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Description

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

29

27

2

2 sec

2

44

42

2

3

1:06

1:04

2

Cylinder-2

Aluminum

1

1:36

34

2

2.33

2

1:47

1:45

2

3

2:00

1:57

3

Cylinder-3

Plastic

1

2:28

2:26

2

2.67

2

2:41

2:38

3

3

2:51

2:49

3

Cylinder-4

Brass

1

3:19

3:17

2

2.33

2

3:38

3:35

3

3

3:48

3:46

2

Solid Sphere-1

Plastic

1

5:21

5:19

3

2.67

2

5:31

5:29

2

3

5:41

5:38

3

Solid Sphere-2

Steel

1

5:57

5:55

2

2

2

6:20

6:18

2

3

6:30

6:28

2

Ring-1

Heavy Tape

1

7:15

7:12

3

3

2

7:26

7:23

3

3

7:37

7:34

3

Ring-2 Light Tape

1

17

14

3

2.5

2

39

37

2

3

OL-15
ROTATIONAL INERTIA BY ROLLING DOWN AN INCLINE.
5/21/2020
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:

Video-2:
Measurements of time for objects to roll down the incline:

EQUIPMENT:
Inclined Plane
Ã°Ââ€˜â€˜
Ã°ÂÂÂ´
Meter stick
Cylinders, spheres
Ã¢â€žÅ½
Ã°ÂÂÂµ
Stopwatch
Vernier Caliper
Figure 1 Purely rolling motion setup
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):
Ã°Ââ€˜Â£ÃŒâ€¦ =
Ã°Ââ€˜Â£Ã°Ââ€˜â€“ + Ã°Ââ€˜Â£Ã°Ââ€˜â€œ Ã°Ââ€˜Â£Ã°Ââ€˜â€œ
=
2
2
(1)
Experimentally we can get this average translational velocity by measuring the objectÃ¢â‚¬â„¢s travel time
(Ã°Ââ€˜Â¡) over a given distance (Ã°Ââ€˜â€˜):
Ã°Ââ€˜Â£ÃŒâ€¦ =
Ã°Ââ€˜â€˜
Ã°Ââ€˜Â¡
(2)
Now, an object with mass Ã°Ââ€˜â‚¬ and radius Ã°Ââ€˜â€¦ 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:
1
1
Ã°Ââ€˜â‚¬Ã°Ââ€˜â€Ã¢â€žÅ½ = Ã°Ââ€˜â‚¬Ã°Ââ€˜Â£Ã°Ââ€˜â€œ2 + Ã°ÂÂÂ¼Ã°ÂÅ“â€Ã°Ââ€˜â€œ2
2
2
(3)
For pure rolling motion, the relationship between translational velocity and rotational velocity is:
Ã°Ââ€˜Â£Ã°Ââ€˜â€œ = Ã°Ââ€˜â€¦Ã°ÂÅ“â€Ã°Ââ€˜â€œ
(4)
The moment of inertia for a cylinder or sphere is given below:
1
Ã°Ââ€˜â‚¬Ã°Ââ€˜â€¦2
(Cylinder)
2
2
Ã°ÂÂÂ¼ = Ã°Ââ€˜â‚¬Ã°Ââ€˜â€¦2 (Solid Sphere)
5
1
Ã°ÂÂÂ¼ = Ã°Ââ€˜â‚¬(Ã°Ââ€˜â€¦Ã°Ââ€˜â€“2 + Ã°Ââ€˜â€¦Ã°Ââ€˜Å“2 ) (Ring)
2
Ã°ÂÂÂ¼=
(5 Ã¢Ë†â€™ 1)
(5 Ã¢Ë†â€™ 2)
(5 Ã¢Ë†â€™ 3)
Using Equations 1 through 4, we can derive the following for the measured value of Rotational
Inertia:
Ã°ÂÂÂ¼=(
Ã°Ââ€˜â€Ã¢â€žÅ½Ã°Ââ€˜Â¡ 2
Ã¢Ë†â€™ 1) Ã°Ââ€˜â‚¬Ã°Ââ€˜â€¦2
