Rotational inertia by rolling down an incline.

 

OBJECTIVE:

To determine the rotational inertia (or moment of inertia) of various cylinders and spheres; comparing with calculated values.

READ THROUGH THIS MANUAL TO UNDERSTAND HOW TO PERFORM THE EXPERIMENT. AFTERWARD, VIEW THE VIDEO OF THE EXPERMENTS BEING PERFORMED AND RECORD THE DATA FROM THE VIDEO. USE THIS DATA TO CALCULATE THE MOMENT OF INERTIA AND WRITE YOUR REPORT.

Video-1:

Measurements of diameters, masses and angle of incline:     https://youtu.be/UlyfoLZEDiY

Video-2:

Measurements of time for objects to roll down the incline:     https://youtu.be/KwKOfROqqqo

 

 

EQUIPMENT:

Inclined Plane

Meter stick

Cylinders, spheres

Stopwatch

Vernier Caliper

Triple beam balance

THEORY:

An object (e.g. a cylinder or sphere) that rolls down an incline from rest without sliding will uniformly speed up. The translational speed of the rolling object is smaller than the speed of the same object sliding down the incline without friction. This is due to the fact that the initial gravitational potential energy is converted not only into translational kinetic energy in the first case, but also rotational kinetic energy.

Since the object rolling down the incline with a constant acceleration, the average translational velocity should equal half its final translational velocity (for an object starting at rest):

Experimentally we can get this average translational velocity by measuring the object’s travel time () over a given distance ():

Now, an object with mass  and radius  rolls down a vertical height from rest as shown in Figure 1. Because static friction between the table and the rolling object does not do work, the initial gravitational potential energy of the rolling object at location A should equal to the sum of its translation kinetic energy and rotational kinetic energy at location B:

For pure rolling motion, the relationship between translational velocity and rotational velocity is:

The moment of inertia for a cylinder or sphere is given below:

 

Using Equations 1 through 4, we can derive the following for the measured value of Rotational Inertia:

Procedure:

  1. Watch the videos in the given link.
  2. From Video-1, record the masses and diameters of the cylinder and solid sphere. The diameters of the rings are taken several times since they are relatively flexible objects. Take their average values.
  3. From Video-1, record the length of the inclined plane and the height at two end points to obtain the slope. The heights are measured several times. Take their average value.
  4. From Video-2, for each rolling object, record the travel time from A to B and fill them into the data table. Note that some rolling motions result in the object colliding with the side, or falling off the incline. Do not use that data.

Data:

Mass and Diameters:

NO OBJECT MASS DIAMETER ROTATIONAL INERTIA FROM EQN. 5
UNITS
1 Cylinder-1: Copper
2 Cylinder-2: Aluminum
3 Cylinder-3: Plastic
4 Cylinder-4: Brass
5 Solid Sphere-1: Plastic
6 Solid Sphere-2: Steel
Inner outer
7 Ring-1: Heavy Tape
8 Ring-2: Light Tape

 

Obtaining the Angle of Incline θ:

Average Height of plank on higher side: ___________

Average Height of table on lower side: ___________

Length of Table: ______________                           Calculated angle θ: __________________

 

Obtaining the Descending Height, h :

Distance traveled  from A to B: _____________         Descending height: _________________

 

Object Time Rotational Inertia

(equation 6)

 Rotational Inertia

(equation 5)

 % error
No Final time Initial time Travel time      t Average travel time t
units
Cylinder-1 copper 1
2
3
Cylinder-2

Aluminum

 1
2
3
Cylinder-3

Plastic
Cylinder-4

Brass
Solid Sphere-1

Plastic
Solid Sphere-2

Steel
Ring-1

Heavy Tape
Ring-2 Light Tape

Calculations:

  1. Calculate the average travel time for each object.
  2. Input the average travel time into Equation 6, and calculate the object’s moment of inertia.
  3. Use Equation 5 to calculate the object’s moment of inertia.
  4. Compare the resultant values from steps 2 and 3, and calculate the % error.

Questions:

  1. Assuming that a person releases both a solid cylinder and solid sphere (with the same mass and radius) from rest at location A, which object do you expect will reach location B first?
  2. If this experiment does not give you mass of the rolling object (solid cylinder/sphere), are you still able to calculate the % error of the moment inertia?
  3. Based on the experimental data in the video, can you calculate the object’s translational acceleration and angular acceleration?