Chapter 10 - Rotation In this chapter we will study the rotational motion of rigid bodies about a...

20
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 (symbol: Δθ) Avg. and instantaneous angular velocity (symbol: ω ) Avg. and instantaneous angular acceleration (symbol: α ) Rotational inertia, also known as moment of inertia (symbol I ) The kinetic energy of rotation as well as the rotational inertia;

Transcript of Chapter 10 - Rotation In this chapter we will study the rotational motion of rigid bodies about a...

Page 1: 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.

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 (symbol: Δθ)• Avg. and instantaneous angular velocity (symbol: ω )• Avg. and instantaneous angular acceleration (symbol: α )• Rotational inertia, also known as moment of inertia

(symbol I )• The kinetic energy of rotation as well as the rotational

inertia;• Torque (symbol τ )• We will have to solve problems related to the above

mentioned concepts.

Page 2: 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 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, also called the axis of rotation or the rotation axis.

• We can describe the motion of a rigid body rotating about a fixed axis by specifying just one parameter.

• Every point of the body moves in a circle whose center lies on the axis of rotation, and every point moves through the same angle during a particular time interval.

Page 3: 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.

Consider the rigid body of the figure.

We take the z-axis to be the fixed axisof rotation. We define a reference linethat is fixed in the rigid body and isperpendicular to the rotational axis.

The angular position of this line is the angle of the line relative to a fixed direction, which we take as the zero angular position.

In Fig. 11-3 , the angular position is measured relative to the positive direction of the x axis.

Page 4: 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.

r

s

From geometry, we know that is given by (in radians) (10-1)

Here s is the length of arc (arc distance) along a circle and between the x axis (the zero angular position) and the reference line; r is the radius of that circle.

This angle is measured in radians (rad) rather than in revolutions (rev) or degrees. The radian, being the ratio of two lengths, is a pure number and thus has no dimension. Because the circumference of a circle of radius r is 2r, there are 2 radians in a complete circle: rad

r

rrev _2

23601 0

(10-2)

1 rad = 57,30 = 0,159 rev (10-3)

Page 5: 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.

Angular Displacement• If the body of Fig. 10-3 rotates about the

rotation axis as in Fig. 10-4 , changing the angular position of the reference line from 1 to 2, the body undergoes an angular displacement given by:

Δθ = θ2 – θ1 (10-4)• This definition of angular displacement

holds not only for the rigid body as a whole but also for every particle within that body.

• An angular displacement in the counter-clockwise direction is positive, and one in the clockwise direction is negative.

Page 6: 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.

Angular Velocity • Suppose (see Fig. 10-4 ) that our

rotating body is at angular position 1 at time t1 and at angular position 2 at time t2. We define the average angular velocity of the body in the time interval t from t1 to t2 to be

tttavg

12

12 (10-5)

Instantaneous angular velocity, ω:

dt

d

tt

0

lim (10-6)

Unit of ω: rad/s or rev/s

Page 7: 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.

Angular Acceleration • If the angular velocity of a rotating body is not

constant, then the body has an angular acceleration, α

• Let 2 and 1 be its angular velocities at times t2 and t1, respectively. The average angular acceleration of the rotating body in the interval from t1 to t2 is defined as:

tttavg

12

12 (10-7)

• Instantaneous angular acceleration, α:

dt

d

tt

0

lim (10-8)

Do Sample Problems 10-1 & 10-2, p244-246

Unit: rad/s2 or rev/s2

Page 8: 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.

Consider a point on a rigid body rotating about

a fixed axis. At 0 the reference line that connects

the origin with point is on the -axis (point ).

P

t

O P x A

Relating the Linear and Angular Variables :

During the time interval , point moves along arc

and covers a distance . At the same time, the reference

line rotates by an angle .

t P AP

s

OP

Aθs

O

The arc length and the angle are connected by the equation

where is the distance . The speed of point is , .

s

d rds dr OP P v rs r

dt dt dt

Relation between Angular Velocity and Speed :

v rcircumference 2 2 2

The period of revolution is given by .speed

r rT T

v r

2

T

1

Tf

2 f

(10-17)

(10-18)

(10-19) (10-20)

Page 9: 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.

rO

The acceleration of point is a vector that has two

components. The first is a "radial" component along

the radius and pointing toward point . We have

enountered this component in

P

O

The Acceleration :

Chapter 4 where we

called it "centripetal" acceleration. Its magnitude is2

2r

va r

r

The second component is along the tangent to the circular path of and is thus

known as the "tangential" component. Its magnitude is

The magnitude of the acceleration vector is

t

P

d rdv da r r

dt dt dt

2 2 .t ra a a

ta r

(10-8)

(10-22)

(10-23)

Page 10: 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.

When the angular acceleration is constant we can derive simple expressions

that give us the angular velocity and the angula

r posi

Rotation with Constant Angular Acceleration :

tion as a function of

time. 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 rotat

Translational Motion Rotational Motion

ional motion using the following correspondence

b

etween the two motions.

x

0

2

0

0

2

0

2 20

2 20

(

eq. 1)

(eq. 2)

2

2

2

2

o

o

t

tt

v

a

v v at

atx x v t

v v a x x

0 (eq. 3)(10-6)

Page 11: 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.

Ori

iv

mi

1 2 3

Consider the rotating rigid body shown in the figure.

