Lab: Conservation of Linear and Angular Momentum

Lab: Conservation of Linear and Angular Momentum

Purpose: Verify the conservation of linear and angular momentum

Equipment:
Rotational dynamics apparatus, rotational accessory kit, ball-catcher, ramp.

Experiment:
1. Set up the apparatus as the pictures show.
2. Measure the height from the starting point to launch point.
3. Line up the end of the ramp with the end of the lab table and release the ball from this starting point and note where the ball strikes the floor.
3. Measure the vertical fall of the bottom of the ball and the horizontal travel distance of the ball.
4. Calculate the launch speed of the ball.
5. Use the aluminum top disk, and mount the ball-catcher on the top of the small torque pulley using a gray-capped thumbscrew.
6. Release a hanging mass, and get the average angular acceleration of descending and ascending.
7. Calculate the moment of inertia based on the average angular acceleration.
8. Set up the ramp, and measure the radius at which the ball struck the ball-catcher.
9. Release the ball at the same height and record the final angular speed.
10. Do the calculation for the final angular speed and compare it with the actual value.

Data:
h=97.5cm        L=51cm            hanging mass=24.7g
ball's diameter=19.0mm             mball=28.3g        diameter of the pulley=50.0mm

Calculation:

Conclusion:
Through the calculation, the theoretical value of omega is 1.74 rad/s.
Through the experiment, the actual value of omega is

Lab: Conservation of angular momentum and energy

Lab: Conservation of angular momentum and energy

Purpose:
Calculate the height that the meter stick can reach after releasing from horizontal position and hitting the clay at the lowest point by applying the concept of conservation of angular momentum and energy.

Equipment:
meter stick, clay, video capturer

Experiment:
1. Set up the apparatus as the pictures show.
2. Calculate the theoretical value before doing the experiment.
3. Put a clay at the lowest point of the meter stick's path.
4. Release the meter stick from the horizontal position with the other end pinned.
5. Record the motion and use LogPro to measure the highest point that the meter stick reaches.
6. Compare the actual value to the theoretical value.

Calculation:

Conclusion:

The theoretical value is 0.108cm, and the actual value is 0.08cm. Within a aspectable error, the angular momentum and energy is conserved.

Lab: finding the moment of inertia of a uniform triangle about its center of mass

Lab: finding the moment of inertia of a uniform triangle about its center of mass

Purpose:
Determine the moment of inertia of a uniform triangle about its center of mass

Equipment:
Disk, holder, triangle, hanging mass, string, sensor.

Experiment:
1. Set up the apparatus as the pictures show.
2. Turn on the air and let the hanging mass, tie the hanging mass with the disk without the triangle.
3. Let go the hanging mass, take the average angular acceleration of descending and ascending.
4. Put the triangle horizontally on the holder, and repeat step three.
5. Put the triangle vertically on the holder, and repeat step three.
6. Based on the hanging mass and the angular acceleration, calculate the moment of inertia of each situation, and use the inertia with triangle subtract the one without triangle to get the inertia of triangle alone.
7. Measure the length and the height of the triangle and calculate the moment of inertia of each situation.
8. Compare the theoretical value to actual value.

Calculation:
Horizontal:
method 1

method 2

Vertical:

 Inertia:

Result:
calculation based on the actual angular acceleration:

Lab: Moment of Inertia

Lab: Moment of Inertia

Purpose: Calculate the moment of inertia and the frictional torque of a disk and apply the value to predict the motion of the system of the disk and the cart.

Equipment:
A large metal disk, a video capture, a cart and a track.

Experiment:
1. Set the video capture and the metal disk.
2. Measure the radii of the small disks and the large disk, and read the mass of the disks.
3. Calculate the mass of each disk by volume and calculate the moment of inertia by the equation I=1/2 m r^2.
4. Spin the apparatus, use video capture to determine its angular acceleration as it slows down.
5. Calculate the torque of friction by the equation of torque=inertia times angular acceleration.
6. Calculate the acceleration of the system of the cart and the disk.
7. Predict the time it needs to travel 1 meter.
8. Do the experiment for three times and calculate the percent error.

Calculation:

Conclusion:
Through our calculation, we get the theoretical time is 7.83s. Through the experiment, we get the average time that the cart travel 1 meter is 7.84s. The percent error is 0.128%, which is lower than 4%。

Lab: angular acceleration

Lab: angular acceleration

Purpose:

Using different  mass of the top disk, different hanging mass and different torque pulley to see what factors affect the angular acceleration.

