Lab 2 Physics

Laboratory #2 Measurement of the Earth's Surface Gravity Authors: Lab Section #: 005 Purdue University Northwest – PHYS 15200 – Spring 2023

1. Abstract In this study, we aimed to determine the acceleration due to gravity, g, by measuring the velocity of a glider on an air track. The velocity was measured by timing the glider's descent over a known distance and using kinematic equations to calculate acceleration. The data was analyzed by calculating the uncertainty in the measurements and propagating it through the calculations. The results showed that the acceleration due to gravity was 5.391 m/s^2, with an uncertainty of approximately 0.2 m/s^2. This value did not match the accepted standard value of 9.80665 m/s^2, but the relatively high fractional uncertainty in the measurement of acceleration makes it difficult to make a strong conclusion. In conclusion, our experiment highlights the importance of accurately measuring and considering uncertainties in experimental data. 2. Introduction The Earth exerts a gravitational force on all objects. Objects under the influence of this force (and other constant forces) accelerate at a uniform rate, independent of their mass (i.e. if we ignore the effects of air resistance, a bowling ball and a ping pong ball fall at the same rate). Galileo was aware of this, but didn't have a complete answer as to why. It would be Isaac Newton who put this together with a bigger picture of how the physics of motion works. Nonetheless, Galileo made measurements of this acceleration due to gravity. Also, given that he didn't have the bigger picture, Galileo did not realize that the value of this acceleration is actually dependent on distance from the Earth's surface. It varies slowly enough that you'd need a sensitive test at sea level and again at a mountain top to see the difference. So, for our purposes, we will assume that it's a constant that has a known value of 9.80665 m/s2, the adopted International Standard. With that mentioned, you should know that the value can be as different as 9.780 m/s2 at the Earth’s equator and 9.832 m/s2 at the Earth's poles due to the oblateness of the Earth (the Earth is an oblate spheroid, flattened at the poles and slightly bulging at its equator). In this lab, you will measure the acceleration due to Earth’s gravity in Hammond Indiana 3. Experimental Procedure An air track was used to provide a ramp with an air cushion which allows a “glider” to move with very little friction. We started by leveling the air track, and then created an inclined plane along the track by raising one side by 1cm using the width of a 100g weight. We will use precise photogate devices to indirectly measure the velocity of the glider at two locations (which are 100 cm or 1 meter apart) as it slides down the track (the photogate devices measure the time it takes the cart to pass and we used this measurement to calculate the velocity at each gate). We

the same as the distance between the leading edges of the carts, allowing you to accurately measure the distance between the photogates. 6. Set the photogates to "GATE" mode. Release the cart from the top of the ramp and allow it to roll down the track, making sure the flag passes through both photogates without making contact. Practice reading the times measured by the photogates. In GATE mode, the photogates will record ∆t1, the time spent in the first gate, and ∆t1 + ∆t2, the combined time in both gates. After each trial, only ∆t1 will be displayed (record it). To find the time in the second gate (∆t2), subtract ∆t1 from the total (∆t1 + ∆t2) which is shown when you press the memory switch button to the "READ" position. Also, make a note of the precision of the photogate devices in your lab report. 7. Conduct an experiment with a minimum of ten trials. In each trial, let the cart go from the top of the ramp and keep track of the time it spends in the first gate (∆t1) and the total time in both gates (∆t1 + ∆t2) in a spreadsheet and lab notebook. Use the recorded information to determine the time spent in the second gate (∆t2). 4. Data Photogate Trials Attempt # T 1 (s) T 1 +T 2 (s) T 2 (s) V 1 (m/s) V 2 (m/s) A 1 .1143 .1635 .0492 .2078 .483 .0951 2 .1147 .1641 .0494 .2071 .481 .0942 3 .1148 .1635 .0487 .2068 .488 .0927 4 .1149 .1646 .0497 .2067 .478 .0924

5 .1149 .1643 .0494 .2067 .481 .0943 6 .1145 .1639 .0494 .2074 .481 .0942 7 .1142 .1634 .0492 .2080 .483 .0950 8 .1146 .1639 .0493 .2072 .482 .0947 9 .1149 .1644 .0495 .2067 .480 .0938 10 .1146 .1640 .0494 .2072 .481 .0942 Attempt # Width of Flag 1 2.374 cm 2 2.374 cm 3 2.376 cm 4 2.376 cm 5 2.376 cm

