Tài liệu Chapter 10: Rotation docx

Chapter 10 Rotation In this chapter we will study the rotational motion of rigid bodies about a fixed axis.. To describe this type of motion we will introduce the following new concepts

Chapter 10

Rotation

In this chapter we will study the rotational motion of rigid bodies

about a fixed axis To describe this type of motion we will introduce the following new concepts:

-Angular displacement -Average and instantaneous angular velocity (symbol: ω ) -Average and instantaneous angular acceleration

(symbol: α ) -Rotational inertia also known as moment of inertia (symbol I ) -Torque (symbol τ )

We will also calculate the kinetic energy associated with rotation,

write Newton’s second law for rotational motion, and introduce the work-kinetic energy for rotational motion

(10-1)

The Rotational Variables

In this chapter we will study the rotational motion of rigid

bodies about fixed axes A rigid body is defined as one

that can rotate with all its parts locked together and without

any change of its shape A fixed axis means that the object

rotates about an axis that does not move We can describe the motion of a rigid body rotating about a fixed axis by

specifying just one parameter Consider the rigid body of

the figure

We take the the z-axis to be the fixed axis of rotation We define a reference

line which is fixed in the rigid body and is perpendicular to the rotational axis

A top view is shown in the lower picture The angular position of the reference

line at any time t is defined by the angle θ(t)t) that the reference lines makes with the position at t = 0 The angle θ(t)t) also defines the position of all the points on the rigid body because all the points are locked as they rotate The angle θ is related to the arc length s traveled by a point at a distance r from the axis via

the equation: Note: The angle θ is measured in radians s

r

 

(10-2)

2 1

In the picture we show the reference line at a time and

at a later time Between and the body undergoes

an angular displacement

An

gular Displace

All the points of th

ment

e rigid

t

body have the same angular displacement because they rotate locked together

 1 

2 1

2 1

2

We define as the instantaneous ang

The SI unit for angular velocity is

An

rad

gular Velocity

ians/second

avg

t t

   

 

0

Algerbraic sign of angular f

tive with

If a rigid body rot

t

t t

t t







 



 









) has a positive sign If on the other hand the rotation is clockwise (CW) has a negative sign





(10-3)

t 1

t 2

d dt



 

If the angular velocity of a rotating rigid object changes with time we can describe the time rate of change of

by defining the

Angula

angula

r Accelerati

r acelera

on

tion



In the figure we show the reference line at a time and at a later time

We define as average angular acceleration fo

 1 2

0

2 1 2

2

1

The SI unit for angular velocity is radia

We define as the instantaneous angular acceleration the limit of

ns/sec

as lim

ond

0

av

t

g

t t

t

t t

t

t









 



 

 















t



(10-4)

ω 1

ω 2

t 1

t 2

d dt



 

For rotations of rigid bodies about a fixed axis

we can describe accurately the angular velocity

by asigning an algebraic sigh Positive for counterclockwi

Angul

se ro

ar Velocity Ve

tation and ne

ctor

gative for clockwise rotation

We can actually use the vector notation to describe rotational motion which

is more complicated The angular velocity vector is defined as follows:

The directi of is along the rotation axis

h

n

T

Curl the right hand so that the fingers point in the direction

of the rotation The thumb of the right

sense

R

hand gi

ight hand rule

ves the sense

:

f

o







(10-5)

When the angular acceleration is constant we can derive simple e

Rotation with Constant Angular Accelerati

xpressions that give us the angular velocity and the angular p

on

osit



We could derive these equations in the same way we did in chapter 2 Instead we will simply write the solutions by exploiting the analogy between translational

and rotational m



Rotatio

otion u

nal Mot

sing the follo

ion

wing correspondance between

Tran

the two motions

slational Motion

x v







0

2

0

2

(eqs.2)

2

(

2

o

a

at

t t

a x

t





  



  

    

 

  

 









Consider a point P on a rigid body rotating about

the origin

Relating the Linear and A

O with point P is on the

ngular Var

x-axis (po

iable

int

s

A) D

t 

 uring the time interval point P moves along arc AP and covers a distance At the same time the reference line OP rotates by an angle

t s



A

θ s

O

 

The arc length s and the angle are connected by the equation:

where is the dista

Relation b

nce OP T

etween angular velocity a

he speed of point P

nd speed

dt dt dt

s r





v r  

The period of revolution is given by:

speed

2





(10-7)

1

T

f

r O

The acceleration of point P is a vector that has two components A "radial" componet along the radius and pointing towards point O We have enountered this component i

The Accele

n chapter

ration

4 where we called it

"centripetal" acceleration Its magnitude is:

2

2

r

v

 

The second component is along the tangent to the circular path of P and is thus known as the "tangential" component Its magnitude is:

The magnitude of the acceleration vector is

t

d r

dt dt dt

 