2Ã°Ââ€˜â€˜ 2
(6)
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
g
Cm
g*cm^2
56g
1cm
66g
2cm
22g
2cm
206g
2cm
10g
2.3cm
66g
2.3cm
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
7
Ring-1: Heavy Tape
8
Ring-2: Light Tape
8.01
33.66
11.33
105.59
5.50
36.29
Inner
outer
158g
7.4cm,
10.6cm
58g
7.4cm
8.4cm
3307.69
891.31
Obtaining the Angle of Incline ÃŽÂ¸:
Average Height of plank on higher side: 6.13
Average Height of table on lower side: 1.955
Length of Table: 89.90311 cm
Calculated angle ÃŽÂ¸: 2.658842Ã‚Â°
Obtaining the Descending Height, h :
Distance traveled Ã°Ââ€˜â€˜ from A to B: 90 cm
Descending height Ã¢â€žÅ½: h = 6.13 – 1.955 = 4.175 cm
Time
Object
Travel
time
Final
time
Initial
time
1
29
27
2
2
44
42
2
3
1:06
1:04
2
1
1:36
34
2
2
1:47
1:45
2
3
2:00
1:57
3
1
2:28
2:26
2
2
2:41
2:38
3
3
2:51
2:49
3
1
3:19
3:17
2
2
3:38
3:35
3
3
3:48
3:46
2
1
5:21
5:19
3
2
5:31
5:29
2
3
5:41
5:38
3
1
5:57
5:55
2
2
6:20
6:18
2
3
6:30
6:28
2
1
7:15
7:12
3
2
7:26
7:23
3
3
7:37
7:34
3
1
17
14
3
2
39
37
2
No
t
Average
travel
time t
units
Cylinder-1
copper
Cylinder-2
Aluminum
Cylinder-3
Plastic
Cylinder-4
Brass
Solid
Sphere-1
Plastic
Solid
Sphere-2
Steel
Ring-1
Heavy
Tape
Ring-2 Light
Tape
3
2 sec
2.33
2.67
2.33
2.67
2
3
2.5
Rotational
Inertia
(equation
6)
Rotational
Inertia
(equation
5)
%
error
CALCULATIONS:
1.
2.
3.
4.
Calculate the average travel time for each object.
Input the average travel time into Equation 6, and calculate the objectÃ¢â‚¬â„¢s moment of inertia.
Use Equation 5 to calculate the objectÃ¢â‚¬â„¢s moment of inertia.
Compare the resultant values from steps 2 and 3, and calculate the % error.
QUESTION:
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?
Ã¢â‚¬Â¢ solid sphere
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?
Ã¢â‚¬Â¢
3. Based on the experimental data in the video, can you calculate the objectÃ¢â‚¬â„¢s translational
acceleration and angular acceleration?
Ã¢â‚¬Â¢ We can calculate the T.A and A.C if we have angle. If Angle =90 then its case of
free fall then the object like simple massive particle.
OL-15
ROTATIONAL INERTIA BY ROLLING DOWN AN INCLINE.
5/21/2020
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:

