Torque & Rotational Inertia

Lecturer: Professor Stephen T. Thornton

Reading QuizReading QuizIn which of the cases shown below

is the torque provided by the

applied force about the rotation

axis biggest? For all cases the

magnitude of the applied force is

the same.

A) F1

B) F3

C) F4

D) all of them

E) none of them

Reading QuizReading Quiz

A) F1

B) F3

C) F4

D) all of them

E) none of them

In which of the cases shown below

is the torque provided by the

applied force about the rotation

axis biggest? For all cases the

magnitude of the applied force is

the same.

The torque is == F d sin F d sin

and so the force that is at 90°90°

to the lever arm is the one that

will have the largest torquelargest torque.

Clearly, to close the door, you

want to push perpendicularlyperpendicularly!!

Last TimeBegan angular motion

Angular position, displacement

Angular speed, velocity

Angular acceleration

Similarities between translation and rotation

Today

Torque

Rotational inertia (moment of inertia)

Rotational kinetic energy

What is torque?

We recognize there is a relationship between tangential force and making something rotate.

first, simple definitionrF

Only the Tangential Component of a Force Causes a Torque

is angle between and r F

The Moment Arm

sin

r

F

F

F

r

r

moment arm

Sign convention for torque according to most textbooks:

> 0 if the torque causes a CCW acceleration.

< 0 if the torque causes a CW acceleration.

Conceptual Quiz:You are using a wrench to loosen a rusty nut. Which of the arrangements below is least effective in loosening the nut? Force is proportional to length of vector.

A.

B.

C.

D.

E. not possible to determine

A

C

B

D

Answer: C

The force vectors are all the same. The arrangement that is the least effective is the one with the shortest moment arm. That is C.

Conceptual Quiz:A mechanic is finding it very difficult to muster enough torque to twist a stubborn bolt with a wrench, and she wishes she had a length of pipe to place over the wrench handle to increase her leverage. Will torque be increased if the mechanic pulls just as hard on a length of rope tied to the wrench handle?

 

A) Yes B) No C) Only in space. D) Not enough information given.

Answer: B (no)The rope placed in this position neither increases the force or the moment arm (length of application of the force causing the torque).

Angular Quantities

If the angular velocity of a rotating object changes, it has a tangential acceleration:

Even if the angular velocity is constant, each point on the object has a centripetal acceleration:

tana Ra=2

cp Ra a Rw= =

tan

dv da R R

dt dt

wa= = =

( )222

R

Rva R

R R

ww= = =

Torque and Angular Acceleration

Torque and angular acceleration

22

/ Newton's 2nd law

(last time )

multiply by ( / )

where and is

called the rotational inertia (or moment of inertia)

Newton's 2nd law for rotatio

t

a F m

a Fa r

r mrr r

r F rFI mr I

r mr mr I

I

n

Linear and angular quantities

Linear Angular

m I

a F

Similarities between linear and angular motion quantities ***

x

v

a

0 0

0 0 0 0

2 20 0 0 0

2 2 2 20 0 0 0

1 1( ) ( )

2 21 1

2 2

2 ( ) 2 ( )

v v at t

x x v v t t

x x v t at t t

v v a x x

Look at system of particles2

2 2 2 21 1 2 2 3 3

for a fixed axis

if ...

then for a fixed axis

i i ii i

i ii

m r

I m r m r m r m r

I

Kinetic Energy of a Rotating Object

massless rod

2 2

2 2

2

1 1( )

2 21

2

1 is the

2

rotational energy

K mv m r

K mr

K I

I is called rotational inertia

Kinetic Energy of a Rotating Object of Arbitrary Shape

21

2 i ii

K m v

Rotational InertiaMoment of Inertia

Rotational kinetic energy2 2 2

2 2

2

1 1

2 2

1

2

where

212

i i i ii i

i ii

i ii

K m v m r

K m r

I m r

I K

I appears to be quite useful!!

The Rotational Inertia (Moment of Inertia) of a Hoop

M

2I MR

The Rotational Inertia (Moment of Inertia) of a Disk

21

2I MR

This is almost certainly an example in textbook.

Use calculus to find this value.

2I R dm=ò

Rotational Dynamics; Torque and Rotational Inertia

The quantity is called the rotational

inertia of an object.

The distribution of mass matters here—these two objects have the same mass, but the one on the left has a greater rotational inertia, as so much of its mass is far from the axis of rotation.

2i iI m R

Rotational Inertia for Uniform, Rigid Objects of Various Shapes and Total Mass M

Do not memorize!!

Rotational Inertia for Uniform, Rigid Objects of Various Shapes and Total Mass M

Demos:Rotational inertia rodsMoment of Inertia wheel

2

2

1

2

where i i

K I

I m r

If a physical object is available, the rotational inertia (moment of inertia) can be measured experimentally.

Otherwise, if the object can be considered to be a continuous distribution of mass, the rotational inertia may be calculated:

2I R dm=ò

The parallel-axis theorem gives the rotational inertia about any axis parallel to an axis that goes through the center of mass of an object:

2CMI I Mh= +

ICMI

Falling Rod. A thin rod of length stands vertically on a table. The rod begins to fall, but its lower end does not slide. (a) Determine the angular velocity of the rod as a function of the angle it makes with the tabletop. (b) What is the speed of the tip of the rod just before it strikes the table?

Conceptual Quiz:A figure skater spins around with her arms extended. When she pulls in her arms, her rotational inertia 

A) increases. 

B) decreases. 

C) stays the same.

Answer: B, decreases

The mass stays the same, but the radius decreases for the mass in her arms. The I must decrease.

Conceptual Quiz A) A) solid

aluminum

B) hollow goldB) hollow gold

C) sameC) same

same mass & radius

solid hollow

Two spheres have the same radius and equal masses. One is made of solid aluminum, and the other is made from a hollow shell of gold. Which one has the bigger rotational inertia about an axis through its center?

Conceptual Quiz

same mass & radius

solid hollow

Two spheres have the same radius and equal masses. One is made of solid aluminum, and the other is made from a hollow shell of gold. Which one has the bigger rotational inertia about an axis through its center?

Rotational inertia depends on mass and distance from axis squared. It is bigger for the shell because its mass is located farther from the center.

A) A) solid aluminum

B) hollow goldB) hollow gold

C) sameC) same

Conceptual Quiz:Two wheels with fixed hubs, each having a mass of 1 kg, start from rest, and forces are applied as shown. Assume the hubs and spokes are massless, so that the rotational inertia is I = mR2. In order to impart identical angular accelerations, how large must F2 be?

A) 0.25 N B) 0.5 N C) 1.0 N D) 2.0 N E) 4.0 N

2

Fr

I mr

Answer: D

The hint on the figure should help. You want Fr/I to be the same ratio. Fr/mr2 = F/mr, so F/r must have the same ratio = 2.