Physical Pendulum REDO

PURPOSE:

To predict the period of a semi-circle and isosceles triangle with some moment of inertia after first calculating the moment of inertia of the semi-circle and isosceles triangle about a certain axis.

To do this, we cut out the two shapes from a foam board, measured their dimensions, calculated the moment of inertia for each shape, and predicted a value for the period. After predicting a value for the period, we measured the period experimentally by using a photogate sensor with the oscillation at relatively small angles.



APPARATUS:

Equipment:

Photogate sensor
foam board
method of cutting the foam board
method of measuring the foam cut-out



EXPERIMENT:

We began by measuring the dimensions of our two shapes
Semi-circle: R = 0.0765-m
Isosceles Triangle: B=0.12-m, H=0.15-m

We began by calculating four moments of inertia.

To calculate the moment for the semi-circle, we first had to calculate the center of mass of the semi circle.

We only needed to find the vertical center of mass for the semicircle, and by symmetry, the horizontal center of mass is exactly half the diameter.
After finding the center of mass we calculated the moment for the semi-circle and the triangle.

From the relationship we found when calculating the center of mass:

For the Triangle:

We then calculated the period of oscillation for each shape.

For the semi-circle:

For the triangle:

After predicting values for the period of each shape with their given orientation, we measured the period with a photogate sensor.

The actual period for a semi-circle with its axis of rotation about the diameter: T=0.5994-s.
The actual period for a semi-circle with its axis of rotation about the outer radius: T=0.5991-s.
The actual period for an isosceles triangle with its axis of rotation about the base: T=0.6091-s
The actual period for an isosceles triangle with its axis of rotation about the apex: T=0.6964-s.

CONCLUSION:

Moment of Inertia REDO

PURPOSE:

To verify the sum-of torques equation, Στ=Iα, by finding the moment of inertia of a certain apparatus and directly applying the equation to predict how long it would take for a cart tied to our apparatus to accelerate down a 1-meter ramp.



APPARATUS:

Equipment -

Large metal disk on a central shaft, m = 4.840-kg
Calipers to measure the dimensions of the apparatus
Camera
1-meter ramp
String
Stopwatch
Rolling Cart, m = 500-g
Tape and marker (or any means of marking a specific point on the disk)


EXPERIMENT:

Using calipers and a ruler, we measured the apparatus' dimensions, which included:
the radius of the cylinder, R1
the radius of the disk, R2
the width of the disk, w
the distance from the surface of the disk to one end of the cylinder, x

We then calculated the moment of inertia for the entire apparatus.
I = 0.021-kg*m^2

Using a piece of tape we marked a point at the edge of the disk.
(red-dotted piece of tape on apparatus picture)
We gave the disk an initial torque to get the apparatus spinning and measured the deceleration of the disk by recording the disk as it slowed to a stop.
We gave the disk an initial torque to get the apparatus spinning and measured the deceleration of the disk by recording the disk as it slowed to a stop.
To calculate the acceleration of the disk, we used logger pro to track the position of the red-dotted tape frame by frame. Tracking this specific point resulted in a graph (not included*) which gave us the acceleration tangential to the edge of the disk.
a = -2.385-cm/s^2 = -0.02385-m/s^2

We converted the tangential acceleration into rotational acceleration using the following equation which expresses the relationship between the two accelerations: a = r*αUsing r = R2 = 0.1002-mα = -0.238-rad/s^2

*Faulty flash-drive
The apparatus spun to stopped due to a frictional torque. We were able to calculate this frictional torque with our calculated value for I and α

Στ=IαΣτ=(0.021-kg*m^2)(0.238-rad/s^2)τf = 5.00*10^-3
We then attached a 500-gram cart to our apparatus with a string, with one end of the string tied to the cart and the other end tied and wound around the small cylinder part of the apparatus (R1). We also set secured a ramp near the apparatus at an angle of 40°(θ).

We derived a new expression for acceleration for an apparatus which now has a tied mass to it.
(Note: In the calculations below, r_s is equal to the previously defined R1 and m is the mass of the cart.)

We calculated the new acceleration to be a = 0.0326-m/s^2
We were able to use a kinematics equation to find a value for the time it would take for the cart to move down the ramp using our new found value for acceleration.

t = 7.83-s is our prediction value.
We then released the cart on the ramp and let it roll toward the ground while simultaneously timing how long it took for the cart to reach the ground.
We did this three times so that we could average the times, thus obtaining a more accurate time measurement.

1) t = 7.53-s
2) t = 8.03-s
3) t = 7.97-s
avg. t =7.84-s

t = 7.84 is our measured value.
We then measured the percentage of how far off we were from the actual measured value and found that we were off by 0.128% 

CONCLUSION:

We verified the sum-of torques equation, Στ=Iα, by finding the moment of inertia of a certain apparatus and directly applying the equation to predict how long it would take for a cart tied to our apparatus to accelerate down a 1-meter ramp. We compared our prediction with the value we actually measured and we were within 0.128% of the actual value.

26-Nov.-14: Physical Pendulum (Simple Harmonic Motion)

PURPOSE:

To predict the period of a semi-circle and isosceles triangle with some moment of inertia after first calculating the moment of inertia of the semi-circle and isosceles triangle about a certain axis.

