2015-April-6 Lab 11: Work-Kinetic Energy Theorem

PURPOSE:  To figure out the amount of work done by a non-constant spring force by proving this is equal to the area under the position vs. force graph.  Additionally, to prove that the amount of work done by finding the area under the position vs. force graph is equal to the change in kinetic energy.

MATERIALS:

• Cart and track
• Spring
• Motion detector - to record position and velocity of cart
• Force probe - to determine spring force

PRE-LAB CONSIDERATIONS:

• For a constant force, the area under the graph is equal to the work done by the applied force because the equation for force is Work = Force*displacement*cos θ.  On a horizontal surface, the angle of displacement is 0 degrees, which cancels out to 1, so it would just be Force times displacement, given in the following graph under the slope:

• For a non-constant force on a horizontal surface, the same equation applies, you would just have to integrate to find the total area because the slope of the line is non-constant. 

EXPERIMENT 1: Work Done by a Nonconstant Spring Force

• Set up the track and cart on a horizontal surface, with the motion detector on one end and the force probe at the other.
• Motion detector was set to "Reverse direction" so that motion towards it was in the positive direction.
• After calibrating the force probe horizontally with no mass and then hanging a 100 g mass on, we attached a spring to it and the other end of the spring to the cart.

Set up of our experiment.

• Using a pre-programmed experiment "L11E2-2 (Stretching Spring) we recorded a position vs. force graph by moving the cart slowly away from the force probe for about a distance of 0.6 meters to achieve this graph:

• Using Hooke's Law, or Force = -kx, k (spring constant) = -F/x
• Since we made the position go in the positive direction, we can omit the negative, and it just becomes the slope of our graph, or force over position, which is 3.349 
• Our lab group did not use the integration routine for this first experiment to find the work done in stretching the spring, but we can use the equation W=1/2kx^2 to get a good idea of how much work was done:
• W = 1/2 (3.349)(0.6)^2 = 0.603 J

EXPERIMENT 2: Kinetic Energy and the Work-Kinetic Energy Principle

• Similar set up to the first experiment, this time also calculating kinetic energy by adding a column in LoggerPro with the equation: Kinetic Energy = 1/2m*v^2
• We weighed the mass of our cart to be 0.496 kg and used this for our equation
• For this experiment, we set zero position when the cart was near the force sensor and spring was unstretched.  We then pulled the cart towards the motion sensor to a distance of about 0.6 meters, ran a test with LoggerPro, and then released the cart.  This gave us a graph of points of spring force applied to the cart vs. position.  
• By using the integration function in LoggerPro, we were able to find the area below the curve between the starting position and various positions of the cart.  This area gave us the total work done on the cart by the spring force.

Graphs that show that the area under the curve of Force x position (work)
should be equal to the Kinetic Energy of the cart at that end position.

• Using the analysis feature, we also found the kinetic energy starting from where we zeroed the position to 3 various positions of the cart:

• At 0.412 meters, the area under the curve was 0.3449 J and kinetic energy was 0.564 J.

• At 0.331 meters, the area under the curve was 0.437 J and kinetic energy was 0.654 J.

• At 0.248 meters, the area under the curve was 0.5108 J and kinetic energy was 0.711 J.

CONCLUSIONS:

• In our experiment, the work done on the cart by the spring did not match up to the amount of kinetic energy the cart had at its corresponding position.  However, in each trial as the cart moves further away from the 0 position and the spring is released, we can see that the amount of work done by the spring force and the amount of kinetic energy both increase. Our errors may have been due to the fact that our force sensor was not calibrated correctly, causing the velocity to be altered in the equation for kinetic energy: KE = 1/2mv^2.  This could have also caused the recorded force to be off.  Another factor of error could have been caused by the notecard we attached to the cart in order for the motion detector to track it better.  It may not have been taped down with enough stability, which could cause it to flutter while the cart was moving.
• Our experiment proves the work-energy theorem, which is that the work done by the net force on a particle equals the change in the particle's kinetic energy.  In this situation, the work done by the net force is done while the spring is being released, which is the amount of elastic potential energy.  Because there is no gravitational potential energy (our experiment is horizontal), we can assume those are zero, and we are left with the equation:

Work = Δ Kinetic Energy

1/2kv^2 = 1/2mv^2

EXPERIMENT 3: Work-KE Theroem

For the last part of this lab, we watched a video of a professor using a machine that stretches a rubber band to measure the force on the rubber band and the distance it is stretched, or the work being done: Work = Force x Distance.  From this, we were able to create our own force vs. distance (m) graph:

We added up the area under the force vs. distance in order to calculate the work done to stretch the rubber band:

Work done to stretch rubber band  - area under Force x distance graph

Then, at the end of the video we were given the mass of the cart, change in position, and change in time.  We used this information to find the final velocity of the cart using v = Δd/t.

Mass = 4.3 kg

Δx = 0.15 m

Δt = 0.045 s

v = d/t  -->  0.15 m / 0.045 s = 3.33 m/s

We can then plug this into 1/2mv^2 to find the final kinetic energy of the cart, which should be equal to the work done.

1/2(4.3kg)(3.33m/s)^2 = 23.89 J

Work done was 25.675 J and final kinetic energy was 23.89 J, which gives us an error of 7%

CONCLUSIONS:

This final experiment again validates the work - kinetic energy theorem even better than our in-class trials.  Our 7% error may stem from the fact that the machine was older, or there were different stretch forces along different places in the rubber band.  There could also be discrepancies at the points where the force changes drastically.