2 2

t

a  r 

(10-8)

O r i

i

v

m i

1 2 3

Consider the rotating rigid body shown in the figure

The part (or "element") at P

Kinetic Energy of Rotatio

has an index and mass

n

Th

i i

m m m m

e kinetic energy of rotation is the sum if the kinetic

K  m v  m v  m v 

 2 2

The speed of the -th element

The term

rotational i

is known as

nertia moment of inertia of rotation The axis

of

its mass, its shape as well as on the position of the rotation axis The rotational inertia of an object describe

mu

st

ho

I

the mass is distributed about the rotation axis

2

i i i

2

(10-9)

2

I r dm

In the table below we list the rotational inertias for some rigid bodies

2

I   r dm

(10-10)

has a discreet disstribution of mass Fo

Calculating the Rotational In

r a continuous distribution of mass the

a

s

erti

i i i

I m r

2

um

We saw earlier that depends on the position of the rotation axis For a new axis we must recalculate the integral for A simpler method takes advantage

Parall

of the

el-Axis Theorem

parallel-axis the

I

I

Consider the rigid body of mass M shown in the figure

We assume that we know the rotational inertia about a rotation axis that passes through the center

of mass O and is perpendicul

ore

o

m

ar t t

com

I

he page

The rotational inertia about an axis parallel to the axis through O that passes

through point P, a distance h from O is given by the equation:

I

2

com

A We take the origin O

to coincide with the center of mass of the rigid body shown

in the figure We assume that we know the ro

Proof of the Parallel-Axis Theo

tational inertia for a

rem

n axis tha

com

through O

 

 

We wish to calculate the rotational ineria about a new axis perpendicular

to the page and passes through point P with coodrinates , Consider

I

a b

2

The distance between points A and P is:

Rotational Inertia about P:

and third integrals are zero The first i

r

I r dm x a y b dm

I x y dm a xdm b ydm a b dm

Thus the fourth integral is equal to

com

h dm Mh

In fig.a we show a body which can rotate about an axis through point O under the action of a force applied at point P a distance from O In fig.b we resolve into two compone

Torq

ts, a

ue

r dial

F





and tangential The radial component cannot cause any rotation because it acts along a line that passes through O The tangential

ob

r

t

F

ject about O The ability of to rotate the body depends on the magnitude and also on the distance between points P and A

The distance is known a

n

t

t

F

r





perpendicular distance between point O and the vector The algebraic sign of the torque is asigned as follows:

If a force tends to rotate an object in

moment arm

the coubterc

F

F





lockwise direction the sign is positive If a force tends to rotate an object in the clockwise direction the sign is negative

F

r F

  

(10-13)

For translational motion Newton's second law connects the force acting on a particle with the resulting acceleration There is a similar r

Newton's Second

elationship betw

Law for Ro

een the to

tation

rque of a force applied on a rigid object and the resulting angular acceleration

This equation is known as Newton's second law for rotation We will explore this law by studying a simple body which consists of a point mass at the end

of a massless rod of length A force F is

m

the system about an axis at the origin As we did earlier, we resolve F into a tangential and a radial component The tangential component is responsible for the



   2

between equations 1 and 2:

t

F F ma

ma r m r r mr I





  

(10-14)

We have derived Newton's second law for rotation for a special case A rigid body which consists of a point mass at the end of a massl

Newton's Second Law for Rotation

ess rod of length We will now

derive the same equation for a general case

O

1 2 3 i

r i

Consider the rod-like object shown in the figure which can rotate about an axis

parts or "elements" and label them The e

net



1 2 3

1 2 3

3 3

and they are located at distances , , , , from O We apply Newton's second

(eqs

n n

m m m m

r r r r

I



.3), etc If we add all these equations we get:

i

Thus we end up with the equation:

n

net I

(10-15)

In chapter 7 we saw that if a force does work

on an object, this results in a change of its kinetic energy

In a similar

Work and Rotational Kin

way, when a torque does

etic Energ

work

on ro

y

a

W

tating rigid body, it changes its rotational kinetic energy by the same amount

Consider the simple rigid body shown in the figure which consists of a mass

The radial component does zero work becau

t r

m

F

  



se it is at right angles to the motion

have a change in kinetic energy

f

i t

mr





  





(10-16)

f

i





 



2 f 2 i

K I I

  

W K

Power has been defined as the rate at which work is done

by a force and in the case of rotational motion by a torque

rotates an object

Power

by an angl

dW d

 

d





Below we summarize the results of the work-rotational kinetic energy theorem

f

i





 

(10-17)

P 

Rotational Motion

Analogies betw

een translational and rotational Motio

Translational Motion

n

x v









0

2 2

2

0

2

2

2 2

a

at

t

t t



  



  

    

 

 

 

 

















 

2

2

mv K

m

F

P Fv

I K I I



 























P 