Conservation of Linear and Angular Momentum

Purpose:
The purpose of the experiment is to test the understanding of all of the concepts that are involved in this collision. The concepts are moments of inertia, gravitational potential energy, torque, and the conservation of momentum.

Equipment Used:

The device is a series of two wheels one on top of the other that has air flow through it and creates a rotating system with a negligible amount of friction. The rotating system has a sensor that reads the rotation of the wheels, both the top and the bottom, since the two wheels can be made to either spin independently or together as a single system. The hanging mass is on a pulley that has a similar design, allowing air to flow through it to create a negligible amount of friction. A small attachment is placed on the device that is going to catch a ball that will be rolled down a ramp.

Data Collected:
A ball of mass .0238kg and radius .019m was rolled down a ramp. The end of the ramp was .975m above the ground and the horizontal distance that the ball rolled away from the end of the ramp was .51m. The height up the ramp that the ball was released from was .192m. Using this data, both the experimental and theoretical velocity that the ball should have can be calculated, but that will be done in the next section. The hanging mass on the system is .0247kg and it is tied around a disk that has a radius of .05m. The average angular acceleration with the disks and the attachment for catching the ball was 5.339rad/s^2. The experimental angular velocity of the system after the ball collided with the device was 1.572rad/s

Calculations:

These calculations are for calculating the experimental horizontal velocity of the ball as it leaves the ramp and the theoretical horizontal velocity of the ball. The experimental velocity was 1.14m/s and the theoretical velocity was 1.4m/s

These calculations are for calculating the moment of inertia of the device that will catch and rotate with the ball. The device is irregularly shaped so there is not formula for deriving the moment of inertia. It was calculated experimentally by comparing the torque and the angular acceleration of the moment of inertia. The moment of inertia was calculated to be .0011191kgm^2. Once the moment of inertia is calculated then a prediction for the angular velocity of the system after the collision can be made. The predicted angular velocity of the system after was 1.712rad/s and the experimental angular velocity was 1.572rad/s.








Conclusion:
The predicted angular acceleration had a percent error of 8.91%. The most likely source of the error is that the location of where the ball collided with the device wasn't exact and that the ball didn't collide exactly in a horizontal manner. That means that the ball was either on a slightly lowered or raised trajectory.

Sincerely,

Swaggy C

Conservation of Angular Momentum

Purpose:
The purpose of the experiment is to use the understanding of angular moment, gravitational potential energy, and elastic collisions to make a prediction about the experiment.

Equipment Used:

A meter stick was pivoted as close to its end as possible and it was placed so when it would swing a piece of clay would collide with it and stick to the ruler. The piece of clay is placed as close to the bottom of the meter stick as possible and a camera is going to capture the maximum height that the meter stick will rise again.














Data Collected:
Length of the meter stick that is around the axis of rotation = .994m
Mass of the meter stick = .137kg
Mass of clay = .00955kg
Distance away from axis of rotation for clay = .994m

Calculations:

The theoretical angle that the meter stick should rise after the collision is calculated, assuming that the ruler is released from a 90 degree angle and it makes a completely inelastic collision with the clay. There is also an assumption that there is no friction and that the system didn't lose any energy when the clay and meter stick collided. The calculations predicted that the angle should have been 74 degrees and the experimental  angle was 66 degrees.











Conclusions:
The experimental angle was smaller than the theoretical angle, which isn't a credit to any error in the experiment. The source for the error was in the assumptions that the system didn't have any friction or that there was no loss of energy or momentum during the collision. The system clearly had friction and the system might have lost a little bit of energy when the clay collided with the ruler because the clay was placed on a rod and it would have taken energy to remove the clay from the rod.

Sincerely,

Swaggy C

Kinetic and Potential Energy of Magnets

Purpose:
The purpose of the lab is to develop a correlation between the function of a force and the kinetic energy of an objected that is acted on by the force.

Equipment Used:

The track is an air track which allows for a glider to move as if it is frictionless, in reality there is a tiny amount of friction but it can be considered to be negligible. There is a magnet at the end of the track and a magnet on the glider. The glider will be sent into the magnet and it will be repelled, the motion sensor at the end of the track by the magnet will be used to calculate the position and to calculate the speed of the glider before, during, and after the collision.




