11 rotation

The rotational variables

Relating the linear and angular variables

Kinetic energy of rotation and rotational inertia

Torque, Newton’s second law for rotation

Work and rotational kinetic energy

11-1,--11-4 the rotational variables:

(1) Translation and rotation:

In pure translation（平动） , every point of the body moves in a straight line , and every point moves through the same linear distance during a particular time interval.

In rotation about the fixed axis（定轴转动） , every point moves through the same angle during a particular time interval.

Rigid body（刚体） : is a body (that can rotate) with all its parts locked together and without any change in its shape.

(2) The rotation of a rigid body about a fixed axis:

Angular position: to describe the rotation of a rigid body about a fixed axis, called the rotation axis,we assume a reference line is fixed in the body, perpendicular to that axis and rotating with the body. We measure the angular position of this line relative to a fixed direction (x).

r

s

Unit of : radian( rad) 弧度Always: =(t)

o

sr

x

s

Angular displacement: a body that rotates about a rotation axis, changing its angular position from 1 to 2 , undergoes an angular displacement:

12

Where is positive for counterclockwise rotation and negative for clockwise rotation.

Angular velocity and speed:

tavg

dt

d

Average angular velocity:

(Instantaneous) angular velocity:

avg and are vectors, with direction

s given by the right-hand rule. They are positive for counter-clockwise rotation and negative for clockwise rotation.

The magnitude of the body’s angular velocity is the angular speed.

Angular acceleration:

Average angular acceleration:

(Instantaneous) angular acceleration:

tavg

dt

d

Both avg and are vectors.

When is positive:

>0, the direction is the same as ;

<0, the direction is reversed to .

(3) Rotation with constant angular acceleration

const.

)(

)(

020

2

221

0

0

2

tt

t

See page 221: Table 11-1 equations of motion for constant linear acceleration and for constant angular acceleration

v

θθR

x

ΔS

0ω,

Δ

dt

ds

t

sv

t

0

lim

(1) Circular motion: description by linear variables

Speed:

Distance: S=r or s=r

11-5 relating the linear and angular variables

O X

R

v t( )

v t t( )

v t( )

v t t( )

v

tv

nv

切向 t 内法向 n

nata

nt

vt

t

vt

va

nt

nt

ˆˆ

ˆˆ

r

v

AB

vn

AB

)()( tvttvv t

v t( )

v t t( )

v

tv

nv r

ABvvn

r

v

t

AB

r

v

t

va

t

n

tn

2

00 limlim

dt

dv

t

va t

tt

0

lim

Radial acceleration:

Tangential acceleration:

rs rrdt

d

dt

dsv

rrdt

d

dt

dvat

rr

van

22

(2) Relating the linear and angular variables

Home work: 5E, 29p

11-6, 11-7 kinetic energy of rotation and rotational inertia

2

2

1 IK

(1)rotational kinetic energy:

Proof: treat the rotation rigid body as a collection of particles with different speeds. We can then add up the kinetic energies of all the particles to find the kinetic energy of the body as a whole. In this way we obtain the kinetic energy of a rotating body,

(Compare with kinetic energy of particle)

2222 )(2

1)(

2

1

2

1 iiiiii rmrmvmK

Let: 2iirmI

2

2

1 IK Then:

I is rotational inertia (or moment of inertia)(转动惯量）

O

ri

z

mi

vi

(2) Rotational inertia :

dmrI 2

2iirmI

For rigid body with continuously distributed mass:

Where r and ri represent the perpendicular distance from the axis of rotation to each mass element in the body.

dm

rM

For particle: I = m r 2

See page 227 table 11-2 some rotational inertias (calculate (e):

1. thin rod about axis through center perpendicular to length;

2. About axis through the end of the rod perpendicular to length.)

Sample problem:11-5

Unit: kg.m2

(3) Parallel-Axis Theorem:

2MhII com

Where I is the rotational inertia of a body of mass M about a given axis.

I com is the rotational inertia of the body about a parallel axis that extends through the body’s center of mass.

h is the perpendicular distance between the given axis and the axis through the center of mass.

M is the mass of rigid body.

C hM

ICom I

平行

Proof of the parallel-axis theorem: (by yourself)

Question: What does the rotational inertia relate to?

Answer: Mass and its distribution.

11-8 torque

Fr

trFrF sin O ×

z

F

P

r

Unit: N.m

Where Ft is the component of perpendicular to , and is the angle between and .

Fr

Fr

Torque is a turning or twisting action on a body about a rotation axis due to a force.

(1)Torque is a vector for rotation about fixed axis, its direction always along the axis, either positive or negative. (see page 230)

(2) If several torque act on a rigid body that rotate about a fixed axis, the net torque is the sum of individual torque.

(3) The net torque of internal forces is zero.

Caution:

11-9 Newton’s second law for rotation (转动定律）

Compare with: amF

We obtain:

For rotation about fixed axis: Inet

Inet

Proof of equation:

f i

Treat the rigid body as a collection of particles, F i and f i are the external and internal forces of mass element mi ,thus:

amfF iii

Then:tiitit amfF

O

ω,α

ri

z

F i

miΔ

v

2iiitiiitiit rmramrfrF

For whole rigid body:

2iiiitiit rmrfrF

Ii 0

That is: Inet Inet

Sample problem: 11-7, 11-8

11-10 Work and rotational kinetic energy

From: rdFdW

For rigid body:

ddrFdsFdW tt

From initial angular position to final angular position, the work is:

dW 0

(1) work:

Caution:

(1) If torque is constant, then:

(2) When several torques act on the rigid body, the net work is sum of individual work.

(3) This work is scalar also.

(2) Work-kinetic energy theorem:

kKK

IIdI

ddt

dIdIdW

if

if

f

i

22

2

1

2

1

W

(4) The gravitational potential energy of rigid body:

cMghU

(1) U can not change when the rigid body rotate about axis that extends through the body’s center of mass.

(2) The conservation of Mechanical energy and conservation of energy can also be used for rigid body.

(3) power:

dt

d

dt

dWP

Sample problem: 11-9, 11-10

Home work: 29p, 65p, 66p