April 6, 2015 Work-Kinetic Energy Theorem Lab

Objective: To show that work equals to the change in kinetic energy.

Exp 1

The Set-Up:

The Apparatus

We place a metal track on the table. At one end of the table is a force sensor stabled by the table with a table clamp. On the other side of the metal track is a motion sensor. Both force sensor and motion sensor is connected to loggerpro. We calibrate the force sensor with a force of 4.9 N applied. Then we set the motion sensor so that toward the sensor is positive.  A cart is place on the metal track with a spring connecting from the cart to the force sensor, so that when the cart is pulled to a certain distance, the force will also be recorded.

Data Collection: 

Force vs Position Graph

We pull the cart and obtain a graph of force vs position. We linearized the force vs position graph. The slope of is the spring constant, k, of the spring in this apparatus. 

The spring constant is 2.904 N/m.

To find the work done by the spring, simply use the integration option in loggerpro. Loggerpro will calculate the area under the curve, that is the work.

Position vs force graph. Area under the curve

In this experiment, the work that the spring have done is 0.08410 m*N.

Exp 2

The Set-Up:

The set-up in part 2 is the same as part 1.

We measure the mass of the cart plus the addition weight on top of it, the total mass is 0.574 kg.

Data Collection:

We use the same force vs position graph from part 1 and integrate it at three different positions. The area under the curve is the work done by the spring.

We want to see how work connects to kinetic energy.  So we analyze the area under the curve and kinetic energy to see the relationship between them. 

Force(purple) and Kinetic energy (blue) vs Position Graph ---1st position

The purple area represents the work that the spring did at that position. The blue line represents kinetic energy. The boxes show that the kinetic energy at this position is 0.071 J, the work is -0.07432 m*N.  Note that m*N= J, from this we see that they are representing the same thing, that is, work is the change in kinetic energy. We see more evidence in the second graph. There is a negative sign in work might because work is a vector, direction matters in work, the negative sign might come from the force is opposite the displacement. 

Force and kinetic energy vs Position Graph--2nd position

Here we have the information of both lines at the same position. Here the kinetic energy is 0.123 J, the area under the purple curve, or work, is -0.1232 m*N. They are very close together, from this we see that work is kinetic energy.

Force and kinetic energy vs Position Graph--3rd position

 This is the third position we choose to analysis the area under the curve and the kinetic energy. Once again we see that kinetic energy is 0.146 J and work is -0.1458 m*N.

 Work is the change of kinetic energy.

Exp 3

The Set-Up:

We open a file called “Work KE theorem cart and machine for Phys L.” It is a video about a professor uses a machine to pull back on a large rubber band. The force is recorded by a force transducer onto a graph. This rubber band is connected to a cart. Once released, the cart passes through two photogates. We want to find the final speed and thus calculate the final kinetic energy of the cart. 

Data Collection:

We first begin by finding the force vs position graph for the robber band. We trace the the force pattern made by the force transducer onto the white board. Then we make a graph using the data.

Stretch of the Rubber Band(m) [x-axis] vs Force(N) [y-axis] Graph

Work is the curve under the force vs position graph. If we want to find the work done by the rubber band, we have to we separate the curve into four sections and find their area one by one.

Area A: ½ (0.27 m) (68 N) = 9.18 m* N

Area B: (0.38-0.27 m)(68 N) = 7.48 m*N

Area C: ½ * (32+68N)(0.42-0.38m) = 2 m*N

Area D: ½ * (32+37)(0.65-0.42m)= 7.935 m*N

Work = total area under the curve= 9.18 + 7.48 + 2+ 7.935=26.6 m*N

Next, we examine the final kinetic energy of the cart attached to the machine.

Here are some basic information given from the video:

mass of the cart = 4.3 kg

Distance between the two photogates =15 cm= 0.15 m

Time that the cart passes the two photogates = 0.045 sec.

KE= ½ m v^2

v=d/t = 0.15 m/0.045 sec= 3.33 m/sec

KE=1/2 (4.3 kg)(3.33)^2= 23.8 J

Since the is initially at rest, its initial kinetic energy is zero, therefore, the charge in kinetic energy of the cart would just be the final kinetic energy. 

Notice that the total work done by the professor is 26.6 m*N while the final kinetic energy of the cart is 23.8 J. Since work= the change in KE, why the value is different? It is different because there is some uncertainty involve, such as friction in the system, or calculation rounding error. 

Summary: 

This lab is separate into three experiments. 

The first experiment seeks to find the spring constant, k, in a spring. We find the spring constant by using the force vs position graph. We obtain the graph by pulling the cart. Then we linearized the graph to give us a slope, the slope of the graph is k. k= 2.904 N/m.

The second experiment deals with the work-energy theorem. Using the force vs position graph from experiment 1, we find the area under the curve. The area under the curve is the work done by the spring. Then we combine kinetic energy graph into the force vs position graph. We chose three positions to analyze them. We find that work is the change in kinetic energy. 

The third experiment involves a video of a professor pulling a rubber band through a machine. The force of the rubber band is recorded. We look at the force curve and find the total work done by the professor. Then the same rubber band is connected to a cart. When the cart is released, it passes through two photogates. The mass of the cart, the distance between the two photogates and the time is given. We use these information to find the change in kinetic energy of the cart. Since work is the change in kinetic energy, they should be the same value. However, we find that they are not the same value, the difference might due to friction in the rubber band.