15 April 2015. Impulse-Momentum Theorem.

Purpose

Part 1
to observe collision forces that change with time and finding out how impulse-momentum theorem works.

Part 2
to observe bigger collision forces that change with time, which also means a larger momentum change.

Part 3
to observe impulse-momentum theorem in an inelastic collision.


Base Theory:

Momentum is defined as:

p = mass * velocity

Then, impulse combines the applied force and the time interval over which that force acts. We call this impulse, which is defined as:

J = Force * time

Therefore, since the momentum of the system cannot change without forces (no external forces: friction and gravity could be disregarded), the magnitude of force before and after a collision must be 0, because there is no change in momentum before and after the collision. Thus, we say that momentum of a system is conserved, regardless of elastic or inelastic collisions (which just vary in force-time). This is shown by the Impulse-Momentum, impulse is equal to the change of momentum.

What we "give" to mass is impulse, and what it "gains" is momentum.

J = Δp = integral of F . dt

Apparatus:

1. Cart and plunger. This set-up is used to model the collision.
2. Clay. Is used in expt. 3 to model inelastic collision.

Procedure and Data analysis

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


Part 1 (observing impulse-momentum theorem)

Procedure

1. Clamp the dynamics cart to a rod clamped to a lab table. Extend the spring plunger on the dynamics cart.
2. Mount a force sensor on another dynamics cart, with a rubber stopper replacing the hook mounted on the protruding part of the force sensor.
3. Do not forget to calibrate the force sensor with hanging some mass and see whether the number in the laptop make sense.
4. Set things up such that the stopper of the moving cart hits the plunger of the stationary cart when the moving cart gets close to the end of the track,.
5. Collide the cart with the plunger several times and observe what happens to the spring plunger.

5. Also don't forget to weigh the mass of the cart.

Data Analysis

• Now, what we have is the data of position and velocity of the cart from the motion detector and the force from the force sensor.
• Before we start analyzing the data, we need to understand this statement:
• That there is no external forces acting in the system; track is leveled and friction is ignored. Therefore the force sensor would detect 0 N.
• The maximum force would be when the plunger is compressed at the maximum by the cart.
• After the collision in complete. There is also no forces acting.
• The collision would take very little time, at max 1/2 second.
• Okay. Now, from the data we have, we can make a momentum column, which equals to the mass of the cart times the velocity.

• Then, we take the integral of the force graph, which will give us the impulse. J = 0.5170 N*s.
• Comparing this to the change of the momentum. 

P final - P initial = 0.224 - (-0.264) = 0.488 kg*m/s

Momentum final = 0.224 kg*m/s

Momentum initial = -0.264 kg*m/s

• The percent error is 

• Notice that this is a nearly elastic collision. When it is elastic, then the initial momentum would exactly equal to the final momentum. In this case, we are losing a little bit of momentum in the final as you can see from the graph. Then, where does this momentum goes? Well, actually it goes to the earth movement, because the mass of the earth is really huge, we can't really see its movement.
• But, anyway the impulse equals to the change of momentum with 5.61 % error. This proof that the impulse-momentum theorem is true.

Part 2 (A larger momentum change)

Procedure

1. It is basically the set up from the first expt.
2. But, we add some more mass into the cart, in this case we add 500 g into it.

We add more mass into it!

Data Analysis

• Now, same as before, what we have is the data of position and velocity of the cart from the motion detector and the force from the force sensor.
• Because our mass also changes, our formula for momentum also changes:

• Next, this is the graph for the force and the integral = impulse. I = 0.5820 N*s

• And the final and initial momentum.

Momentum final = 0.260 kg*m/s

Momentum final = - 0.323 kg*m/s

• the change of the momentum. 
P final - P initial = 0.260 - (-0.323) = 0.583 kg*m/s
• With percent error:

• Notice that this is a larger momentum change than before (this makes sense because we add more mass to the cart, automatically the momentum will also increase), and yes the change of momentum and impulse still equal to each other (with some error) even though we add more mass to the cart.
• This just proof that this law is universal and can be applied regardless the change of the mass.

Part 3 (Inelastic collision)

Procedure

1. It is basically the same set up from part 2
2. but we change the plunger into a clay, which will produce the inelastic collision.
3. DO NOT change the mass of the cart, and try to set up the speed of the cart as the same as in part 2.

Data Analysis

• Now, same as before, what we have is the data of position and velocity of the cart from the motion detector and the force from the force sensor.
• Turns out, the cart stick to the clay and generate a velocity = 0 at the end.
• In this experiment, we fail to set up the speed of the cart as the same as in part 2, due to limitation of time. Therefore, we can't really compare the impulse and momentum from part 3 to part 2.
• Based on the graph, the impulse is going to be I = 0.4825 N*s.
• and the final and initial of momentum.

