Lab 3- Little g - PHY 207-lab3 PDF

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Physics 207

Lab #3- Little g Introduction The objective of this lab was to determine the value of acceleration due to gravity on Earth experimentally. The expected result of little g is 9.8 m/s^2, as determined by Newton’s universal law. This experiment is important because it is an essential part of our lives since we always deal with gravity. We used different methods and experiments to calculate the value of gravity. In the lab we also learned how to experiment for gravity when an object is placed on a slopped surface by calculating the angle of the surface. Procedure First, we calculated the value of gravity by free fall. We held a wooden block on a height of 1 meter and calculated the time it took from when it was released until when it touched the ground. Then we used the equation, g =(2Δy)/t^2, and plugged in 1 meter for “Δy” and the time we calculated for “t” (time) and calculated g (gravity). We repeated this experiment three times to get three recorded values for time and calculated gravity by plugging different time values each time. However, Δy stayed constant in all the trials. Finally, we took the average of the three values of gravity, average gravity= (1st gravity+ 2nd gravity+ 3rd gravity)/3, to come with our experimented value for acceleration due to gravity for this experiment. After that, we calculated gravity from a slo-mo free fall video. We were able to find the time when he dropped the ball and recorded it. Then, we played the other frames until we got to one where the height was exactly on one of the meters recorded on the side, to have closer value to exact Δy and therefore decrease the error. We then changed the height from centimeters to meters. Finally, we subtracted the final time from the initial time, and calculated gravity using the following equation, g =(2Δy)/t^2.

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Then, we measured gravity more accurately by leveling a ramp. We were given the length of the track to be 1.22 m. We then started increasing the height of the track until the cart starts the moving. To calculate the angle we used the equation, θ=sin^-1(h/L). The angle that we calculated is the angle required for the cart to start moving. And to calculate gravity, we found acceleration from the time vs. velocity graph and plugged it in the equation, a=g*sin(θ). After that, we connected the track to the computer and used the application LoggerPro to record and sketch graphs of the movement of the cart. We held the cart to the high end of the track and started recording at the moment we let the cart go. We repeated the same procedure under three different angles, and measured the height to calculate the angle using the equation, θ=sin^-1(h/L). Then, we found acceleration from the slope of time vs. velocity graph and plugged it in the equation, a=g*sin(θ) to calculate gravity. Finally, we repeated the previous experiment with the LoggerPro connected to the track. However, this time we added mass on top to the cart to see the effect of mass on the acceleration of the cart down the track. Then, we sketched graph for different masses and their acceleration and were able to find the relationship between the mass and the acceleration of the cart on the track using the equation a=g*sin(θ). Data and graphs Experiment: A rough measurement Trial Height Δy (m) Time (s) 1 1 0.50 2 1 0.45 3 1 0.48 Average of the three measurement= 8.85 m/s^2

Gravity (m/s^2) 8.0 9.88 8.68

Experiment: Slo-mo free fall Acceleration due to gravity = 9.796 m/s^2

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Experiment: Leveling a ramp Angle when it begins to move = 0.20 degrees Time (s) 0.05 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Position (m) 1.151382 1.12751 1.093758 1.052598 1.002109 0.948326 0.887135 0.818535 0.746642 0.672829 0.573222

Velocity (m/s) -0.05305 -0.05945 -0.07409 -0.08781 -0.1061 -0.11388 -0.12851 -0.13903 -0.14955 -0.16418 -0.27714

Acceleration (m/s^2) 0.065551 -0.0343 -0.03354 -0.05945 0.124242 -0.0061 0.094516 0.137962 -0.0404 -0.31327 -1.31661

Experiment: The Rolling Cart at different angles Angle 1: 0.50 degrees

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Little g = 10.20 m/s^2 Time (s) 0.05 0.5 1 1.5 2 2.5 3

Position (m) 1.07702 1.000737 0.894818 0.76146 0.607247 0.41407 0.211837

Velocity (m/s) -0.14909 -0.19345 -0.24422 -0.29041 -0.35489 -0.38782 -0.09467

Acceleration (m/s^2) 0.037095 -0.01753 -0.14787 -0.00305 -0.18903 -0.15778 3.548144

