Apr, 6th 2014: Lab 11: Work-Kinetic Energy Theorem Activity

EXPT 1: Work Done by Non-constant Spring Force

Purpose:

To measure spring constant and the work done by displacing the string with a non-constant applied force using LoggerPro.

Experiment:
We know that Work= Force* displacement. If we were to graph F and displacement, for constant force, we can see a rectangle for constant force. Area of rectangle = FΔx, which is exactly the same as how you calculate work. In the case of non-constant spring, we know that the spring is harder to pull when it's almost reaching its maximum stretch. Hence, the force to pull is not constant. In fact, it increases gradually, making the graph of F and displacement look like a triangle. Since F spring= kx,  we get W= (1/2)(kx)(Δx) = (1/2)kx^2

Left: Work for constant force; Right: Work for non-constant force

1. Set up cart, ramp, motion detector, force probe, and spring. The force probe will be used to measure the applied force and motion detector will detect the change in position of the cart-spring system.

The spring is hooked to the black force probe and the cart. Motion detector is on the other side.

• Leave at least a half a meter space between the motion detector and the cart+spring when they move toward the detector.

2. Calibrate the force probe with 500g mass, so the force of (0.5)(9.8)= 4.9 N should be entered into LoggerPro calibration. The probe's hook is attached to the spring and the spring to the cart. The cart has be heavy (we have to put extra masses on our plastic-made cart) so that we can count the mass of spring negligible.
3. Since we're pulling toward the detector, the displacement decrease with respect to it. So, we will get a graph that has negative slope. We can change this default by setting the detector to "reverse direction", so we can get a positive slope for F vs. X.
4. Make sure that the spring is relaxed, and parallel to the track before start stretching.
5. Zero the force probe before start, then begin to pull slowly toward the detector, hit collect data and LoggerPro will graph F vs. X. Stop when the spring is stretched to about 0.5m.
6. A good data will present in LoggerPro if the motion detector sees the cart the whole time and that nothing else (like your hands) interfere the detection.

Data:

1. As mentioned, since we're dealing with non-constant force, our graph has a slope and since the force requires to stretch the spring increases when the spring approaches its maximum stretch, we a get a graph that approximately forms a right triangle with x and y axes.
    * If force is constant, we get a straight line graph of F vs X. and that would make a rectangle shape with x and y-axes. The area of the rectangle is the work done by the applied force.
2. Follow the equation of Hook's Law: F= kx, we then know that the slope of F vs. x is k, the spring constant.

Initial position=0.04, final position= 0.51; k= 2.96N/m

3. LoggerPro provides an integration command. Highlight the graph and use integration, we get

w=  ∫ kx dx = 0.3799 J ~ 0.38 J

    * Note that the initial position is not at zero but 0.04.
4. To check whether our work done is accurate or not,

• Use integration by hand since we know spring constant from the previous graph. Therefore, 

          or,

Comparing the work done from the area under the graph and integration by hand we get approximately similar answer: 0.38J is the work done by applied force on spring.

Conclusion:
    In most cases of non-constant applied force, the graph of force vs. position doesn't perfectly resemble any known geometrical shape. That's why we have to use integration, which is a sum of many areas of small rectangle under the graph, to find the work done.

EXPT 2: KE and Work-Kinetic Energy Principle

Purpose:

  We want to collect and compare the data of Δk to W (by spring) and use work-energy theorem to associate that comparison.

Experiment:

1. Although we already calibrated the force probe and the setup remains the same, there are still a few extra steps to setup.
2. Measure the mass of the cart
3. In LoggerPro, we enter a new formula under "Data", then “New calculated column”. This formula allows us to calculate kinetic energy of the cart.
4. Since motion detector also collects velocity, we can use this in KE= (1/2)mvelocity^2
5. Pull the cart toward the motion detector until spring is about 0.5m stretched. Then, click collect and release the cart so the spring could it back, ideally, to its relaxed position and save the graph. The experiment has to be in this precise order or the detector might collect negative value of position, or force at the beginning. This data collecting process only takes a little more than 1 second.
6. After we've obtained F vs. position, now we want another graph, which is KE vs. Position in the same file as F vs. position.

         *You don’t want the region (at the end) where the data table collect negative force nor position value. This should be obvious since the shape of the graph in that region is a scribble and not an (approximately) straight line.

Negative position and force are crossed out and will not be shown in the graph

Data and Analysis:

1. As mentioned, we're able to calculate KE, with a measured velocity at any position. Since both KE and Force graphs have position as an x-axis, we can place the two on y-axis next to another and compare. 
2. From force done by spring vs. position, we know that the area under this curve is the work done. Hence we want to compare the integral of force of spring to KE at that same position.
3. Use Integral command for only force. vs position. Also, under "Analyze", click "examine" to compare Work and KE at one point.

example of one comparison at one position

As we choose area to integrate, make sure that you highlight from right to left. The reason is that motion detector detect decrease in position as the cart moves away from it. That's why we have to integrate in the same direction, from larger position to smaller position.

The screenshot below is a comparison between change in kinetic and work done by spring at several different positions. Note that as position becomes smaller (cart further away from motion detector), work and change in kinetic energy increase.

Conclusion:
 The relationship between work and change in kinetic energy is presented in the ratio of 1, in theory. In our case, it's not quite there. We ended up with change in kinetic energy= 0.97*Work, although this is not logical. Since we put less work to gain slightly more change in KE, so there's a systematic error. Nonetheless, our result does agree with the work-energy principle that work done by applied force to stretch the spring equals to the spring's change in kinetic energy (and that we attach the cart to make it easier to detect change in speed). In this experiment, we don't count in friction. Hence, work total = work to stretch the spring.

EXPT 3: Work-KE

Procedure:

Since the work total = change in KE according to the work-KE theorem, we want to prove this by using experimental data to compare the two.

Experiment:

1.    In the video "Work KE Theorem cat and machine" (in school's computer), the professor uses a machine to pull back a large rubber band. A force transducer indicate the different applied force used. As the professor pulls back the rubber band, the transducer makes a graph that compare applied force vs. position. As she repeats the pull several times, the overlap of similar point by point reassure that we have a fairly precise data. The graph shown in the video is then approximated into known geometrical shapes and used to calculated the work done (areas of those shapes) by the professor as shown below,

Add all the areas and we get Work= 26.595J

2.    In the second part of the lab, the cart is attached to the rubber band. As the professor pulls the cart back, she releases it making the cart go through 2 photo gates.

The two photogates

• Before this experiment, we know: mass of cart, Δx between photogates.
• By the end of the experiment, we would obtain Δt ( the change in time from when the front of the cart gets to first photogate to front of the cart gets to the second photogate)

Now we should be able to use work- KE theorem (and kinematics) to calculate final speed, and ultimately to find final kinetic energy of the cart.

Data and Analysis:

1. From the sketched graph, we add the area of 1 triangle, 2 trapezoids, and 1 rectangle and we get W (total) by the professor= 26.595 J.
2. m cart= 4.3kg; Δt= 0. 045sec; Δx= 0.15m, 

Kinetic energy final = ΔKE; cart starts from rest.

According to Work-KE theorem: W_total = ΔK, hence our error is:

Conclusion:
       Since the work done of 26.595J only give out 23.9 J as ΔKE, the other 2.695 J is, by most logical explanation, is lost in form of heat. As far as uncertainty is concerned, there are measured uncertainty (in mass, time, and Δx), negligible mass of rubber band (neglecting the elastic potential energy it might store), and the fact that approximate the graph into straight lines to find the work done by the professor. Nonetheless, within this 10% error, we can conclude that the rubber band-cart system follows the work-kinetic energy theorem.