The Work-Kinetic Energy Theorem As an object slides down an incline, it gravity does an amount of work =, where is the change in the y coordinate as the object moves and friction does an amount of work = cos The total work done is = + That work translates into an increase in kinetic energy = ( ) / 2, where v A is the object's velocity at the greater height and v B is its velocity at the lower height. The Work-Kinetic Energy theorem guarantees that =. We will use an inclined air track, which reduces friction to a minimum, two photogates, a counter/timer, a meter stick, a micrometer and a balance. In the previous experiment we saw that ignoring friction on the lowest air supply setting (1) would generate an error of roughly 3-4 %. This error is expected to be reduced as the air supply generates more pressure. We will set the air supply for the air track on high (above 3) so that we can ignore the effects of friction. The results you obtain from this experiment are highly dependent on the accuracy of your measurements, so be careful! Name: Lab Partners:
When entering numeric data, use exponentials: ie., 1.6 * 10-19 = 1.6E-19. Procedure 1. Before elevating the track, make sure it is level. Do this by placing a cart on the track in the middle, turning the air supply on, and adjusting the feet (evenly on each side of the track) so that the cart does not move toward either end. 2. Use the Vernier calipers to measure the thickness of the block used to elevate the track:! " = m 3. Place a photogate over the level track and measure the effective length of the cart. This is the length the cart moves while the photogate is active. To do this, plug the photogate into the counter/timer, turn the counter/timer on, turn the air supply on, and slowly move the "floating" cart past the photogate until the light turns on. Note the position of the edge of the cart: # $ = m Then move the cart until it turns off, and note the new position of the same edge: # = m The difference of these positions is the effective length: l eff = x 2 - x 1 = m 4. Measure the mass of the cart: m = kg 5. Elevate the air track at the end with the single foot, and place the photogates EXACTLY 0.5 m apart and at the appropriate heights to be triggered by the cart as it rides down the track. Set the counter/timer to "split timers", "gate mode" and turn off "input hold" and "memory". Plug the timer at the high end of the track into the jack marked "A", and the timer for the low end into jack "B". If using a Smart Timer, you will need two timers, one for each photogate. Plug them into input channel 1 on each timer, and set the timers for "Time/Stopwatch" mode. Press Start before each trial. Record the times for gates A and B for each of ten trials. Be sure to reset the timer before each trial, and be sure to release the cart from the very end of the track on each trial. To do this reliably, hold the cart against the end stop with the eraser end of a pencil, and
release the cart by quickly moving the eraser away from and above the cart. Practice for a few trials before recording data will improve your consistency. Record all digits in the timer display. Analysis t A,1 = s t B,1 = s t A,2 = s t B,2 = s t A,3 = s t B,3 = s t A,4 = s t B,4 = s t A,5 = s t B,5 = s t A,6 = s t B,6 = s t A,7 = s t B,7 = s t A,8 = s t B,8 = s t A,9 = s t B,9 = s t A,10 = s t B,10 = s 1. Compute the angle of elevation using the fact that when elevated, the distance between the track feet is the hypotenuse of the triangle whose side opposite the angle of elevation is the thickness of the elevating block. Do not round this number! θ = sin -1 ( th / x feet ) = degrees 2. Compute y = 0.5 sin θ = m and W = m g y = J using 9.81 m / s 2 for g. An Joule is a unit of energy, equal to one kg m 2 / s 2. 3. For each of the ten trials compute v X,i = l eff / t X,i :
v A,1 = m / s v B,1 = m / s v A,2 = m / s v B,2 = m / s v A,3 = m / s v B,3 = m / s v A,4 = m / s v B,4 = m / s v A,5 = m / s v B,5 = m / s v A,6 = m / s v B,6 = m / s v A,7 = m / s v B,7 = m / s v A,8 = m / s v B,8 = m / s v A,9 = m / s v B,9 = m / s v A,10 = m / s v B,10 = m / s 4. For each of the ten trials compute K X,i = m v 2 X,i / 2: K A,1 = J K B,1 = J K A,2 = J K B,2 = J K A,3 = J K B,3 = J K A,4 = J K B,4 = J K A,5 = J K B,5 = J K A,6 = J K B,6 = J K A,7 = J K B,7 = J K A,8 = J K B,8 = J K A,9 = J K B,9 = J K A,10 = J K B,10 = J 5. Compute K i = K B,i - K A,i for each trial: K 1 = J K 6 = J K 2 = J K 7 = J K 3 = J K 8 = J
K 4 = J K 9 = J K 5 = J K 10 = J 6. Compute the average of the K i : K avg = J 7. Determine the standard deviation in K avg. 8. Compare the average change in kinetic energy with total work done. Answer the following questions: 1. Does the work done by gravity lie within one standard deviation of K avg? 2. Was the Work-Kinetic energy theorem verified? Why or why not?