Description

second table. please provide the answeres

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°Ââ€˜â€˜

Ã°ÂÂÂ´

Meter stick

Cylinders, spheres

Ã¢â€žÅ½

Ã°ÂÂÂµ

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°Ââ€˜Â£ÃŒâ€¦ =

Ã°Ââ€˜Â£Ã°Ââ€˜â€“ + Ã°Ââ€˜Â£Ã°Ââ€˜â€œ Ã°Ââ€˜Â£Ã°Ââ€˜â€œ

=

2

2

(1)

Experimentally we can get this average translational velocity by measuring the objectÃ¢â‚¬â„¢s travel time

(Ã°Ââ€˜Â¡) over a given distance (Ã°Ââ€˜â€˜):

Ã°Ââ€˜Â£ÃŒâ€¦ =

Ã°Ââ€˜â€˜

Ã°Ââ€˜Â¡

(2)

Now, an object with mass Ã°Ââ€˜â‚¬ and radius Ã°Ââ€˜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°Ââ€˜â‚¬Ã°Ââ€˜â€Ã¢â€žÅ½ = Ã°Ââ€˜â‚¬Ã°Ââ€˜Â£Ã°Ââ€˜â€œ2 + Ã°ÂÂÂ¼Ã°ÂÅ“â€Ã°Ââ€˜â€œ2

2

2

(3)

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

Ã°Ââ€˜Â£Ã°Ââ€˜â€œ = Ã°Ââ€˜â€¦Ã°ÂÅ“â€Ã°Ââ€˜â€œ

(4)

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

1

Ã°Ââ€˜â‚¬Ã°Ââ€˜â€¦2

(Cylinder)

2

2

Ã°ÂÂÂ¼ = Ã°Ââ€˜â‚¬Ã°Ââ€˜â€¦2 (Solid Sphere)

5

1

Ã°ÂÂÂ¼ = Ã°Ââ€˜â‚¬(Ã°Ââ€˜â€¦Ã°Ââ€˜â€“2 + Ã°Ââ€˜â€¦Ã°Ââ€˜Å“2 ) (Ring)

2

Ã°ÂÂÂ¼=

(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°ÂÂÂ¼=(

Ã°Ââ€˜â€Ã¢â€žÅ½Ã°Ââ€˜Â¡ 2

Ã¢Ë†â€™ 1) Ã°Ââ€˜â‚¬Ã°Ââ€˜â€¦2

2Ã°Ââ€˜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°Ââ€˜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°Ââ€˜â€˜

Ã°ÂÂÂ´

Meter stick

Cylinders, spheres

Ã¢â€žÅ½

Ã°ÂÂÂµ

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°Ââ€˜Â£ÃŒâ€¦ =

Ã°Ââ€˜Â£Ã°Ââ€˜â€“ + Ã°Ââ€˜Â£Ã°Ââ€˜â€œ Ã°Ââ€˜Â£Ã°Ââ€˜â€œ

=

2

2

(1)

Experimentally we can get this average translational velocity by measuring the objectÃ¢â‚¬â„¢s travel time

(Ã°Ââ€˜Â¡) over a given distance (Ã°Ââ€˜â€˜):

Ã°Ââ€˜Â£ÃŒâ€¦ =

Ã°Ââ€˜â€˜

Ã°Ââ€˜Â¡

(2)

Now, an object with mass Ã°Ââ€˜â‚¬ and radius Ã°Ââ€˜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°Ââ€˜â‚¬Ã°Ââ€˜â€Ã¢â€žÅ½ = Ã°Ââ€˜â‚¬Ã°Ââ€˜Â£Ã°Ââ€˜â€œ2 + Ã°ÂÂÂ¼Ã°ÂÅ“â€Ã°Ââ€˜â€œ2

2

2

(3)

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

Ã°Ââ€˜Â£Ã°Ââ€˜â€œ = Ã°Ââ€˜â€¦Ã°ÂÅ“â€Ã°Ââ€˜â€œ

(4)

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

1

Ã°Ââ€˜â‚¬Ã°Ââ€˜â€¦2

(Cylinder)

2

2

Ã°ÂÂÂ¼ = Ã°Ââ€˜â‚¬Ã°Ââ€˜â€¦2 (Solid Sphere)

5

1

Ã°ÂÂÂ¼ = Ã°Ââ€˜â‚¬(Ã°Ââ€˜â€¦Ã°Ââ€˜â€“2 + Ã°Ââ€˜â€¦Ã°Ââ€˜Å“2 ) (Ring)

2

Ã°ÂÂÂ¼=

(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°ÂÂÂ¼=(

Ã°Ââ€˜â€Ã¢â€žÅ½Ã°Ââ€˜Â¡ 2

Ã¢Ë†â€™ 1) Ã°Ââ€˜â‚¬Ã°Ââ€˜â€¦2

2Ã°Ââ€˜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°Ââ€˜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¬ÂÂ¼

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

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