Momentum final = 0 kg*m/s

Momentum initial = - 0.483 kg*m/s

• the change of the momentum. 
P final - P initial = 0 - (-0.483) = 0.483 kg*m/s
• With percent error:

• the change of momentum and impulse still equal to each other (with some error) even though this is an inelastic collision.
• This just proof that this law is universal and can be applied regardless whether its elastic or inelastic.

Extra

• Because we fail to set up the same speed. I find some sources that do the elastic and inelastic collision with same mass and speed from http://www.vernier.com/innovate/impulse-comparison-for-elastic-and-inelastic-collisions/.
• This is the graph.

Trial	Change in Momentum (kg m/s)	Impulse (N s)
Elastic Collision	0.32	0.32
Inelastic Collision	0.17	0.18

• The impulse from the elastic collision was very close to twice the impulse of the inelastic collision. This is the result that we sought. The elastic impulse is a little less than twice the inelastic impulse. That the ratio is just under 2 could be due to the fact that the “elastic” collision is losing some energy. We saw this as a slightly smaller speed after the impulse. Perhaps a magnetic bumper would see a more nearly elastic collision, and yield a ratio closer to 2 (http://www.vernier.com/innovate/impulse-comparison-for-elastic-and-inelastic-collisions/)

Conclusions

• Impulse is always going to be equal to the the change in momentum regardless its an elastic or inelastic collision, faster or slower speed, or lighter or heavier mass.
• Faster and heavier mass will produce larger impulse.
• The inelastic collision will produce half as big impulse as the elastic collision.
• The error for this experiment are from:
• the integral of the force graph is done from a recorded point to point (the force sensor read the input by frame per frame, and each frame has milliseconds gap between them), so it is not purely from what really happening.
• there might be very small friction and air resistance acting on the cart as the external forces, but we ignore that.

15 April 2015. Magnetic Potential Energy Lab.

Purpose

Part 1
Deriving an equation for magnetic potential energy of the system described in the section apparatus through several experiments.

Part2
Using the equation found in part 1, we are going to prove that the conservation of energy is indeed true once again.

Apparatus:

1. Air track glider. This apparatus is designed to have a moving mass (the glider) in horizontal motion without friction. On the track there are holes that emit air from the reversed vacuum cleaner connected to the system.

2. Magnets. We are going to measure the force between these two magnets.
3. Some books. These are used to varies angle of the track.
4. Angel Measurement. Your phone.
5. Motion detector.

Base Theory

• What do we know about Force and Potential Energy that we get from that force? We recall that there is no relationship whatsoever between them. We just memorize the formula. Turns out, there is indeed a relationship between them! (see picture)

• Yay! The relationship is:
• F = - du/dx
• Note that this is true for potential energy of any kind.

Procedure and Data analysis

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


Part 1 (determining the equation of the Potential Magnetic Energy between magnets)

Procedure

1. Set up the air track on a horizontal table. Make sure that it is really level out, meaning the glider stays still unless we give it a push (when the vacuum is on).
2. Attach a magnet at the end of glider, and another magnet with the same polarity at the end of air track as well. It is important to check whether the magnet works since it will affect the entire calculation later.

3. Connect the other end of air track to the reversed vacuum system
4. Measure the mass of the air track glider.

0.343 kg

5. Use books or blocks to create at least five different angle, so that the air track glider will have five different forces of gravity acting. Record each angle.

6. Measure the Separation Distance (the distance between two magnets) created by each angle each time. Record it.

Data Analysis

• Now, we have five different separation distance. What affect these lengths to be different? Yes! It's the gravitational force acting on the air track glider.

• Now, if we plot this Force vs the Separation Distance. We will get points that seem to match a power fit. Thus, Lets do a power fit in logger pro and see what we got.

F = 8.464 x 10^-7 times R^-2.213

• Now, we basically got the equation of the Force of the Magnetic.

• What do we do with it? In the Base Theory section, we know that F = - du/dx. This means that to get U (potential energy), we need to integral force over dx. Lets do this!

• Yes. Now we got our equation for Potential Energy between Magnets.

Part 2 (Conservation of Energy: PE magnet and KE)

Procedure

1. We use the same set-up as part 1 and plus we add a motion sensor behind the magnet to record the velocity of the glider. This later will be used as Kinetic Energy.
2. Place the glider on the air track, reasonably close to fixed magnet. Run the motion detector. Determine the relationship between the distance the motion detector reads and the separation distance between the magnets. Record.
3. Give the air glider a push and record the whole motion on logger pro.

Data Analysis

• Now, we need to understand that the conservation of energy playing in the system is the Kinetic Energy of the glider and the Potential Energy of the Magnets.
• Our goal is to add up these Energy and expect a constant value of the Total Energy.
• The motion detector will gives us these data: Time, Velocity, and Position.
• We need to determine the formula for KE and PE.
• For PE, the formula we got from part 1 is: F = 8.464 x 10^-7 times R^-2.213. The separation distance (R) is the distance between magnet, which we acquire by the procedure number 2. (see picture)

• So the PE magnetic is:

• And the KE is:

• Thus, total energy is:

• Plotting all of these together, we got:

• Yes. we got the total energy to be "almost" constant with some error.

