on either side of the central maximum are equal in width. If we set the center of the central maximum be y = 0, the location of mth intensity minimum can be represented by the equation

where is the wavelength of the light, L is the distance away from the slit, and a is the slit width.

Now what would happen if you have two very narrow slits separated a distance d apart? The answer isn’t as simple as it would seem. The two overlapping patters of light waves produce something like this:

We call this double-slit diffraction (DSD). In DSD with very narrow slits, there would be alternating bright and dark fringes, with the bright fringes supposedly being of equal intensity. If we once again set a wall across from these slits and let the center of the diffraction pattern be y = 0, then the distance of the mth intensity peak be

This equation solves for the location of intensity peaks. Note that this differs from the intensity minimum found by the previous equation.

In the previous image, a large central series of bright fringes is evident, and other groups of fringes are present afterward. This is because this is an image of double slit diffraction with two identical slits of finite width, the resulting pattern being a combination of SSD and DSD with very narrow slits. When this is the case, intensity fringes are grouped together in the outline of a SSD pattern and are known as the diffraction envelope.

The width of the interference fringes are controlled by the slit separation d, while the diffraction envelope is controlled by the slit width a. Remember that this kind of pattern is observed when the slits are of equal width.

In this experiment, we observed how the different factors such as slit width affect SSD and slit width and slit separation affect DSD.

METHODOLOGY

In our experiment to understand how these factors play a role in pattern formed by light, we used an optical bench equipped with a laser of wavelength 650 nm and a holder for an SSD or DSD experiment disk. A white sheet was attached to the opposite wall set at 185.5 cm away from the experiment disk. It was made sure that the projected diffraction pattern and the slit were at the same level vertically. Lights were turned off to make the projected pattern more visible with a desk lamp pointed downwards being used to illuminate without disturbing the pattern.

To measure the effects of slit width on the diffraction pattern, we equipped an SSD disk on the setup. A 0.04 mm slit was selected. On the projected pattern, the boundaries of the dark fringes were marked on the white sheet.

An mth intensity minimum would be located at the center of these boundaries. The locations of these intensity minima were found. The distance between first order (m = -1,1) was noted as y1. y2 was also measured. The distance of these intensity minimum from the central maximum were found as y1 = ½(y1). The same was done for y2. This was repeated for a slit width of 0.02 mm. The pattern for a slit width of 0.08 was also noted qualitatively. The wavelength for the laser would be calculated, and the result would be compared with the wavelength marked on the laser diode used.

To measure different factors for DSD, a DSD disk was attached to the holder instead. Firstly, the variable double slit of slit width 0.04 mm was selected. The slit separation was varied from 0.125 mm to 0.75 mm. Qualitative observations of the diffraction pattern, including the interference fringes and the diffraction envelope, were noted.

The double slit of slit width 0.04 mm with slit separation 0.25 mm was selected. The interference fringes and diffraction envelope were observed. Taking a look at the diffraction envelope, the boundaries of the dark fringes were marked. The mth intensity minimum would be located at the center of a dark fringe. The locations of these intensity minima were marked.

y1 and y2 were measured using the same method. With the data, the slit width was calculated along with the percent difference.

We then observed the effects of changing the slit width and slit separation. The succeeding steps were done for slit width a and slit separation d groups in mm, (a,d) = (0.04, 0.25), (0.04, 0.50), (0.08, 0.25), and (0.08, 0.50). For each DSD setup used, the number of interference fringes within the central maximum were recorded. The width of this central maximum was measured and then divided by the number of fringes to obtain an estimate for fringe width.

RESULTS AND DISCUSSION

From the first part of our experiment, these were our results

We found that this proved a relatively accurate method of determining the wavelength of the laser used. Some percent deviation could have been a result of the difficult in identifying where the dark fringes begin as they appear to have some faint hints of light from the bright fringes.

Both methods of using first and second order fringes produced accurate results. Precision could further be improved in this experiment with multiple trials, but the experimentation sufficed to provide a rough description of the relationship between factors.

As was observed, an increase in slit width caused a decrease in the width of the central maximum.

For the experimentation on double slits, these were our results

Using the given equations above, this also proved a fairly accurate way of verifying the slit width. Slight deviation could have arisen from the difficult to determine the start and end of the dark fringes. It is worth noting that some lateral diffraction were also observed, probably a result of the width of the slit on the other axis.

The succeeding experiment helped us verify the effects of slit width and slit separation on double slit diffraction. It was noted that an increase in slit separation increased the number of fringes within a fixed diffraction envelope. An increase in slit width had similar results as in single slit diffraction, with an increase in slit width decreasing the diffraction envelope.

