Momentum of Inertia of a Triangular Plate

Purpose:
To use all of the concepts of moment of inertia to predict and make a reasonable experiment.

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. The black triangular plate is held from the center of mass and is the object that's moment of inertia will be calculated both theoretically and experimentally.

Data Collected:
The moment of inertia of an object is based on 3 things, the mass of the object, the axis of rotation, and the orientation and distance of the object from the axis of rotation. The triangular plate is treated as a thin plate.
The mass of the triangle = .456kg
Length of the base of triangle = .10m
Length of the height of triangle = .15m
The mass of the hanging object = .025g
Diameter of disk that is connected to hanging mass = .05m
In order to experimentally calculate the moment of inertia with the triangular plate in either position, the moment of inertia of the system must be calculated first. So a hanging mass is let loose and the angular acceleration of all three are recorded.
Angular accelerations
No plate = 1.337rad/s^2
Plate with base extended = 1.041rad/s^2
Plate with height extended = .989rad/s^2

Calculations:

The theoretical moment of inertia of the triangular plate in both positions are calculated first. The moments of inertia are found at the edges of rotation, so along the height or base and then the parallel axis theorem is used to move the center of mass to the center of the plate.

Using the angular accelerations, the numerical value for the moment of inertia can be calculated using the angular acceleration. Once the moment of inertia of the system without the plate is calculated, then the plate can be introduced to the system and in both positions it can be calculated.















The theoretical value for the moment of inertia rotated with the base out = .0019
The experimental value for the moment of inertia rotated with the base out = .0013
The theoretical value for the moment of inertia rotated with the height out = .0022
The experimental value for the moment of inertia rotated with the height out = .0016

Conclusion:
The predictions for the moment of inertia seemed to be over the value that was experimentally calculated. The likely source for the error was that the moment of inertia couldn't calculate for the hole in the center that is used to hold the triangle to the system.

Sincerely,

Swaggy C

Conservation of Momentum

Purpose:
The purpose of the experiment is to experimentally prove that momentum is conserved in a collision.

Equipment Used:

The picture is of an instrument that records the activity on its level glass surface. The reason the surface is level is because it is designed for small collisions between things like marbles and steel balls. It has to be level since the conservation of momentum in only two directions can be observed by this instrument so if it isn't level then forces such as gravity might be applied to the objects in a manner that will alter their momentum. In this case, one marble is put on a trajectory that will make it collide with another stationary marble in a glancing impact, then both marbles will be in motion and the data should prove that the momentum in both axis are conserved.








Data Collected:
Mass of marble 1 = .0864kg
Mass of marble 2 = .0857kg

The image shows the path of the marbles through the entire experiment. The blue dots are for the marble that is initially in motion and the red dots are for the marble that will be collided with marble that is initially in motion.

The image shows the data of the dots that were taken from the collision. The slopes of the data shows the velocity in both the x-axis and the y-axis for both marbles, before and after the collision. If momentum is conserved then the momentum in the x-axis and y-axis should be conserved.








Calculations:

The momentum in the x-axis before the collision = .0533
The momentum in the y-axis before the collision = 0
The momentum in the x-axis after the collision = .0747
The momentum in the y-axis after the collision = .0054

Conclusion:
Obviously the momentum in either axis isn't conserved, but that isn't proof that momentum isn't conserved. The source of error in this experiment is that the collision between the marbles wasn't a glancing collision, however that alone isn't enough to create a collision where momentum isn't conserved. The table isn't perfectly level, the marbles collided at too high of a velocity and because of the collision the rotation of the marbles might have changed. When all of these factors are combined it can be very easy for it to seem as if momentum isn't conserved.

Sincerely,

Swaggy C

Periods of Oscillation of Pendulums

Purpose:
The purpose of the experiment was to improve the ability to derive moments of inertia, and then use those moments of inertia to predict the period of oscillation as if the object were a pendulum.

Equipment Used:

The object, either a half circle or an isosceles triangle, is hung from a pivot and its period of oscillation is recorded by the photogate which records the time it takes for one oscillation to occur.











Data Collected:
First, the half sphere was oscillated, both from its base and at the apex of its curve. The only data that is needed to calculate both of the periods of oscillation for this experiment is the radius of the half circle and the location of its center of mass. The mass isn't important since the equation for torque and angular acceleration ensure that the oscillation is independent of the mass when it is acting like a pendulum.
Second, the isosceles triangle was oscillated both from the center of its base and from the tip where both equal sides come together. The only data that is needed to calculate both of the periods of oscillation for this experiment is the base and height of the triangle and the location of the center of mass.
Half Circle Data:
Radius = .125m
Center of mass = 4R/3pi
Theoretical period of oscillation from base = .77s
Experimental period of oscillation from base = .768s
Theoretical period of oscillation from apex = .755s
Experimental period of oscillation from apex = .764s

Triangle Data:
Base = .078m
Height = .145m
Center of mass = H/3
Theoretical period of oscillation from tip = .670s
Experimental period of oscillation from tip = .676s
Theoretical period of oscillation from base = .560s
Experimental period of oscillation from base = .565s

Calculations:

Half circle calculations:

These calculations are for deriving the moment of inertia of the half circle as if the axis of rotation. The moment of inertia turns out to be .5MR^2

These calculations are for deriving the theoretical period of oscillation, which is .77s. The second portion of the calculations are for deriving the moment of inertia if the axis of rotation was at the apex of the curve. It is done by using the parallel axis theorem twice, once to move the axis of rotation to the center of mass from the base of the half circle, then again to move the axis of rotation to the apex of the circle. The parallel axis theorem only works for when the axis of rotation is at the center of mass.

These calculations are for deriving the theoretical period of oscillation at the new location, which is .7548s

















Triangle calculations:

These calculations are for deriving the moment of inertia of the triangle if the axis of rotation was at the tip of the triangle. The moment of inertia was M((1/24)B^2 + (1/2)H^2).

These calculations are for the period of oscillation for the triangle if oscillated from the tip. The second portion of the calculations are for calculating the moment of inertia if the axis of rotation was from the base of the triangle. The final portion of the calculations are for calculating the theoretical period of oscillation which is .5596s














Conclusion:
For both the half circle and the triangle, the predictions were very close to experimental value, and that is because there was only a single variable to calculate for. The only reason that the prediction didn't match the actual period was because sin(angle) is approximated as being only the angle, in radians.

Sincerely,

Swaggy C

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