Conclusions

• We manage to get the Potential Energy of Magnet from taking the integral of the forces playing in the system. And this actually works!
• We prove that the conversation of energy is indeed true. The KE and the PE magnet sums up to be a constant value with, we admit, quite a huge error=29.5%
• These errors are from:
• When we acquire the distance measure and the angle, we have our uncertainties (0.5 cm and 0.1 degrees).
• This lead into a bigger uncertainties in our Force model, and thus our Potential Magnetic Energy.
• There are some air resistance that we do not calculate over the system.
• There are some human errors in measuring the distance etc
• But, beside these errors, we find the PE from integrating the force, and actually manage to get a quite good result!

8 April 2015. Conservation of Energy. Mass-Spring System.

Purpose

to prove the theory of conservation of energy: that the initial energy is always going to be in the same amount of the final energy by calculating the energy of a moving "mass" spring (in addition to some masses attached to it).

Apparatus:

1. Spring and some masses. In this experiment, we basically use these aparratus as an object to determine its potential and kinetic energy
2. Motion detector, logger pro, laptop. For recording and analyzing data.

Procedure:

1. Set up the clamp,  the spring and the mass as shown in the picture. (pic)

2. Measure the spring constant by hanging some masses and record the distance stretched. Then you will be able to plot the distance vs mass, and the slope is your spring constant.
3. Then, attached the mass hanger so that it is vertical and the spring is just unstretched. Zero the motion sensor with the mass hanger in this position. Don't forget to reverse the direction, so that towards the motion sensor is positive. Also measure the this distance from the floor.
4. To verify that the motion detector is working, pull the spring down slowly, and check if the graph make sense.
5. Now, record the spring movement. Collect the data for at least 5  or 10 seconds. Make sure that the motion detector could read the movement perfectly by attaching a post-notes below the mass and carefully pull the spring in a perfectly vertical position. Otherwise, the spring will sway away.

Base Theory

• From the procedure, we do realize that we are going to plot every energy there are in the moving spring, and calculate the TOTAL ENERGY of all - expecting a constant value (based on the theory of conservation of energy).
• Let's us evaluate the energy in this moving "mass" spring.
• The mass: Gravitational Potential Energy and Kinetic Energy.
• The spring: Elastic Potential Energy, Gravitational Potential Energy, and Kinetic Energy.
• How do we calculate the Gravitational Potential Energy and Kinetic Energy of moving spring? We usually neglect that the spring actually has mass, but now we are not going to ignore it!
• This is how to calculate Gravitational Potential Energy of the moving spring.

• This is how to calculate Kinetic Energy of the moving spring.

Data Analysis:

• We measure the spring constant by hanging some masses and record the distance stretched (look at the pictures below). First we measure the length of the spring at equilibrium. Then add some masses and measure the additional distance created.

x= 37.1 cm

• From this data, we plot our graph of force vs distance; and the slope is our spring constant!

k = 11.01 N/m

• Then, we measure the mass of the spring itself. And we also measure the distance from the floor when the spring is unstretched.

• Realize that now, we have these data:
• spring constant = 11.01 N/m
• m spring= 0.064 kg
• m mass= 0.2 kg
• distance when the spring is equilbrium from the floor= 0.887 m
• Next we record the spring movement. The motion detector will give us data, including position and velocity over time (these are the rest data that we need).
• Now, we create several new calculated column, which are all the energy that varies with the movement of the spring (picture).

• This is how we input it into logger pro.

• Before we actually graph the real condition into logger pro. Let us predict several graph first.

• Now let's plot the energy playing in this system and the TOTAL ENERGY for real.

• Yes! We can see that the total energy is constant! (with some errors, calculation below)

EXTRA:

• Let us predict these graphs (Explanation in the pic).

• Turns out, the prediction matches the real graph.
• Now, we plot all the energy vs position and time.

• Form these graphs, we can see that the energy always go through a cycle, an in every point of each cycle energy is conserved! Look at the "almost" constant value of total energy playing in this system.

Conclusions:

• In this experiment, we manage to proof that the theory of conservation of energy is true and amazingly applicable in every second!
• There are indeed some errors in this experiment. The possible sources are:
• When we acquire the spring constant, we notice that the correlation of the line is not one. Thus its already producing some errors.
• When we zero the motion sensor we didn't actually get the value zero precisely but near it. 
• There are some air resistance that we do not calculate.
• There are some human errors in measuring the distance etc.
• Mass of the spring can affect the calculation quite massively. The fact that we always omit the mass of the spring as negligible when we calculate is quite disturbing.