Data Collected:
First, a function for the force the magnet exerts on another magnet needs to be created before any experiment can be performed. This was done by raising the far end of the air track, thus having the glider have the acceleration of gsin(angle). When the glider is stopped by the magnet, then the force that the magnet is exerting at the distance away from the magnet is equal to the mass of the glider times the acceleration. As the angle became larger, the distance that separated the two magnets became smaller.

Using computer software an equation can be created for the force the magnet exerts based on the distance between the magnets. The integral of that function gives the potential energy of the magnet. The potential energy and the kinetic energy are equal to the total energy of the system, so the total energy of the entire system through the entire experiment should be a constant value. The formula for the potential energy of this magnet = (.0001344/1.011)x^-1.011.

Conclusion:
From the graph it can be seen that the red line, which is used to represent the total energy of the system, stays at a fairly constant level. The yellow line, which represents the kinetic energy of the system, is level with the total energy of the system at the beginning and at the end. The magnetic potential energy of the system is very small in the beginning and in the end of the experiment as it should be. Since the two follow the relationship that is expected between potential energy and kinetic energy then it can be said that the formula derived for the potential energy of the magnet is very accurate.

Sincerely,

Swaggy C

Work and Power

Purpose:
To develop the understanding of work and the properties of work. Then the new found concept of work will be compared and contrasted with the concept of power.

Equipment Used:

 The picture is of the stairs that the students will walk and run up. Next to the stairs over the railing there is a pulley system that has a backpack hanging over it and the students will raise it.
















Data Collected:
The data that was collected was the height of each step in a stair case and then the amount of steps in the stair case. This calculation gave the height, the delta y, that the force will be applied over.The force that is going to be applied will be the weight of the individual that is either walking or running up the stairs. The same concept is going to be applied to a pulley system with a weight at the bottom that is raised to the top. However the height of the stairs isn't going to be the distance that it will be raised will be different.

Calculations:
The force that is being applied over the height of the staircase gave work, however the time it took to accomplish that work was recorded as well. By dividing the work done by the time it took to do the work, then power is given.

Sincerely,

Swaggy C

Impulse versus Momentum

Purpose:
The purpose of this experiment was to experimentally show that impulse is equal to the change in momentum to the objected which has the force acted on it.

Equipment Used:

The cart was set with a a force sensor on top. First the cart was ran into a spring and sent back. The motion sensor recorded the position and velocity relative to time. Then a nail was attached to the force sensor and the spring was replaced and a piece of clay.









Data Collection:
For the first portion of the experiment, the cart was sent into the spring and the force sensor recorded the amount of force it took to send the cart back. The motion sensor recorded the velocity of the cart before and after the collision. The change in the momentum of the system should be equal to the integral of the force that was applied to the cart.
mass of cart + force sensor = .449 kg
integral of force = -.8135
initial velocity = 1.153m/s
final velocity = -.911m/s
Then more mass, specifically .5 kg, was added to the cart and the procedure was repeated.
mass of cart + force sensor + additional mass = .949kg
integral of force = -.8336
initial velocity = .524m/s
final velocity = -.390m/s
Then a nail was attached to the force sensor and the nail was sent into a piece of clay. The collision is considered to be completely elastic.
mass of cart + force sensor = .449kg
integral of force = -.2214
initial velocity = .760m/s
final velocity = 0m/s

Calculations:
Calculating the change in momentum is done by multiplying the change in velocity by the mass of the object. Change in momentum = m(Vf - Vo)
For the first part of the experiment the change in momentum was -.9267 and the integral of the force was -.8315. % error = 10.27%
The second portion of the experiment where more mass was added the change in momentum was -.8674 and the integral of the force was -.8336. % error = 4.05%
The final portion of the experiment where the nail stuck to the clay the change in momentum was -.3412 and the integral of force was -.2214. % error = 35%

Conclusion:
The percent error for the first portion of the experiment was fairly high. There are several reasons that could contribute to the high amount of percent error. The force sensor could have ran into the spring at an angle so the force sensor would not be able to read the entire force of the spring and only managed to read a portion of the force. The more likely reason for the error was that the cart was moving at a high velocity and it returned at a high velocity as well. This hypothesis has some validity since the second portion of the experiment, which was essentially the same except more mass was added to the cart, the cart was sent into the spring at a smaller velocity and returned at a small velocity as well. The final portion of the experiment had an extremely high percent error at 35%. The most likely source of error is that the force sensor didn't read all of the force that was being applied to the nail since the nail was taped to the force sensor.