From the equations above, it would also be interesting to point out that increasing the wavelength of the laser or the distance of the laser from the wall would increase the separation of dark fringes of the same order, thereby increasing the width of the central maximum.

Light confounds us in its dual nature, but this experiment allowed us to at least grasp light a bit better in this one aspect. With more experimentation, we could hopefully find more about the strange phenomenon that is light.

References:

1.Tipler, P., Physics For Scientists and Engineers, Chapter 9, W.H. Freeman and Company, England, 2008.

2. National Institute of Physics, Philippines. O4 Interference and Diffraction. Lab Manuals.

[matophysics-blog]

Light Moves...In Mysterious Ways

It comes from our sun, an electromagnetic wave traveling millions of miles in mere minutes to get to our Earth. It is strange, giving life the energy it needs to live yet expecting nothing in return. It is unique, behaving as both a wave and a particle and so much more at the same time. It is light.

Though its uses range from fueling photosynthesis in plants to powering electronics through solar panels, I would argue that light’s most important role in our lives is to allow us to see. It does this by bouncing off of objects and reflecting back into our eyes to be processed. 

However, it doesn’t give light justice to say that it simply “bounces.” Light is unique in that it is a wave that can travel without a medium. When it hits a surface, light can do several things depending on the medium it hits. Some light is absorbed by the object, some light passes through and is refracted, and some light is reflected. 

When light strikes the surface, it hits it at a point called the point of incidence and the ray in which it strikes the surface forms the angle of incidence with the surface’s normal. The reflected ray from the medium forms an angle of reflection, and the refracted ray through the medium forms an angle of refraction, both measured from the perpendicular line to the point at which it strikes the medium.

For a medium that can both reflect and refract light, a certain angle called the critical angle exists. When the angle of refraction is 90 degrees, the critical angle is achieved. For all angles greater than the critical angle, no refracted ray exists, and the entire ray of light reflects from the surface. This phenomenon is also called total internal reflection.

It’s not often that we get to observe the way something so integral to our life works. That is why I feel blessed as a Physics major to be able to experiment so actively with phenomenon such as light. In our study, we attempted to discover how exactly light interacted with the world around it, specifically how it reflects and refracts. Using an optical disk, we wanted to investigate the reflection and refraction of light. We also wanted to measure the index of refraction of a material using our optics set-up. Through our experimentation with light, we also aimed to learn how to trace the path of light as it emerged from optical materials of different geometries. 

Alignment of Optics

To investigate how light interacted with different media and surfaces, we used an optics set-up to harness light for such purposes. Our first challenge was to obtain a focused ray of light so that we could clearly observe what happens. We obtained a single beam using the optics set-up shown here:

The optics set-up, from right to left, consisted of a light source, a parallel ray lens to focus the  light in the direction we wanted, a slit plate to form the specific rays of light needed, a slit mask to isolate one such ray, and an optical disk to measure the specific angles of the light ray, all resting atop an optical bench.

We first aligned the optics to produce a single beam of light shining on the 0-0 degree line of the optical disk by adjusting the different items. After alignment, the light shining on the optical disk looked like this:

Once this was accomplished, we moved on to finding out more about how light interacted when it struck surfaces.

Our Methodology

In our experimentation, we would be investigating how light interacted with different media. First, we explored how it reflected on plane and spherical mirrors by allowing the single beam of light to strike the mirrors at different angles against the center of these mirrors. This was done by aligning the surfaces of the mirrors with the 90-90 axis of the optical disk and rotating the disk to strike the mirrors at different points. In this process, we measured the angle of reflection at different angles of incidence, conducting three separate trials for each of plane, convex, and concave mirrors. We then investigated how several rays of light would reflect after striking these mirrors at different points.

Then, we observed the reflection and refraction of light as it struck glass at different angles using a single beam of light and a semicircular glass (cylindrical lens). The light beam was made to strike the semicircular glass at its flat side. The angles of reflection and refraction were then measured for varying angles of incidence at 10 degree intervals from 10 to 50 degrees. From these, the index of refraction of the glass was computed. The same procedure was conducted but instead for light striking the glass at its curved surface.

We also searched for the glass’s critical angle by rotating the optical disk from 0 to 90 degrees to create varying angles of incidence with the medium. The critical angle was recorded as the angle in which the ray of light was parallel with the surface of the glass. From this recorded angle, the index of refraction of the glass and the speed of the ray inside the glass were identified.