Position (m) 1.056714 0.942564 0.780119 0.570203 0.312816 0.211288

Velocity (m/s) -0.21495 -0.2808 -0.35306 -0.50764 -0.56801 -0.07683

Acceleration (m/s^2) -0.10773 -0.06022 -0.04573 0.105187 -0.20428 4.743563

Angle 2 = 0.9 degrees Little g = 8.91 m/s^2 Time (s) 0.05 0.5 1 1.5 2 2.2

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Angle 3 = 0.59 degrees Little g = 11.65 m/s^2 Time (s) 0.05 0.5 1 1.3

Position (m) 0.84131 0.661853 0.39788 0.223636

Velocity (m/s) -0.37959 -0.45459 -0.55246 -0.30413

Acceleration (m/s^2) -0.06403 -0.63569 -0.23476 5.180824

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Angle = 0.59 degrees (without mass) Little g = 11.65 m/s^2 Time (s) 0.05 0.5 1 1.3

Position (m) 0.84131 0.661853 0.39788 0.223636

Velocity (m/s) -0.37959 -0.45459 -0.55246 -0.30413

Acceleration (m/s^2) -0.06403 -0.63569 -0.23476 5.180824

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Angle = 0.59 degrees (with mass) Little g = 9.71 m/s^2 Time (s) 0.05 0.5 1 1.5 1.85

Position (m) 1.00156 0.861067 0.653621 0.391569 0.207172

Velocity (m/s) -0.2808 -0.36495 -0.48477 -0.55154 -0.02378

Acceleration (m/s^2) -0.0813 -0.20199 -0.44438 -0.38873 3.419329

Sample calculations 

To calculate gravity, given the height and time, use the following equation: g =(2Δy)/t^2. g= (2*1)/(0.45)^2=9.88 m/s^2

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

To calculate the average of the three measurements of gravity, use the following equation: average gravity= (1st gravity+ 2nd gravity+ 3rd gravity)/3. Average gravity = (8+9.88+8.68)/3 = 8.85 m/s^2



To change centimeters to meters, divide the centimeters by 100. 60cm=60/100=0.6m.

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To calculate the angle of the ramp given the length and the height, use the following equation: θ=sin^-1(h/L). θ=sin^-1(0.0043/1.22)=0.20 degrees.



To calculate gravity from knowing the angle and acceleration use the following equation, a=g*sin(θ). 0.10= g*sin (0.59), g=9.71 m/s^2.

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To find the uncertainty of the angle use the following equation, uncertainty= arcsine (0.0001/L). uncertainty = arcsine (0.0001/1.22)=0.004 m.

Questions 1. Why is this method not very good? What are the limitations? This method produces inaccurate measurement because it has many sources of error. First and most importantly there is an error in recording time. There is always a delay between when the person with the watch says “start” and when the person drops the wooden block. Similarly, there is a delay in stopping the watch, and that is due to the time of human reaction between seeing the block touching the ground and stopping the watch. Time produce a great source of error because when calculating gravity we raise time to the second power, which makes any error in time more effective to the result. The second source of error is the uncertainty in the measurements of the wooden meter stick which effect our value for the height. 2. Let's consider how 'level' this track really is. Using uncertainty analysis, what is the uncertainty in the angle measurement of the track? Based on this uncertainty, our tracks

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are probably not exactly at θ = 0.000… In an ideal physics set up, even a very small angle should create an acceleration. So, why can you get the car to stand still? The length of the lab was given to be 1.22 meters. Since the length was given, therefore the uncertainty is

± 0. The uncertainty of the height of the track has to be measured with

±

0.005 m because the height was measured with a ruler. Therefore, the inaccuracy of the sin ratio is 0.0065 ± 0.004. The inaccuracy of the angle then is 0.6016 - 0.37243 =