We divide the body into parts of masses , , ,..., ,....

The part (or "element") at has an index and mass .i

i

m m m m

P i m

Kinetic Energy of Rotation :

2 2 21 1 2 2 3 3

The kinetic energy of rotation is the sum of the kinetic

1 1 1energies of the parts: ....

2 2 2K m v m v m v

22

2 2 2 2

1 1 The speed of the th element .

2 2

1 1 The term

rotational i

is known as

nertia moment of inerti

2 2

or about the axis of rotation. Tha e axis

i i i i i ii i

i i i ii i

K m v i v r K m r

K m r I I m r

of

rotation be specified because the value of for a rigid body depends on

its mass, its shape, as well as on the position of the rotation axis. The rotational

inertia of an object describ

mu

e o

st

s h

I

w the mass is distributed about the rotation axis.2

i ii

I m r21

2K I 2 I r dm

(10-31)

(10-32) (10-33)(10-34)

(10-35)

Page 12: 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.

2The rotational inertia . This expression is useful for a rigid body that

has a discreet distribution of mass. For a continuous distribution of mass the s

i ii

I m rCalculating the Rotational Inertia :

2

um

becomes an integral, .I r dm

Page 13: 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.

In the table below we list the rotational inertias for some rigid bodies.2 I r dm

(10-10)

Page 14: 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.

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 parallel-axis t of the

I

I

Parallel - Axis Theorem :

com

Consider the rigid body of mass shown in the figure.

We assume that we know the rotational inertia

about a rotation axis that passes through the center

of mass and is perpen

heorem.

dicular t

M

I

O o the page.

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

through point , a distance from , is given by the equation

I O

P h O2

comI I Mh (10-36)

Page 15: 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.

A

com

We take the origin

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

in the figure. We assume that we know the rotational inertia

for an axis that i

O

I

Proof of the Parallel - Axis Theorem :

s perpendicular to the page and passes

through . O

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

to the page and passing through point with coordinates , . Consider

an element of mass at point with coordinates

I

P a b

dm A x

2 2

2 22

2 2 2 2

, . The distance

between points and is .

.

2 2 . The second

and third integrals are zero. The first

y r

A P r x a y b

I r dm x a y b dm

I x y dm a xdm b ydm a b dm

Rotational Inertia about :P

2 2 2com

2 2

integral is . The term .

Thus the fourth integral is equal to

I a b h

h dm Mh

2

comI I Mh

Page 16: 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.

Do Sample Problems 10-6 to 10-8, p254-256

Page 17: 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.

In fig. a we show a body that can rotate about an axis through

point under the action of a force applied at point a distance

from . In fig. b we resolve into two com

Tor

ponents, radia

que:

O F P

r O F

l and

tangential. The radial component cannot cause any rotation

because it acts along a line that passes through . The tangential

component sin on the other hand causes the rotation of the

o

r

t

F

O

F F

bject about . The ability of to rotate the body depends on the

magnitude and also on the distance between points and .

Thus we define as torque

The distance is

s

in

k n

.

now

t

trF rF r

O F

F

r

F

r P A

as the and it is the

perpendicular distance between point and the vector .

The algebraic sign of the torque is assigned as follows:

If a force tends to rotate an objec

moment a

t in the count

rm

O F

F

erclockwise

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)

(10-39)

(10-41)(10-40)

Draaimoment

Page 18: 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.

For translational motion, Newton's second law connects

the force acting on a particle with the resulting acceleration.

There is a similar relationship between the torqu

Newton's Second Law for Rotation :

e 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 that consists of a point mass at the end

of a massless rod of length . A force is

m

r F

applied on the particle and rotates

the system about an axis at the origin. As we did earlier, we resolve into a

tangential and a radial component. The tangential component is responsible

for the

F

2

rotation. We first apply Newton's second law for . (eq. 1)

The torque acting on the particle is (eq. 2). We eliminate

between equations 1 and 2: .

t t t

t t

t

F F ma

F r F

ma r m r r mr I

(compare with: )F ma I (10-45)

Page 19: 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.

We have derived Newton's second law for rotation

for a special case: a rigid body that consists of a point

mass at the end of a massless rod of length . We will

now

m r

Newton's Second Law for Rotation :

derive the same equation for a general case. O

12

3

i

ri

net

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

through point under the action of a net torque . We divide the body into

parts or "elements" and label them. The

O

1 2 3

1 2 3

1 1 2 2

3 3

elements have masses , , ,...,

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

law for rotation to each element: (eq. 1), (eq. 2),

(eq

n

n

m m m m

r r r r O

I I

I

21 2 3 1 2 3

1 2 3 net

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

... ... . Here is the rotational inertia

of the th element. The sum ... is the net torque applied.

T

n n i i i

n

I I I I I m r

i

1 2 3he sum ... is the rotational inertia of the body.

Thus we end up with the equation nI I I I I

net I (10-15)(10-42)

Page 20: 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.

Rotational Motion

Translational Motion

x

v

Analogies Between Translational and Rotational Motion

0 0

2

0 0

2

2

0

2 20

20 0

2

2

2

2

o

o

a

v v at

atx x v t

v v a x x

t

tt

2 2

2

2

mvK

m

F ma

F

P Fv

IK

I

I

P (10-18)