Reflection by Plane and Spherical Mirrors

To investigate the reflection of light, we had the light strike the mirrors at their centers at different angles of incidence. It looked something like this:

Doing this for different angles of incidence and with the convex and concave mirrors, we got the results:

As expected the angles of reflection were equal to the angles of incidence in the plane mirror. Surprisingly, the same was true for the convex and concave mirrors as well, even if they were slightly bent. The law of refraction held true. This could be attributed to the fact that the light ray struck these mirrors at their centers, where their geometry would be similar to that of a plane mirror.

Ray Tracing for Plane and Spherical Mirrors

We then saw the effects of shining several rays onto the mirrors and got the following results for the plane mirror:

We noticed how the reflected light rays were perfectly in line with the incident light rays.

We repeated this for the convex mirror:

The outer light rays hit the mirror at an angle, causing these ray to converge at a point known commonly as the focal point.

Lastly, this was also done for the concave mirror:

The light rays also struck the mirror at an angle, but this time causing the rays to diverge.

We then summarized our results through the drawing:

Reflection and Refraction of Glass

The next part of the experiment involved the semicircular glass, which was set up as observed

Light would hit the semicircular lens depending on if it hit the flat or curved sides,

When the light hit the curved side of the glass, it was refracted as if it hit a plane mirror. This is due to the light creating its point of incidence with an imaginary plane tangent to the glass surface (represented by the dotted lines).

We then measured the angles of reflection and refraction of the glass at different angles of incidence. We first measured using the flat side of the glass.

We were stumped because we could only observe the refracted ray. However, under closer inspection, we also observed the reflected ray as such:

For the flat side, we obtained the results:

We computed the angle of refraction using Snell’s law, having the medium of incidence be air and the medium of refraction being the glass. It was discovered that the index of refraction of the glass was around 1.4514 (averaging the values we got for the different angles). The data was precise, being within small ranges from one another.

We then did this for the angles from the curved side:

Altogether, obtaining the results:

It should be noted that the equation used for getting the index of refraction of the glass here is different. This is because we treat the incident ray (causing the ray of refraction) as being in the medium of glass and the refracted ray (causing the angle of incidence) being in the medium of air. Hence, we use the reciprocal of sines used in the earlier equation. It should be noted that the index of refraction obtained in this part of the experiment is close to that obtained using the flat side of the glass. The angle of refraction would not be the same if the exact same equation was used, because the angles and indices would not correspond. 

It should also be noted that there was no angle of refraction obtained for an incidence angle of 50 degrees. To explain this phenomenon, we proceeded with our experiment.

Total Internal Reflection

We then found the angle at which no refracted ray occurred. This happened when the critical angle was reached. Finding this, we found:

It should be noted that the data shows a different index of refraction, rather far from the previous indexes obtained. It could have been caused by the difficulty in measuring the exact angle wherein the refracted ray would disappear, as it vanished suddenly within a marginal range. Using the given equations derived from Snell’s Law, we obtained the index of refraction and the speed of light within the glass. These results show how light changes speed when not in a vacuum, causing it to refract as such.

Ray Tracing for Different Refracted Media

Lastly, we wanted to observe the pathway of light in various shapes of media. Using a variety of objects, we observed for a right triangle in one of its legs,

(rays exited the other leg)

a right triangle at its hypotenuse,

(rays were reflected back, but created an interesting glowing tip)

(even if it was not in the experiment, we thought it would be interesting to observe what would happen when light struck the glass at the right triangle’s tip),

(the rays split)

a trapezoid,

(the rays were “moved”)

a convex glass,

(note how the rays converge)

and a concave glass,

(note how the rays diverge)

summarized in the ray tracing:

In all media, it should be noted that some light escaped even when the light was internally reflected. This means that light is both reflected and refracted when it hits a surface (when it can).

After the experiment, I understand that even though light is mysterious, it follows set rules in its movement. It interacts with different objects through reflection and refraction. We might not always understand how it works, but understanding these basics would allow us to appreciate light more, and hopefully drive us to explore more of its unknown. References:

1.     Tipler, P., Physics For Scientists and Engineers, Chapter 9, W.H. Freeman and Company, England, 2008.

2. National Institute of Physics, Philippines. O2 Reflection and Refraction. Lab Manuals.

[matophysics-blog]

Balancing Masses

https://docs.google.com/spreadsheets/d/1GJSW5ifdSFf9JYLiphTSV_h2hu3lM9Gq2-aS1hbG8Q8/edit?usp=sharing

[matophysics-blog]

Acceleration Due to Gravity

Today’s lab class had us experimentally finding the value of the acceleration due to gravity using two (2) different methods that utilize different tools within the lab.