To do this, we cut out the two shapes from a foam board, measured their dimensions, calculated the moment of inertia for each shape, and predicted a value for the period. After predicting a value for the period, we measured the period experimentally by using a photogate sensor with the oscillation at relatively small angles.



APPARATUS:




Equipment:

Photogate sensor
foam board
method of cutting the foam board
method of measuring the foam cut-out



EXPERIMENT:

1. We began by measuring the dimensions of our two shapes
   Semi-circle: R = 0.0765-m
   Isosceles Triangle: B=0.12-m, H=0.15-m
2. We began by calculating four moments of inertia.
   
   
   
   To calculate the moment for the semi-circle, we first had to calculate the center of mass of the semi circle.
   
   
   
   We only needed to find the vertical center of mass for the semicircle, and by symmetry, the horizontal center of mass is exactly half the diameter.
   After finding the center of mass we calculated the moment for the semi-circle and the triangle.
   
   
   
   From the relationship we found when calculating the center of mass:
   
   
   
   For the Triangle:
   
   
   
   
3. We then calculated the period of oscillation for each shape.
   
   For the semi-circle:
   
   
   
   
   
   For the triangle:
   
   
   
   
4. After predicting values for the period of each shape with their given orientation, we measured the period with a photogate sensor.
   
   The actual period for a semi-circle with its axis of rotation about the diameter: T=0.5994-s.
   The actual period for a semi-circle with its axis of rotation about the outer radius: T=0.5991-s.
   The actual period for an isosceles triangle with its axis of rotation about the base: T=0.6091-s
   The actual period for an isosceles triangle with its axis of rotation about the apex: T=0.6964-s.

CONCLUSION:

24-Nov.-14: Mass-Spring Oscillations

PURPOSE:

To show how the period of oscillation in a spring changes with a variation in the mass and the spring constant for a mass-spring system.

This task was accomplished with three other groups and each group equipped with their own spring. Each group's spring had a different spring constant.
We then added the same mass* to our systems so that the only variable would be the spring constant.


After sharing those findings with other groups, we added different masses to our own spring so that the mass is the only variable.
This entire process should allow us to find how the period changes with a varying mass or spring constant.



APPARATUS:







Equipment:

A spring, secured in a fixed position
A mass, to match the masses of the other groups
Three different masses
A motion sensor, to measure the spring constant and period



EXPERIMENT:

1. We began by measuring the spring constant of our spring.
   To do this we attached a 50-g mass to the spring and let the motion sensor detect the position of the bottom end of the spring with no mass attached to the spring and compared it to the position of the bottom end with a mass attached to it, also detected by the motion sensor.
   
   After finding that the position without a mass attached to be 33-cm
   and the position with a mass attached to be 22.9-cm.
   We used Hooke's Law to find k.
   
   
   
   The other three groups also did this same measurement and shared their values.
2. Using the motion sensor, each group measured the period of oscillation with the same effective mass attached to the their spring. We created a graph with a power fit for the four values for the spring constant and the period.
   
   
   
3. We then individually measured the period of oscillation with three new masses attached to the spring.
   
   
   

CONCLUSION:

Our group was on the "left side"

19-Nov.-14: Conservation of Linear and Angular Momentum

PURPOSE: 

To verify that both Linear and Angular Momentum are conserved in a rotational dynamics apparatus.

This will be shown by rolling a small ball down a ramp off the edge of a table. We then practice the kinematics of this trajectory to find the velocity of the ball at the bottom of the ramp using measuring tape and carbon paper. We then place at the bottom of the ramp a ball catcher which will allow the ball to stick at the point of impact. When the ball collides it begins spinning with the ball catcher which is mounted onto a disk. We measure the Angular velocity of the disk after the collision and determine if momentum is conserved.



APPARATUS:








Equipment:

Rotational Dynamics Apparatus
Solid ball - mass = 28.3-g, diameter =19.0-mm
carbon paper
measuring tape
mass to hang on the apparatus to measure the moment of inertia - mass = 24.7-g
torque pulley - radius = 25.0-mm




EXPERIMENT:

1. We began by setting up and measure the necessary dimensions of our apparatus, which included:
   the distance between the top of the ramp and the table top,
   the distance between the bottom of the ramp and the table top,
   the distance between the table top and the ground.
2. We then calculated the moment of inertia of the disk/torque-pulley/ball-catcher by hanging a mass on a string attached to the apparatus and getting the average angular acceleration of the rotation as the mass descends and as it ascends through the graph.
   
   
   
   
   
   
3. After finding the moment of inertia for the disk/torque-pulley/ball-catcher system we calculated the velocity of the ball at the bottom ramp. This was done both theoretically and experimentally. To find it theoretically, we used conservation of energy. To get an experimental value we started by letting the ball shoot off the ramp, from the table to the ground and using carbon paper measured the horizontal distance of the trajectory. Using this new horizontal distance and those values found previously for the various heights, we calculated the velocity at the bottom of the ramp.
   
   

Now that we'd found initial velocity at the bottom of the ramp, the moment of inertia of the apparatus and with known mass of the ball, we calculated what the final rotational velocity of the entire disk/torque-pulley/ball-catcher system along with the ball.



We then compared our calculated value to the experimental value for angular momentum.





CONCLUSION:

The calculated value for the angular velocity was 1.74-rad/s and the experimental value was 1.572-rad/s.

A Great Day for Physics

(11/19/14, 7:09am)