6 April 2015. Work-Kinetic Energy Theorem Activity

Purpose

1. To find the work done by a nonconstant spring force, and from that the spring constant (k).
2. To proof that work and kinetic energy principle (W=ΔKE) is true.

Apparatus:

1. Spring. In this lab blog, we basically determine the spring constant of several springs.
2. Cart. This act as the mass pulling the string and as an object to prove that the work - KE theorem is indeed true and applicable.
3. The sledge and clamp. To set up various conditions that is suitable for each experiment.

Procedure, data analysis, and conclusions

for this lab blog only, the procedure, data analysis, and conclusions will be described individually under one part section. Since there are three parts of this experiment.

Part 1 (work done by a nonconstant spring force)

Base Theory

• Let's look at the definition of work in physics.

• Now, when the force is constant, looking by the picture below, we now know why the area under the graph of F vs x should be equal to the work done by the applied force.
• When the force is not constant, looking by the picture below, we now know why the area under the graph of F vs x should be equal to the work done by the applied force.

Procedure

1. We set up the cart as the picture below.

2. Do not forget to calibrate the force sensor first by hanging a mass, and calculated the force, which is the weigh= the mass times the gravity; see if the value correspond to the data displayed.
3. Set up the motion detector to zero when the spring is unstretched and uncompressed. Also set up so that when you pull toward the detector, its a positive direction.
4. Then, collect the data when you carefully and slowly pull the mass toward the motion detector.
5. The motion detector will record many values, one of them is position (Δx). And the force sensor will record the Force applied by your pull.

Data Analysis

• Now, our job is to find the work done by the spring force.
• Remember from the base theory section above that Work of a nonconstant force equals the area under the graph, which also can be found through integration. Look at the picture below.

• So the work done is 0.5695 Joule.
• Now, lets find the spring constant! The spring constant varies from one spring to others. It is unique and basically describe how force varies over distance. So the unit is N/m.
• Remember, from Hooke's law that the force of the spring is equal to the spring constant times the distance. So F= k.x.
• That means, we can plot F vs x, and find the slope. This slope is our spring constant!

k = 6.056 N/m

Conclusions

• Work can also be defined as the area under the graph of Force vs Distance, or in other word the integral of F over x.
• If we plot  Force vs position, the slope will be our spring constant

Part 2 (KE and Work-KE principle)

Procedure

1. Using the same set up as part 1, but we also measure the mass of the cart this time.
2. Do not forget to calibrate the force sensor first by hanging a mass, and calculated the force, which is the weigh= the mass times the gravity; see if the value correspond to the data displayed.
3. Set up the motion detector to zero when the spring is unstretched and uncompressed. Also set up so that when you pull toward the detector, its a positive direction.
4. Then, carefully and slowly pull the mass toward the motion detector. Then, when you release the cart, start collecting the data.
5. The motion detector will record many values, one of them is position (Δx) and the velocity (v). And the force sensor will record the Force applied by your pull.

Data Analysis

• Now, our job is to compare the work done by the spring, and the kinetic energy.
• Remember from the base theory section above that Work of a nonconstant force equals the area under the graph, which also can be found through integration.While the kinetic energy is half time the mass of the cart times velocity squared.
• Now, we need to create the "column" of kinetic energy, which will be 0.5*0.916*V*V

• We need to compare it in each moment, which means comparing them (Work done and KE) in every certain amount of distance.
• That is why we combined the graph of KE vs positions and Force vs Position. Remember to crossed out the data when the position is negative, which is when the spring went compressed and stretched repeatedly (figure to the right).
• Why Force and not Work? Because we learn from part 1 that we can find the work through integrating Force over distance. This is easier, since we do not need to calculate the work first, and then plot it with distance, then finally compare it to KE vs Distance.

• As we can see, the value almost correspond with each other. The Force is always a little bit high because when we do the set up, the Force sensor is not equal zero (almost but not), so when the logger pro do the integrating, there is little bit of area that is not supposed to be added.

Position (m)	Work (N*m=J)	Kinetic Energy(J)
0.361	0.1606	0.150
0.309	0.2689	0.242
0.130	0.4744	0.461

• This means that the work done by the spring is equal to its kinetic energy. This corresponds with the the theory about "what we 'give' to mass is the work, and it is equal to what the mass 'gain' (which is the kinetic energy in this experiment)"

Conclusions

• From this experiment we get that W=ΔKE. W being the area under the graph of F vs x.
• Again, this corresponds with the the theory about "what we 'give' to mass is the work, and it is equal to what the mass 'gain' (which is the kinetic energy in this experiment)"
• The uncertainties of this experiment comes from the fact that:
• the spring actually have some mass, but we ignore it.
• it is quite hard to adjust the force sensor to zero at the first place.
• there must be some friction even though very small on the sledge, but we ignore it
• the position of the cart when we say its zero, is actually only close to zero, since it is very hard to adjust the string into the condition unstretched and uncompressed. The spring will tend to go down vertically because of its weigh and it will automatically affect the horizontal position.

Part 3 (Work-KE theorem)

Procedure and Data analysis

• We watch a video about Work-KE theorem. In the video, the professor uses a machine to pull back on a large rubber band. The force being exerted on the rubber band is recorded by an analog force transducer onto a graph.
• The stretched rubber band is then attached to a cart of a known mass. The cart, once released, passed through two photogates a given distance apart. By knowing the distance and the time interval between the front of the car passing through the first photogate and then the second photogate, we can calculate the final speed and thus the final kinetic energy. 

• This is the graph of Force vs the stretch of the rubber band (m). And the area under the graph is the work.

• Then the calculation of KE and the percent error.

Conclusions

• The percent error of  11.327% is quite big, because we are not exactly doing a very good job in calculating the area under the graph.
• When we divide the graph into parts, we basically letting some part of the graph uncalculated, and some part that are not in the graph calculated.
• We can do integral if this graph is drawn in logger pro, but in the past we do not have the technology yet.
• So, dividing into parts is basically the way of how people calculate the area in the past.