Lab 11: Work-Kinetic Energy Theorem

Work-Kinetic Energy Theorem 

I-Shou Lin, May Soe Moe, Roya Bijanara

4/10/17

The purpose of this lab is to investigate and examine the work-kinetic energy theorem and how the work done is equal to the change in kinetic energy of an object. 

Energy is all around us, and there's work to be done everywhere ( no pun intended). The work-kinetic energy theorem relates energy to work by stating that the work done on an object causes a change in the object's kinetic energy. In order to compare work done and the change in kinetic energy, graphs of force vs distance and kinetic energy vs position are needed. The area under the curve of the force vs distance graph is the work done on the object. The area which is equal to the integral, can then be compared to the value of the kinetic energy at the corresponding points. 

Experiment 1: 
we set up the track, cart, motion detector, force probe, pulley, cart stop and hanging mass. The force sensor was zeroed first, and the 500 grams of mass was added to the cart. Then, we hung a 50 grams of mass from the end of the string, pulled the cart back and then released the cart to get the data. The idea is that the work done on the cart by the tension force in the string should equal the kinetic energy gained by the cart at any point during the acceleration of the cart.

Experiment 2: 
We measured the work done by a spring on the cart when the spring is stretched. Set up the ramp, cart, motion detector, force probe, and spring. Then, we zeroed the force probe and the motion detector so that toward the detector is the positive direction. In order to measure the work done by a constant force, a pull is required on the cart as it moves slowly to 0.6 m of stretched position.

Experiment 3: 
Part 3 of the experiment uses the same setup as part 2. However, this time we will measure the work done by a non-constant spring force on the cart. We measured the mass of the cart and entered a formula that allows us to calculate the kinetic energy of the cart at any point. We zeroed the force probe and the motion detector to be zero at the starting position. We pulled the cart so that the spring is stretched about 0.6 m from the equilibrium position. The cart is released, we begin graphing.

Experiment 4: 
In this part we justr watched a movie of a professor using a mchine to pull back on a large rubber band. The graph of force vs position is shown in the video.


Data Table

part 1:

	Integral value ( joules)	Kinetic energy ( joules)
Area 1 of the graph	0.1562	0.195
Area 2 of the graph	0.2755	0.232

part 2: 

Spring constant ( N/m)	Work done in stretching the spring ( joules)
3.511	0.21585

part 3: 

	Trial 1	Trial 2
Mass of the cart ( kg)	0.549	0.549
Δ kinetic energy ( joules)	0.109	0.150
Work done ( joules)	0.1287	0.1775

part 4: 

Mass of the cart ( kg)	Δ X photo-gate ( cm)	Δ T photo-gate ( milliseconds)
4.3	15	45

Calculated Results

part 4 of the experiment: 

k.e.( final)=0.5 * mass* velocity^2 ( final)= 0.5 * 4.3 kg* (0.15 m/0.045 s)^2= 24 joules 

For experiment 1 of the lab, the graphs depict force vs position, position vs time,  and velocity vs time, These graphs depict the relationship between work and the change in kinetic energy of the cart as well as the velocity and acceleration of the cart. The area under the curve of the force vs position graph shows the work done by the tension force in the string because work is force * distance. An integral is used because the force isn't constant. Velocity is the ratio of position divided by time, and acceleration is the ratio of velocity divided time.

For experiment 2, the two graphs depict the work done by a spring on a cart when the force is constant. Again, the area under the curve of the graph is equal to the work done by the spring because work is force multiplied by distance. In this case simple algebra is all that's needed to calculate the work done due to the force being constant. In addition, the last graph shows the work done by a constant spring force with a linear fit so that the spring constant can be determined.

For experiment 3, the five graphs transmit the work done by a spring force on a cart when the force isn't constant. Again, the area under the curve graph of the curve is equal to the work done by the spring because work is equal to force multiplied by distance. This time, an integral function is required because the force is not constant. In addition, the last 3 graphs of the 5 graphs analyzes the work done through the area under the curve and the change in kinetic energy of the cart. These two values should be equal in theory according to the work-kinetic energy theorem.

In conclusion, this lab was divided into four parts with the first three parts requiring collecting data to be graphed and then analyzing that data. For us it was a particularly confusing and unpleasant laboratory experience because of the number of technical functions involved in the computer software and graphing. We made many mistakes and had to perform each experiment many times. But, in the end we were able to produce good data and good results. Part four of the lab is just watching a video and then taking data in order to find the work done and the final kinetic energy. With respect to the comparison of the data, the results for experiment 1 for the integral and kinetic energy are within 20% of each other. Since they're fairly close, the original work-kinetic energy theorem holds. The work energy principle state that the work done on the cart-spring system is equal to the change in the cart-spring system's kinetic energy. And the work done on the cart and the change in the cart's kinetic energy are off by 16-17%. For the last experiment of the lab, the final kinetic energy of the cart is almost equal to the work done by the professor. The work done by the professor is 23.25 joules, and the cart's final kinetic energy is 23.9 joules. That is a percent difference of 2.79. These two values won't be the same because of the presence of friction and air resistance when the cart was accelerating through the two photo-gates. Also, the graph of force vs position isn't entirely consistent, and this would cause the calculations for the work done by the professor to be subject to errors and propagated uncertainties.