± 0.229. Also,

the reason why the cart was still even when the angle is not 0 degrees because the track is not completely frictionless, but there is a small coefficient of static friction. 3. Based on this angle, estimate the static (or rolling) friction coefficient that is acting on the car when it's on the ramp. The mass of the cart is ~ 500 grams. A 500 grams cart has a coefficient of static friction of 0.006 that is acting on the cart when it is placed on the ramp. Therefore, it does not move at first when the angle is changed . 4. Your task is to obtain a value for based on the measurements. You'll need to know the angle for each run so make sure you've carefully noted that variable. The procedure will be to import the position data into excel, then turn this data into a velocity vs. time graph, then find the slope of this v(t) graph, which will be the acceleration. Then use the relationship to obtain a value for . Remembering the kinematics relationships, the velocity of an object is given by the derivative of the position with respect to time. To do this in excel, you'll need to create a second column of data that contains the change in x over the between each data point. We found the value of little g by using the equations v(t)=v0+at, and a=g*sin(Ɵ), little g = a/ sin(Ɵ ). By using those equations, we were able to find the value of little g. The measurements for the different trials were close in the value of little g. However, there was a

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little difference due to the uncertainty of the measurements. In addition, also the fact that there is a small coefficient of friction acting on the cart when it is on the track affects the experimented values of little g. 5. Doing a similar analysis to this data as you did in the question above and determine the acceleration of the cart with and without the mass. Use your analysis to make a claim either that the mass affected the acceleration or that it did not. Would you expect it to based on our understanding of kinematics? The mass should not affect the outcome value of little g because gravity is only determined by the acceleration and the angle of the ramp. However, since the track is not completely frictionless, there is a small difference between the value of little g with and without mass due to the friction force which depends on mass, since friction force depends on the mass of the cart. In an ideal frictionless track, the mass would not have an influence on the value of little g. 6. More than likely, there are differences between some groups' estimation of shown in the table above. Comment on these discrepancies. If everyone had access to the same raw data (i.e. the video), shouldn't their results be the same? What could lead to variations in these results? Calculate the average value from based on these measurements. Is it within the uncertainty you would expect from the experiment? Different groups had a noticeable difference in the experimented values of little g given the same data. That is because there were many sources of error. First, in the video, some group could have converted the frames to seconds incorrectly, recorded time = 0 as the initial time instead of t = 20, or miscounted the frames. Secondly, different groups could have used different units to measure the distance other than a ruler. And finally, there could have been an error in

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measuring the height of the track at any given time interval. The video is recorded at 60 frames per second. By calculating acceleration due to gravity, we were able to get 9.796 m/s^2. We were able to obtain a value of little g close to the accepted value because we were able to determine the time when the ball was dropped in addition to converting frames to seconds accurately. The results were within the uncertainty of the experiment, and were not greatly off, given the various sources of error occurring. Conclusion The results that we obtained from the free fall experiment were close to the accepted value of little g. One of the trials we got 9.88 m/s^2 which is 0.08 m/s^2 off from the accepted value. However, there were many sources of error. First, using the wooden ruler to measure the height had a level of uncertainty. In addition, the person holding the wooden block could have moved unintentionally and slightly changed the height which affects the outcome of the experiment. Also, the delay in starting and stopping the watch would contribute to a great error in the experiment value of little g. the error in time would have a greater effect on the value of little g since we raise it to the second power when plugging it in the equation to obtain little g. we could make the experiment more accurate by using a certain height marked on the wall, and using more efficient watch to get less error in the measurement of time. The results for this experiment was expected, we even got on one of the experiment a value of 9.796 m/s^2 which is so close to the accepted value of gravity which is 9.8 m/s^2. Our most accurate method was the slo-mo video, because we were able to determine the exact point when the ball was released and did our calculations based on the time from the frame. In addition, converting the frames to seconds correctly helped us get to the value close to the accepted value of gravity. Error could occur when we were determining the time when the ball

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was dropped. The video was not clear as to when exactly the ball was dropped. To avoid that error, we could use a video that clearly extinguish the time when the ball is dropped, to get more accurate value of little g. In the air track experiment, we were able to get a value of gravity within the level of uncertainty on the experiment. However, the fact that the track was not completely frictionless and had a small coefficient of friction affected our results. To avoid that error, we can use a perfectly frictionless track so there is no friction force acting on the cart. Another error that can affect the result is delay in letting the cart go when recording the graphs. That would result in an error in the initial height and distance. To avoid this error, we made sure that the cart is on the start point, the highest end of the track, and we let the cart go exactly when starting the graph.

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