Lab: Simple Harmonic Motion

Lab: Simple Harmonic Motion

Purpose: 
Predict and verify the period of a semi-circle foam and a isosceles triangle foam oscillating around a certain pivot.

Equipment:
Photo-gate sensor and foam

Experiment:
1. Set up the apparatus as the pictures show.
2. Calculate the Inertia of the semi-circle around the pivot.
3. Get the expression of angular acceleration and omega.
4. Calculate the value of the period.
5. Hanging the semi-circle around the pivot, give a initial push so that the foam oscillate in a small angle. Record the period.
6. Repeat the step with a different pivot and for the triangle.

Calculation:
Center of mass:

 Moment of Inertia:

 Period at the center of the circle

 Period at the middle of the arc length

Triangle:

Result:

Theoretical and actual value:

 Conclusion:
 Theoretical Period at the center of the circle: 0.602s
Actual Period at the center of the circle: 0.5994s

Theoretical Period at the middle of the arc length: 0.59s
Actual Period at the middle of the arc length: 0.5991s

Theoretical Period of the triangle at the apex: 0.691s
Actual Period of the triangle at the apex: 0.69s

We successfully predict and verify the period of a semi-circle foam and a isosceles triangle foam oscillating around a certain pivot.

Lab: Spring-Mass Oscillation

Lab: Spring-Mass Oscillation

Purpose:
Find the relationship , between the period and the mass with the same spring and between the period and the spring constant with a constant mass.

Equipment:
Spring. various mass, motion sensor, ruler

Experiment:
1. Set up the apparatus as the pictures show.
2. Hang a 50g mass on the spring and measure the length of the spring L1; Hang a 100g mass on the spring and measure the length of the spring L2
3. Calculate the spring constant based on the mass and the difference of length.
4. Calculate the mass needed to reach the same total mass with other group.
5. Hang the calculated mass on the spring and stretch a little so that the mass oscillates, and record the period.
6. Write down the data from other group, and create a graph of spring constant vs. period with the same mass.
7. Hang different mass on the same spring and record the period for each oscillation.
8. Create a graph of mass vs. period.

Data: 

 Result:

Conclusion:
Graph of Period vs. Mass: y=2.686*x^0.4632
Graph of Period vs. Spring constant: y=2.570*x^-0.5702

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.