Once again, all the data gathered from the experiment is compiled in this spreadsheet:

https://docs.google.com/spreadsheets/d/1RR2Pa5f3Ps2IdPvDujANgqWFcq7FfirPtkHoERujcCo/edit?usp=sharing

Ball Drop

The first method for this experimentation to discover the acceleration due to gravity (g) utilizing the Lab Quest tool and a simple motion detector. The procedure went as follows:

We would place the motion detector on the table, after which we would hold the ball a distance away from the motion detector. Once the motion detector starts recording, the ball is thrown upwards and caught repeatedly to produce graphs of the motion of the ball over time. To minimize error, the ball is caught at its sides so hands would not interfere with the sensor. The acceleration of the ball is then taken from both the distance vs time graph as well as the velocity vs time graph recorded. 

From the velocity vs time graph (v-t graph), we saw linear lines within given parameters. These would result in equations of y=mx+b. From our knowledge of kinematic equations, we know that v=u+at, where v is final velocity, u is initial velocity, a is acceleration, and t is time. Taking t as the value of x in our linear equation, we see that  m=a and u=b. Thus, g=m from this graph.

From the distance over time graph (x-t graph), we saw quadratic lines within given parameters. These would results in equations of y=Ax^2+Bx+C. From our knowledge of kinematic equations, we know that s=x0+vt+(½)at^2, where s is displacement, x0 is initial displacement, v is velocity, t is time, and a is acceleration. Taking t as the value of x in our quadratic equation, we see that x0=C, v=B, ½a=A. Thus, g=2A from this graph.

As seen from data gathered, the experimental value of the acceleration due to gravity, at  -8.37 m/s^2 from linear graph and -9.31 m/s^2 from quadratic graph, was distant from its true value.Worth noting is that the value found from the linear graph taken from the v-t graph was particularly lower than the value from the quadratic graph from the x-t graph. This can be attributed to error on our part as researchers. The experiment itself was prone to much error. It proved difficult to throw the ball perfectly upwards and catch it in such a way that the sensor could still detect the ball. The graph that produced this data could have also been parametrized more to produce more conclusive results. These errors accumulate into the inaccuracy of the result.

Nevertheless, the data still produces an acceleration within a 2 unit range from the actual known value, and we see how this method of experimentation could possibly produce results for the value of g. 

Photogate

To simulate another method that could experimentally determine the value of g, we used the Lab Quest tool again, a photo gate, and a strip of plastic that had alternating clear and opaque markings.The experiment would work as follows:

We would hang the bottom tip of the strip of plastic over the photo gate. Once recording starts, we simply drop the strip into a box below. The photo gate records every time the switches from opaque to clear and vice versa. This is done as light is transmitted from one side of the photo gate to another. If it is detected on the other side, that means the section is clear; if not, then it is opaque.

Similar method for gathering the acceleration was used on the graphs produced by this method. However, it should be of note that the graphs produced were only in the positive direction as this was the frame of reference used by the photo gate. In this case, the downward direction is positive, explaining the positive value for g gathered.

We can see here that at 9.74 m/s^2 and 9.72 m/s^2, this method produces more accurate and precise results. Although not perfectly accurate, the experiment is more precise and accurate because there is less influence of human error. Our role was simply to drop the plastic and the photo gate would do the rest. 

Thoughts

There are many ways to achieve the same goal, and it is certainly made clear by these experiments that there are experimental ways to discover the values of constants. Know with the aid of more advanced technology, we can find these values with higher precision and accuracy; however, knowing how people used to do these experiments before is really insightful into the struggles of physics research in years prior.

[matophysics-blog]

National Institute of Physics Lab Tour

Instead of a regular Physics 101.1 class today, we instead had a tour of the different labs located within the National Institute of Physics. 

As the National Institute of Physics, NIP is the leading institute of physics research in the entire Philippines. That being said, taking a look into the different labs that the building had to offer was both an honor and a privilege.

The labs within NIP are located in the Research Wing of the building. This is area of NIP is normally closed off to regular students, to allow the different people performing research there the proper environment for scientific discovery; however, during a week within the annual Physics Month held every February, the NIP Research Wing opens up its doors to people, allowing others to view the inner workings of leading physics research within the country.

The last time I had the opportunity to tour the labs within NIP was at the start of the school year as sort of an introductory tour for pre-Physics freshmen. This day I had the chance to take another tour of the facilities as an actual student of the institution. In the days to follow, I would have the opportunity to take the tour once more, this time as a tour guide as an applicant of the UP Physics Association, who handles the lab tours for the different participants.

The Labs

There are six (6) labs within NIP, each of them having a specialization within the field of Physics. Since topics are interrelated, there is some overlap between the labs; nevertheless, each lab is unique in the topics they are primarily interested in and in the way they tackle these problems. The 6 labs are the Condensed Matter Physics Laboratory (CMPL), which separates into the two labs SuperCon and SemiCon, Instrumentation Physics Laboratory (Instru/IPL), Structure and Dynamics Group (SanD), Photonics, Theory, and Plasma.