The magnitude of uncertainty sources in a laboratory experiment can come from various sources, including the precision of measurement instruments, the accuracy of the data collected, and the repeatability of the experiment. To calculate the magnitude of these uncertainties, it is important to consider the sources of error and to propagate them appropriately through any calculations. For example, if the measurement instruments have a certain degree of precision (e.g. 0.001 cm for a Vernier caliper), this precision should be taken into account when measuring quantities such as the width of a flag or the height of an inclined plane. Similarly, any inaccuracies in the data collected (e.g. incorrect readings of times or distances) should also be taken into account when performing calculations. When performing calculations, it is important to propagate the uncertainties appropriately. For example, when calculating velocity, the width of the flag on the glider and the time through each photogate are used. The uncertainties associated with these measurements should be propagated through the calculation to obtain the uncertainty in the final result. This can be done by using error propagation equations, which take into account the uncertainties in the inputs to the calculation and produce an estimate of the uncertainty in the output. It is important to record the necessary uncertainties for the equipment settings and data, and to document the calculation of any unknown propagation of uncertainty equations. This will help to ensure the accuracy and reproducibility of the results, and will provide a clear understanding of the sources of error in the experiment. 6. Conclusion The goal was to measure the acceleration of the glider down the air track using photogates and a timer. The results showed that the acceleration was 5.391 m/s^2, which did not match the agreed upon standard value of g (9.80665 m/s^2). One possible reason for this discrepancy could be the limitations of the lab equipment. The photogates and timer may not have been precise enough to accurately measure the velocity of the cart, which would impact the final calculation of acceleration. Additionally, the air track may not have been level, which could have affected the results. To improve the experiment, it would be helpful to ensure that the air track is level and to use a more precise timer and photogates. It may also be helpful to perform multiple trials to average out any potential sources of error. Overall, while the experiment provided a measured value for the acceleration of the glider, it is not clear if this value is accurate or if it matches the expected result based on physics theory. Further investigation and refinement of the experiment is necessary to make a strong conclusion.

Questions a. Does your result agree with the agreed upon standard value of g (9.80665 m/s2 )? Make sure to directly state your measured value from step 16 when comparing to the accepted value. i. The measured value of g is 5.391 m/s^2, which does not agree with the agreed upon standard value of g (9.80665 m/s^2). This difference between the measured and standard values indicates that there may be systematic errors in the experiment, or that other factors (such as temperature, atmospheric pressure, or instrument errors) influenced the results. Further investigation and analysis may be required to determine the source of the discrepancy. b. What measurements did you make in order to calculate velocity? To what accuracy did you make these measurements? If you increased the angle of the air track to 90 degrees, would the relative uncertainty in your measurement increase or decrease? Explain why. i. To calculate the velocity of the glider, the time (Δt) it took for the glider to pass through the photogates was measured and the width of the flag attached to the glider was used. The velocity was then calculated as the width of the flag divided by the time taken to pass through the photogates. The accuracy of these measurements we took were up to four decimal places. If the angle of the air track was increased to 90 degrees, the relative uncertainty in the measurement could increase. This is because a steeper incline would result in the glider moving faster, which would require more accurate timing of the photogates to ensure the proper measurement of velocity. Additionally, the increased speed of the glider could introduce additional sources of error, such as friction and air resistance, which would affect the accuracy of the velocity measurement. c. Define fractional uncertainty to be the uncertainty on your measured value divided by the measured value. Calculate the fractional uncertainty of your measurements of a and g (what is and )? Which has smaller fractional uncertainty, your measurement of a or g? Why do you think this is?

i. Yes, you could measure the acceleration of the glider down the air track using only a stopwatch by measuring the time it takes for the glider to reach the bottom of the track, assuming you know the length of the track. The equation used to calculate the acceleration in this experiment would be a = 2d/t^2, where d is the distance traveled by the glider and t is the time taken. f. The gravitational force of the moon and the sun affect the apparent acceleration of gravity for an object in freefall on the scale of ± 2 x 10-6 m/s2. Is it possible that your apparatus to measure g could detect these effects? Why or why not? If you did have a device capable of measuring these effects, describe an experiment which would measure this variation i. It is not possible for the apparatus used in the experiment to detect the effects of the moon and the sun on the acceleration of gravity, as the magnitude of these effects is on the scale of ± 2 x 10^-6 m/s^2 which is much smaller than the precision of typical laboratory equipment. ii. To measure this variation, a highly sensitive and precise apparatus would be needed, such as an instrument capable of detecting changes in the gravitational field. One such experiment could involve using a gravimeter, a device that measures changes in the Earth's gravitational field, to detect the variation in acceleration due to the moon and sun. The gravimeter could be positioned in a location where it is not influenced by other factors such as earthquakes, tides, or underground fluid movements, and measurements could be taken over time to determine the effects of the moon and sun on the acceleration of gravity