Apparatus:

The diameter and mass of the top steel disk: 126.6mm      1356g

The diameter and mass of the bottom steel disk: 126.6mm      1348g

The diameter and mass of the top aluminum disk: 126.6mm      466g

The diameter and mass of the small torque pulley: 12.5mm      10.0g

The diameter and mass of the large torque pulley: 25.0mm      37.0g

The mass of the hanging mass supplied with apparatus: 

Experiment:

1. Set up the apparatus as the pictures show.

2. Set up the Pasco rotational sensor and computer, set the equation on the sensor setting to 200 counts per rotation.

3. Make sure the hose clamp on the bottom is open so that the bottom disk will rotate independently of the top disk when the drop pin is in place.

4. Turn on the compressed air so that the disks can rotate separately.

5. Wrap the string around the torque pulley to the highest point, start measurements and release the mass. 

6. Use the graph of angular velocity vs. time to measure the angular acceleration of upward motion and downward motion.

7. Follow the instruction and change the factors to get angular acceleration under each condition.

Result:

Conclusion:
EXPTS 1,2, and 3: Effect of changing the hanging mass----the heavier the hanging mass, the larger the angular accerleration
EXPTS 1 and 4: Effect of changing the radius and which the hanging mass exerts a torque----The larger the radius, the larger the angular acceleration.
EXPTS 4,5, and 6: Effect of changing the rotating mass----the lighter the rotating mass, the larger the angular acceleration.

Lab: Collisions in two dimensions

Lab: Collisions in two dimensions

Purpose:
Look at a two-dimensional collision and determine if momentum and energy are conserved.

Experiment:
1. Set up the apparatus as pictures show.
2. Connect the camera and follow the instruction to set up.
3. Gently set the stationary ball on the leveled glass table. Aim the rolling ball so that it his the side of the stationary ball. The balls should ideally roll off at some decent angle from one another.
4. Input the video in LogPro and add point series following the motion of the balls.
5. After adding two series, set the origin at the point the rolling ball left my hand. And rotate the axes to be the same direction as the ball rolled.
6. Click linear fit for each portion of the graph to get the slopes.
7. Based on the value of slope, record the initial velocity and final velocity of the rolling ball and stationary ball.
8. Calculate the momentum and energy before and after the collision.

Data and analysis:
Steel ball and marble:
m(steel ball 1)=0.0671kg  m(marble)=0.0053kg

Two steel balls:
m(steel ball 1)=0.0671kg   m(steel ball 2)=0.0603kg

Result:
Steel ball and marble:

Two steel balls:

Calculation:

Conclusion:

Based on the graph, within acceptable error, the momentum and energy are both conserved in the collision in two dimensions.

Verification the relationship of Impulse and momentum

Verification the relationship of Impulse and momentum

Purpose: 
Through the bounce of the cart, verify the relationship of Impulse and change of momentum.

Eqiupment:
track, cart, force sensor, motion sensor, and mass.

Experiment:
Part One:
1. Set up the apparatus as the pictures show.
2. Measure the weight of the cart and the force sensor.
3. Give the cart with the force sensor a initial velocity and let the force sensor bounce upon the other cart.
4. Record the motion and the force.
5 Integrate the force that the cart received and examine the velocity right before the collision and after the collision.

 Part Two:
1. Set up the apparatus as the pictures show.
2. Measure the weight of the cart with the force sensor and the mass.
3. Give the cart with the force sensor a initial velocity and let the force sensor bounce upon the other cart.
4. Record the motion and the force.
5 Integrate the force that the cart received and examine the velocity right before the collision and after the collision.

 Part Three:
1. Set up the apparatus as the pictures show.
2. Put a nail in front of the cart and stick a clay at the same height of the nail on the other side of the track.
3. Measure the weight of the cart with the force sensor and the nail.
3. Give the cart with the force sensor a initial velocity and let the force sensor bounce upon the other cart.
4. Record the motion and the force.
5 Integrate the force that the cart received and examine the velocity right before the collision and after the collision.

Result:

Conclusion:
We verify the relationship of Impulse and change of momentum J=F*t=delta p= m(Vf-Vi)