CMPL contains two main divisions. The first of the CMPL labs is SuperCon which deals with superconductors. Superconductors are materials that gain magnetic properties at very extreme temperatures. The research in this lab finds ways to harness the potential of these materials as well as identify the properties of potential superconductors. The other CMPL lab is SemiCon, which deals with other conductors and conducting materials. This lab had the most unique rooms. The Yellow Room, a room that uses yellow light and plastic covers for shoes, utilizes these specific features to provide a pristine environment that allows the researchers to build and develop their semiconductors. SemiCon also houses the largest and most expensive machine in the entire NIP building.

The Instrumentation Physics Laboratory contains the most varied topics tackled within their research. From complex systems to optics and art to biology, IPL, as the name would suggest, uses a variety of different instruments to tackle many problems faced by physicists. Many of IPL’s research has immediate real world applications, making the lab an efficient contributor to Philippine society.

The Structure and Dynamics Group is probably my favorite. The lab itself uses computers to employ simulations that answer some of the most complex physics problems. Technology is SanD’s biggest asset as they do not require physically conducted experimentation to produce results. Fields of interest include entanglement and quantum computing, parallel and high-performance computing, spin transport and spin-orbit interactions, and two-dimensional electron systems. This is the lab I hope to join in my future here in NIP.

The Photonics Group deals with the many different aspects of photonics. Light is their primary concern as they harness its properties and find practical uses for the ever complicated wave-particle. Image imprinting and laser manipulation were interesting things of note in this lab.

Theory has the simplest room, containing only table, desks, and the shelves of books needed to build up their thoughts. The Theory Group tackles the different problems of theoretical physics, those that cannot be answered by experimentation as they are too difficult to devise experiments for. Equations fill their boards and coffee fills the air of this lab.

Last but not the least is the Plasma laboratory, which harnesses the power of the 4th state of matter for practical applications. In a large room containing all their equipment and machinery, the Plasma laboratory uses plasma in its different states to do whatever they need it to do.

My Thoughts

Seeing the different labs once more, I have strengthened my love for physics. Just seeing the different things that physics research can accomplish and the wide variety of fields that physics actually affects drives me to do my best as a student. Hopefully, I can one day join a lab of my own and help contribute to the pool of knowledge that is ever expanded by those in the labs of the National Institute of Physics.

[matophysics-blog]

Density of Philippine Coins

As a continuation of our lesson on measurements and uncertainties, our activity today had us trying to find the density of Philippine coins. This activity, however, had us learn the concept of error and error propagation in experimentation. The goals of today’s class were to perform error propagation and statistical analysis, as well as use estimated uncertainties for measurements.

The measured data for this activity are all located in the following spreadsheet:

https://docs.google.com/spreadsheets/d/17DwUOpJhx4cDNdcSTp34lz6m7V5C_AOZXeGc67xjBp4/edit?usp=sharing

Measuring

Our task for this activity was to experimentally determine the densities of the 5¢, 10¢, ₱1, and ₱5 coins. We were given basic measuring instruments: a vernier caliper with smallest measurement of 0.02 mm to measure the diameters and thicknesses of the coins and an electronic balance with smallest measurement of 0.1 g to measure the mass of the coins.

We started taking the measurements using the coins and the instruments provided. To measure the diameter as well as the thickness of a coin, we took 3 trials at different places along the coin. This is to take into account that the coin might not be uniform all throughout. Also note that to get the proper measurements of the 5¢ coin, which has an inner hole, we took both the the inner and outer diameters of the coin. 

Given that the measuring instrument used was the vernier caliper, with least count of 0.02 mm, the uncertainty for the measurements in each trial was ±0.01 mm. The average of all the trials was what we used as the final measurement for that specific quantity. Since the measurements were averaged, the final uncertainty was equal to the average of all uncertainties used. Since the absolute uncertainty for each of the three trials for one measurement was ±0.01 mm, the average was also equal to ±0.01 mm.

The measurements for diameter were then converted to measurements for radius, as the radius of the coins would be the quantities used in later computations. Since we divided the quantity by 2, we would also have to divide the absolute uncertainties for these measurements. Thus, the uncertainty for the value of the length of the radius was equal to ±0.005 mm.

For the mass of the coin, we used the electronic balance, I took the convention that the uncertainties for the measurements taken here were equal to the lowest count on the electronic scale. On non-electronic measuring devices, the norm is to take half the least count on the instrument’s scale. This uncertainty accounts for the discrepancies a researcher may have while reading the measurement. This method does not particularly work when taking the measurement from an electronic measuring device since the measurement is stated there for you to see, no room for human error. As such, the uncertainty used is instead the least count of the electronic measuring device.