Sincerely,

Swaggy C

Kinetic Energy of a Spring Cart

Purpose:
The purpose of the lab is to see if there is a correlation between a varying force applied over a distance and the kinetic energy of the object that is being acted on by the force.

Equipment Used:

The image is of a spring that is attached at one end to a force sensor and the other end to a cart with a block on top. At the other end of the track is a motion sensor. The force sensor will be used to measure the amount of force the spring is exerting on the cart and the motion sensor will measure the position and velocity of the cart. The point where the cart is attached to the spring's unstreched position will be the zero of the position graph. The direction towards the motion sensor is considered the positive direction.



Data Collected:

The cart was pulled towards the motion sensor and then released. The spring that was attached to the cart then exerts a force on the cart, accelerating it towards the define zero. Logger pro collected the force on the cart and position of the cart. The integral, or area under the curve, of the force graph should be equal to the amount of kinetic energy of the cart, which is half of the mass times the velocity^2.


Calculations:
When x = .193m the area under the curve was .4403 N*m and the kinetic energy is .346 J
When x = .153m the area under the curve was .5464 N*m and the kinetic energy is .433 J
When x = .110m the area under the curve was .6396 N*m and the kinetic energy is .501 J
The error at x = .193m was 21.4%
The error at x = .153m was 20.8%
The error at x = .110m was 21.7%

Conclusion:
There is a very high amount of error for every point at which data was collected, however the data is off by the same large margin for each point. This would mean that at some point during the experiment one aspect of the equipment was not properly calibrated. Perhaps the force sensor wasn't zeroed correctly or the position of the unstretched spring was not set to zero.

Sincerely,

Swaggy C

Relationship Between K, Period, and Mass

Purpose:
The purpose of the experiment is to develop a tentative relationship between the spring constant, oscillating mass, and the period of oscillation of a spring. After the tentative relationship is found then a small discussion of where this relationship comes from will be held.

Equipment Used:

The set up is of a spring and hanging from it is a mass and there is a motion sensor to record position versus time. Three other groups each had a different spring but had the same "effective" mass on the spring to oscillate.
















Data Collected and Calculations:
First, the spring constant, k, of our group's spring needed to be calculated. This was done by hanging a mass and then recording how much the spring stretched. This was repeated twice more with more mass each time this was done and then the spring constant can be calculated.
Mass = .05kg, stretch of spring = .074m
Mass = .1kg, stretch of spring = .153m
Mass = .15kg, stretch of spring = .228m
Spring Constant of our group was 6.37N/m
Second, the relationship between the period of oscillation and the spring constant was compared. Four separate groups had different springs with different spring constants but the same effective mass was hung from the springs. The period of oscillation, T, were then compared to the spring constant, k.
k = 2.39N/m, T = 1.369s
k = 6.32N/m, T = .90s
k = 14.01N/m, T = .53s
k = 26N/m, T = .42s
From the data it is seen that as the spring constant increased, then the period of oscillation became smaller. Now that the relationship between spring constant and period of oscillation has been explored, the relationship between mass and the period of oscillation will be explored.
Mass = .105kg, T = .90s
Mass = .150kg, T = 1.033s
Mass = .200kg, T = 1.125s
Mass = .250kg, T = 1.25s
From the data it is seen that as the mass increases the period of oscillation increases as well.

Conclusion:
There were no predictions in this experiment but there should be an understanding of how the relationship between the two different variable affects the period of oscillation. From previous physics problems, the period of an oscillating mass is equal to 2pi divided by omega of the system. For a spring system, omega is the square root of the spring constant divided by the mass. So T = 2pi*(m/k)^1/2. So from the equation it should be predicted that as the mass increased, then the period of oscillation should increase but if the spring constant increases then the period should be smaller and that is what occurred in this experiment.

Sincerely,

Swaggy C