## Rotational Motion

Rotational Motion;

Introduction

In this lab a constant torque will create a constant angular acceleration for a rigid body rotating

about its center of m

ass. You will see that the moment of inertia depends on the rotation axis for

a given object and finally you will look at ro

tational motion from an energy viewpoint.

Applications of the moment of inertia concept include the design of crankshafts, rotary rides at a

fair, or the baton twirling of a marching band

leader. Text Reference: Young & Freedman § 9.1-

5.

Theory

A stone dropped into a well falls a distance

2

2

1

tgy

?=?

. If we convert this equation to the

angular dom

ain by considering a disk turning about some axis it says

2

2

1

t

?=?

a?

.

In more detail, a rigid object free to rotate about a principle axis with moment of inertia

I

will

accelerate according to

where

t

is the net torque (given by the sum of

Fr

×

for each force applied to the object), and

a

is the angular acceleration in rad/sec

2

. Recall

p

radian = 180

°

. Take care that

t

,

a

, and

I

are in

consistent units. The moment of inertia

I

is readily calculated for symmetric objects, and has the

form

I

=

ß

MR

2

, where

ß

is a dimensionless fraction between 0 and 1 that depends on the shape of

the object and the rotation axis. A 3-dimensional object has three principal values of

I

,

corresponding to three independent possible axes of rotation. These values would all be the

same for a sphere through its center, but all three values would be different for a low symmetry

object. Thus, for example, a rectangular block of dimensions

a

×

b

×

c

has moment of inertia

for rotation about the axis normal to the

a

×

b

face and passing through its center.

In this lab, we use the arrangement shown in figure 1. A rectangular block of mass

M

and

dimensions

a

×

b

×

c

is mounted with a rotation axis through the center of one face. A mass

m

is

tied to a string wound around a massless pulley of diameter 2

r

mounted on the same axis. The

string exerts a torque on

M

given by

where

T

is tension in the string which can be found using Newton’s second law and is

Note we have used

a

=

a

r

, and both

g

and

a

are positive. Combining these equations we have

(

)

1

a

t

I

=

(

)

()

2

12

1

22

baMI

+

=

(

)

3

rT

=

t

(

)

(

)

(

)

4

.

rgmagmT

a

–

=

–

=

where we have written

I

d

(dynamic moment of inertia) to distinguish it from the static moment of

inertia

I

s

determined from the geometry and mass, as in equation 2. These are entirely equivalent

in principle.

Figure 1. Apparatus for the dynamic measurement of rotational moment of inertia.

It is also useful to look at this problem

from an energy viewpoint. Thus we have

and for our specific problem this can be written as

where

?

and

?

are the angular velocity and position of the pulley. The terms in

m

and

?

are due

to the kinetic energy of the small mass as it falls. To get the signs right, note that the change in

height

?

h

is negative, and the change in potential energy is negative. Note that the kinetic

energy includes both the spinning block (

M

) and the falling mass (

m

), while the potential energy

involves only the falling block, and all terms can be written in terms of pulley angle, as above.

Think about the signs of the two terms in equation 7 needed to keep the sum zero.

Procedure

()

5

?

?

?

?

?

?

–

=

a

a

rg

mrI

d

(

)

6

0

=

?

+

?

UK

()

(

)

(

)

()

7

0

2

1

222

=+-+-

+

if

if

s

mgr

mrI

??

??

2r

m

M

rotation axis

4. Research Statement will be attached. It is actually my paper in process. You will have a better idea how the format will be.

LAB REPORT GUIDLINESS

Lab Report is written from the third person; in the passive form, in the past tense. Signed data sheets

need to be stapled to the back of the report. All graphs must be labeled.

Lab report without the signed data sheets attached will automatically get a zero score.

Lab Report includes the following parts, each of them needs to be clearly marked in the lab report:

Title of the experiment; Student’s name (prominent); lab partners’ names; Group Number (attached to the lab

computers); TA’s name; Section number (SLN number from ASU catalog). (1 point)

Objective section – (3 points) It is a statement of what physics concepts/theory/law was investigated or tested.

What physics quantities had to be measured? It is one or two sentences.

Experimental Data (3 points)– raw data collected during the lab experiment. All experimental data should be

tabulated. Tables should have appropriate headings and contain units. All plots should be clearly labeled and scales

adjusted to present graph appropriately.

Data analysis section – (10 points)

The section where the experimental data used to calculate the physics quantity that needs to be found to achieve

the objective. All necessary calculations have to be present, which also includes all equations. All intermediate

quantities are also provided. All the work for calculating uncertainties of the experiment is also present.

Microsoft equation is one of the tools that can be used for typing the equation. The equation can either be typed

or it will also be acceptable to neatly hand write these calculations.

Results section – (3 points)

Results must be tabulated. Results include the physics quantities that were stated to be found in the objective. The

results should be organized in clearly labeled tables along with their errors if applicable. Results and uncertainties

have to be reported with the correct number of significant figures and appropriate unit.

The raw experimental data collected during the lab and signed is not what needs to be reported in the result

section.

Discussion and Conclusion section – (10 points)

This is the most important part of the lab report. Begin discussion with the purpose of the experiment. Briefly

explain the physics theory/concept that was tested. State only the key results (with uncertainty and units)

quantitatively with numerical values; does not provide intermediate quantities.

Briefly explain the procedure that was used to collect the experimental data. Describe difficulties and

shortcomings that were encountered during the experiment and it caused the uncertainties in the final results.

Discuss “statistical errors” that affect your measurements, but which you can’t do anything about given the time

and equipment constraints of this laboratory. Suggest experimental redesigns to eliminate and overcome these

troubles. Include a description of sources of “systematic errors” in your measurement that bias your result (e.g.

friction in pulleys that are assumed frictionless in the formula). Describe the qualitative effect of each source of

error (e.g. friction slowed motion, causing a smaller value of acceleration to be measured). Describe only the 5

prominent sources of error in the experiment.

Address questions such as: Are the deviations in the experimental results due to uncertainty in the

experimental method, or are they due to idealizations inherent in the theory (or both)? If the deviations are due to

experimental uncertainties, can you think of ways to decrease the amount of uncertainty? Discuss how the

apparatus could be redesigned to reduce uncertainties or to streamline experimental technique. If the deviations

are due to idealizations in the theory, what factors has the theory neglected to consider?

All questions from the Lab Manual should be also answered in the discussion section. The conclusion

made if the objective of the experiment achieved.

Overall the full lab report is expected to be about 4-5 pages.

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