Computations

The relative uncertainties were then taken, as the succeeding computations would involve multiplication, division, and powers. The relative uncertainty for each measurement is equal to the absolute uncertainty divided by the measurement. Since relative uncertainty is expressed as a percentage (%), these quantities were multiplied by 100 to get the relative uncertainty as a percentage.

Relative uncertainties are used for these mathematical processes because using absolute uncertainties decreases the uncertainty. When having uncertainties, the goal is to be the most “pessimistic” or to have the highest value of uncertainty possible. Thus, we would want a larger representation of the uncertainty which is achieved by using the relative uncertainty rather than the absolute uncertainty.

Area of the coin’s face was computed by the formula Area = πr^2 leaving units of sq. mm. It should be taken note that for the 5¢ coin, the appropriate area was taken by finding the area of the larger circle using the outer diameter and subtracting from this the area of the smaller circle that uses the inner diameter. Uncertainty for these quantities is taken by multiplying the relative uncertainty for measurements of the radius by 2. (See Reference 1 for detailed rules on uncertainty propagation in mathematical operations.)

The volume of the coin is taken by multiplying the area of a coin’s face by its thickness. The uncertainty for this quantity is taken by simply adding the relative uncertainty for the area and for the thickness. The resulting volume now has units of cubic mm. This is then converted to cubic cm as this is the standard g/cm^3 is standard in relating the density of an object.

The density is then computed by taking the mass of the coin and dividing it by the total volume of the coin. This results in units of g/cm^3 with uncertainty equal to the sum of absolute uncertainties for the involved measurements.

The results gathered are summarized as the following (to 3 sig figs):

5¢ - (7.22 ± 6.21%) g/cm^3

10¢ - (7.09 ± 4.62%) g/cm^3

₱1 - (7.05 ± 2.19%) g/cm^3

₱5 - (7.51 ± 1.59%) g/cm^3

References:

1. Basic rules for uncertainty calculations. Retrieved on March 15, 2015, from http://web.uvic.ca/~jalexndr/192UncertRules.pdf.

[matophysics-blog]

A Grain of Rice

In an effort to apply what we learned about measurements and uncertainty, we, in our Physics lab class, decided to measure grains of rice using two different measuring tools, namely the vernier caliper and the micrometer caliper.

Measuring Devices

Before I begin, let me discuss the tools we used to take the measurements. The vernier caliper is a measuring device that allows one to take the length of objects. It is specifically useful to take the diameters of round objects as it has provisions and clamps to both accurately measure inner and outer diameters. Some vernier callipers also allow depth to be measured. A vernier caliper comes in two parts, the main scale (MS) and the fractional scale (FS). The MS works just as any ruler would, with the 0 mark on the FS being the mark you base your reading off of. Like a ruler, you're measurement won't lie exactly on one number on the MS – this is where the FS comes in. The FS is used to increase the precision of the measurement. There are many lines on the FS, each corresponding to a different fractional unit (in our 1 mm vernier caliper, the FS came in gradations of 0.02 mm). You have to find the first line on the FS that matches with a line on the MS. That is your FS reading. To get your total measurement, you add your MS reading with your FS reading.

The micrometer caliper works the same way, except that it has a circular MS and FS and its FS has gradations of 0.01 mm as the MS has gradations of 0.5 mm. You put the object in between the appropriate section and turn the dial until you hear a click that signifies that the caliper is tightened. Once this happens, you read the mark on the MS, and then you check the number on the FS that "hits" the marking line. Adding these two values up, you get your main reading.

One thing of note is that you have to make sure your instruments do not have any zero error. This is the error caused by an uncalibrated caliper and can lead to systematic error. To check for zero error, you put your caliper all the way to its lowest measurement and see if the 0 on the MS aligns with the 0 on the FS. If it does, then you are good. If it does not, the take note of how much it differs and account for this in recording your measurements. Luckily for us, there was no zero error for either of our instruments.

Measuring Rice

Actually measuring rice was a bit tricky. For the vernier caliper, it took us a while before we managed to get the rice properly between the "teeth" of the vernier caliper. It was difficult to ensure that what we were trying to measure was the entire length of the rice grain. It proved even more so with the micrometer caliper as we had to place the rice in between a rather thick compression part. Nevertheless, after a few tries, we managed to get a hang of it and proceed with our measurements.