Video-2:
Measurements of time for objects to roll down the incline:

EQUIPMENT:
Inclined Plane
Ã°Ââ€˜â€˜
Ã°ÂÂÂ´
Meter stick
Cylinders, spheres
Ã¢â€žÅ½
Ã°ÂÂÂµ
Stopwatch
Vernier Caliper
Figure 1 Purely rolling motion setup
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):
Ã°Ââ€˜Â£ÃŒâ€¦ =
Ã°Ââ€˜Â£Ã°Ââ€˜â€“ + Ã°Ââ€˜Â£Ã°Ââ€˜â€œ Ã°Ââ€˜Â£Ã°Ââ€˜â€œ
=
2
2
(1)
Experimentally we can get this average translational velocity by measuring the objectÃ¢â‚¬â„¢s travel time
(Ã°Ââ€˜Â¡) over a given distance (Ã°Ââ€˜â€˜):
Ã°Ââ€˜Â£ÃŒâ€¦ =
Ã°Ââ€˜â€˜
Ã°Ââ€˜Â¡
(2)
Now, an object with mass Ã°Ââ€˜â‚¬ and radius Ã°Ââ€˜â€¦ 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:
1
1
Ã°Ââ€˜â‚¬Ã°Ââ€˜â€Ã¢â€žÅ½ = Ã°Ââ€˜â‚¬Ã°Ââ€˜Â£Ã°Ââ€˜â€œ2 + Ã°ÂÂÂ¼Ã°ÂÅ“â€Ã°Ââ€˜â€œ2
2
2
(3)
For pure rolling motion, the relationship between translational velocity and rotational velocity is:
Ã°Ââ€˜Â£Ã°Ââ€˜â€œ = Ã°Ââ€˜â€¦Ã°ÂÅ“â€Ã°Ââ€˜â€œ
(4)
The moment of inertia for a cylinder or sphere is given below:
1
Ã°Ââ€˜â‚¬Ã°Ââ€˜â€¦2
(Cylinder)
2
2
Ã°ÂÂÂ¼ = Ã°Ââ€˜â‚¬Ã°Ââ€˜â€¦2 (Solid Sphere)
5
1
Ã°ÂÂÂ¼ = Ã°Ââ€˜â‚¬(Ã°Ââ€˜â€¦Ã°Ââ€˜â€“2 + Ã°Ââ€˜â€¦Ã°Ââ€˜Å“2 ) (Ring)
2
Ã°ÂÂÂ¼=
(5 Ã¢Ë†â€™ 1)
(5 Ã¢Ë†â€™ 2)
(5 Ã¢Ë†â€™ 3)
Using Equations 1 through 4, we can derive the following for the measured value of Rotational
Inertia:
Ã°ÂÂÂ¼=(
Ã°Ââ€˜â€Ã¢â€žÅ½Ã°Ââ€˜Â¡ 2
Ã¢Ë†â€™ 1) Ã°Ââ€˜â‚¬Ã°Ââ€˜â€¦2
2Ã°Ââ€˜â€˜ 2
(6)
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
7
Ring-1: Heavy Tape
8
Ring-2: Light Tape
outer
Obtaining the Angle of Incline ÃŽÂ¸:
Average Height of plank on higher side: ___________
Average Height of table on lower side: ___________
Length of Table: ______________
Calculated angle ÃŽÂ¸: __________________
Obtaining the Descending Height, h :
Distance traveled Ã°Ââ€˜â€˜ from A to B: _____________
Descending height Ã¢â€žÅ½: _________________
Time
Object
No
units
1
Cylinder-1
copper
2
3
1
Cylinder-2
Aluminum
2
3
Cylinder-3
Plastic
Cylinder-4
Brass
Solid
Sphere-1
Plastic
Solid
Sphere-2
Steel
Ring-1
Heavy
Tape
Ring-2 Light
Tape
Final
time
Initial
time
Travel
time
t
Average
travel
time t
Rotational
Inertia
(equation
6)
Rotational
Inertia
(equation
5)
%
error
CALCULATIONS:
1.
2.
3.
4.
Calculate the average travel time for each object.
Input the average travel time into Equation 6, and calculate the objectÃ¢â‚¬â„¢s moment of inertia.
Use Equation 5 to calculate the objectÃ¢â‚¬â„¢s moment of inertia.
Compare the resultant values from steps 2 and 3, and calculate the % error.
QUESTION:
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 objectÃ¢â‚¬â„¢s translational
acceleration and angular acceleration?
No.
DATE
I I = ight?
L2d2 -1] MR?
– fletom
-1.360.056 kg 1.05ml
equation cei –
Copper: solution
f (9.8 m/570.06m (2 sec)?
2 (0.9m)?
kg
Equation 5:
copper Ceylinder)
Solutim :
I: 360.056 kg)(0.0005m
I= 78 109 kg.
I = – – MRÃ‚Â²
.m
om?
% error :
Value from Equation 6 Ã¢â‚¬Â¢ Value from Eq. 5
Value from Eq. 5
ko. 83 870-9-47410434
** 100%
7×10-9
12
9.57%
>
lo error =
CAFTLE JOTE
=Ã¬ÂÂ¼
Experimentally we can get this average translational velocity by measuring the object’s travel time
(t) over a given distance (d):
d
Ã…Â«=
(2)
t
Now, an object with mass M and radius R 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:
Mgh = Ã¢â‚¬Å¾Mv} + {w}
(3)
For pure rolling motion, the relationship between translational velocity and rotational velocity is:
Vj = RWF
(4)
The moment of inertia for a cylinder or sphere is given below:
1
1 = = MR2 (Cylinder) (5 – 1)
2
2
I = MR2 (Solid Sphere)
5
(5-2)
= 3M(R? + R3) (Ring)
(5-3)
Using Equations 1 through 4, we can derive the following for the measured value of Rotational
Inertia:
ght?
I =
— 1)MR?
(6)
2d2