The following is a link to our measurements and the subsequent sub results to the questions:

https://docs.google.com/spreadsheets/d/1DsqfxJWet7PCCeKjzrh4j0L5fUz7opAenDcDDEFlfLk/edit?usp=sharing

*note: the measurements listed do not show the proper significant digits of 3 for all readings as a result of limitations on excel

1. What is the length of a rice grain?

The length of a rice grain varied in between a range of measurements from 6.33 to 8.00 mm from a total of 110 different measurements of different samples. The uncertainty of the measurements varied between the two instruments used, with ±0.01 being the uncertainty of the vernier caliper and ±0.005 being the uncertainty for the micrometer caliper. This answer was obtained through direct measurement of said samples. The validity of the measured values is known to be true as it was observed firsthand by the researchers.

2. What is the average length of a rice grain?

The average length of a rice grain is 7.14±0.86 mm. This answer was obtained through the formula 

AVERAGE = (SUM OF ALL MEASUREMENTS)/(TOTAL NUMBER OF MEASUREMENTS)

The uncertainty for this was taken as the maximum difference between a measured value and the mean of the set of data values.

It is known to be true because it follows a certain formula that utilises the measured data to produce its result.

3. What is the acceptable range for the length of a grain of rice?

The acceptable range for the length of a grain of rice, found through the formula Q=μ±σ, is  7.14±0.32 mm. The mean is taken as the average of the lengths of the grains of rice. The standard deviation, within which 67% of all measurements are found in, is taken through an equation utilising excel.

The precision can be commented on by comparing this to the known value of the length of a grain of this particular type of rice, l=7.09±0.36 mm taken from Hobson, Carter IMTC 2007 paper. As you can see, the values found are close to the known value, making the measurements accurate, the value for acceptable range being the most accurate as it encapsulates the known value of the actual measurement. It should also be noted that the questions are being asked in increasing orders of approximation which results in asking for an increasing level of precision.

This activity allowed me to apply what I had learned on uncertainty and measurement into an actual process.

[matophysics-blog]

Measuring the Fundamentals

When eager students rush into the field of Physics, they have their minds sent on concepts and theories. They would like to shake the world with their innovative ideas and groundbreaking suppositions, having their eyes set on becoming the next Einstein or Hawking. I know I was one of them.

However, as my Physics 101.1 class has made me aware, there are places everyone, even the greatest of minds, need to start. In our case, where we need to begin is a study on measurements.

As I mentioned in my previous post, in physics experimentation, there are a lot of measurements that we have to take. Measurement is the fundamental tool for gathering data so undoubtedly a lot of it is necessary as we test different hypotheses in the different studies within the field. When taking measurements, we need to note that what we are doing is finding quantitative descriptors about the objects we are looking at. Take a ball for example. We can describe it as red and bouncy, but that does not give us as much info as saying it has a diameter of 40.0 cm and a mass of 8.33 g in Physics.

What is a measurement?

A measurement comes in two parts, the magnitude and the unit. Take for example the 40.0 cm diameter of the ball. We have 40.0 as the magnitude. The magnitude is the number in a measurement. This tells us how much of a given quantity is present.Then we have cm as the unit. This tells us what aspect of an object we are measuring. Centimetres (cm) is a unit for measuring length. Other units are available for measuring length such as meters (m) and inches (in). Units are also present for other factors we wish to measure: there's seconds (s) for time, kilograms (kg) for mass, and many others.

Measurements are inherently uncertain. When measuring an object and seeing that it is 40.0 cm, how are we not sure that it is not actually 40.01 or 39.99 cm? Or how can we tell apart 40.0 cm from 40.0000000000000001 cm? We can with a very high precision measuring device, but for the most part, we can't. However, there are some conventions that we follow to properly show what we are certain and uncertain of in a measurement.

Given a 30 cm ruler with 1 cm gradations (a gradation is a line or a mark on the measuring device), we can measure an object surely until the smallest gradation. However, we know that sometimes, the object falls in between two lines, and we're not sure how to report the measurement. Say the object is between 5 and 6 cm. We can say that the object is 5.4 cm long, the .4 being the uncertain measurement. However, we cannot say that the object is 5.42367 cm long as we are unsure of even the first decimal number. The convention is noting measurements is that we are allowed one (1) uncertain digit past the smallest certain digit provided by the measuring device.

Error in Measurement

As we gather data though, we have to be aware that we are limited, both in ourselves as researchers and in the instruments that we use. Error is a major part in dealing with measurements. Slight mistakes to measurements taken, although seemingly small at first, can have great implications on experimentation as a whole. When scientists wanted to land a probe on a comet, they had to be exact with the measurements they took: time for the comet to travel around the sun, distance of the comet from Earth at given points in time, etc. 

There are two types of error that can be made when taking measurements. The first is random error. This is the error that comes from the circumstances in which the measurement was taken. This is mostly attributed to human error caused by the researcher taking the measurement. Sources of random error include how the researcher views the measurement along the gradations and how the researcher handles the measuring device, to name a few. The second type of error is systematic error. This is the error that comes from the experiment itself. This could include outside factors entering the set-up, an apparatus was slightly askew, or the measuring device was poorly calibrated.

Although error cannot be avoided, there are certain steps that can be taken to minimize their effects on the results of the experiment. Mistakes of random error cannot be avoided. However, since they are caused by random events and vary from one testing to another, a way to decrease the influence of random error is to increase the number of trials done. Mistakes of systematic error, on the other hand, can be avoided. By ensuring the experiment is in optimal conditions so that it is not affected by external factors, apparatuses calibrated properly, and experiment conducted properly, systematic error can decrease.

Precision and Accuracy

Another integral concept of measurements is differentiating between precision and accuracy. These descriptors are determined by measurements' distance away from each other and a known value. Measurements are precise if the values taken from them are close to one another. A measurement can be more precise if it has more decimal places. Measurements are accurate if they are close to the known value.

Levels of Approximation

Lastly, we cannot be infinitely precise with our measurements, and thus, we have to give way to certain levels of approximation.

The lowest level of approximation is orders of magnitude. This level of approximation utilizes powers of ten (10) to provide an approximation. Using this, we can say that an apple is 100 g or 10^3 g (using scientific notation). We cannot provide specific numbers as these are only general estimates that we conjecture from observation.

The first level of approximation is significant figures. Having the same apple as an example, we can say that the apple is 1.01 x 10^3 g.

Higher levels of approximation are referred to as best estimates.  A best estimate for a measurement Q is given in the form of Q=⟨Q⟩±ΔQ, where ⟨Q⟩ is the expectation value of the measurement and ΔQ is the uncertainty, where ±ΔQ is a range away from ⟨Q⟩ where the true value of the measurement lies.

There are several cases for best estimates on what determines both ⟨Q⟩ and ΔQ.

If the measurements being taken are regular or non-fluctuating, where repeated measurements produce the same value for the magnitude, then ΔQ is given as half the least gradation. In a 1 mm gradation ruler, the uncertainty ΔQ is ±0.5 mm.

If the measurements being taken are fluctuating and there are a limited number of them, the ⟨Q⟩ is given as the average of all taken measurements and ΔQ is the maximum difference between ⟨Q⟩ and any value within the set of measurements.

If the measurements being taken are fluctuating and there are a very large number of them,then Q=μ±σ. This is due to the idea that multiple measurements tend to follow a Gaussian normal distribution, with μ being the central value or average and σ being the standard deviation away from the central value.

With all of this knowledge of measurement in hand, I can be ready to face on the reset of the things I would learn in this class.

References:

 R.C. Batac, Different levels of approximation in measurement, blog post from http://rcbteaching.blogspot.com/2015/02/physics-1011-different-levels-of.html, accessed: February 15, 2015

[matophysics-blog]

Intro to Physics 101.1/Measurements and Uncertainties

As a Physics Major, I was going to have to get used to having lab classes. Within the field, or any field of science in fact, the most available way to produce results and findings is to conduct experiments. Thus, these lab classes become necessary not only in developing our skills as physics students, but also as scientists as a whole.

Our lab professor told us students that the reason that Physics 101.1 and all other Physics lab classes are separated from their corresponding lecture classes is that there is a need to hone the experimental skills needed in physics along with the conceptual knowledge we gain. In Physics 101.1 specifically, there is a need for us to have the basic skill set we need in conducting physics experiments if we were to properly develop in this field. There is a need for us to master the basics such as measurement and simple experimentation before we move on to the more complex topics that physics as a whole has to offer

In today's class, we discussed measurements and uncertainties. In physics experimentation, there are a lot of measurements that we have to take. Measurement is the fundamental tool for gathering data so undoubtedly a lot of it is necessary as we test different hypotheses. And as we gather data, we have to be aware that we are limited, both in ourselves as researchers and in the instruments that we use – that is what we'd call uncertainty. 

Aside from this, we also learned about precision and accuracy. Measurements may be precise when they are close to each other. Measurements are accurate when they are near the value that is commonly known. In taking measurements, they can be accurate but not precise, precise but not accurate, both accurate and precise, or even neither of the two.

In our lab, we learned how to account for uncertainty in measurements as we measured grains of rice. We used the vernier caliper for our measurements. It was difficult at first to get the rice exactly horizontal, but in the end, we were able to get it and make 60 measurements of different samples of rice grains.