Chapter 10

Rotation

10 Rotation

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10.1_2 The Rotation & Rotational Variables

 We now look at motion of rotation.

 We will find the same laws apply.

 But we will need new quantities to express them

o Torque

o Rotational inertia

 A rigid body rotates as a unit, locked together.

 We look at rotation about a fixed axis.

 These requirements exclude from consideration:

o The Sun, where layers of gas rotate separately

o A rolling bowling ball, where rotation and translation occur

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Translation and Rotation

• Motion of Translation : Object moves along a straight or

curved line.

• Motion of Rotation: Object rotates about a fixed axis

(object rotates about center of mass) .

Rotation

Variable

Symbol

Unit

Position

meter (m)

Displacement

x

meter (m)

Velocity

meters/sec (m/s)

Acceleration

meters/sec

(m/s

)

Motion of Translation:

i.e. Motion along a straight line

(along x-axis)

Variable

Symbol

Unit

Angular Position



radians (rad)

Angular Displacement



radians (rad)

Angular Velocity



radians/sec

(rad/s)

Angular Acceleration

radians/sec

(rad/s

)

1 rad = 2π = 1 revolution

Motion of Rotation:

i.e. rotation of tires about a fixed axis

10.1_2 The Rotation & Rotational Variables

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• A rigid body is a body that can rotate with all its parts locked together

and without any change in its shape.

• A fixed axis means that the rotation occurs about an axis that does not

move. The fixed axis is called the axis of rotation.

Rigid body

Fixed axes

Follows a

circular line

TOP VIEW

 Reference Line: The angular position of this line

(and of the object) is taken relative to a fixed

direction, the zero angular position.

10.1_2 The Rotation & Rotational Variables

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1- Angular Position,

s: length of a circular arc that extends from the x

axis reference line.

r: radius of the circle.

An angle () measured in radians (rad)!

• Changing the angular position of the

reference line from 

to 

, the body

undergoes an angular displacement 

given by;

2- Angular Displacement ()

• An angular displacement in the

counterclockwise direction is positive,

and one in the clockwise direction is

negative.

• Clocks are negative!

10.1_2 The Rotation & Rotational Variables

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3- Angular Velocity ()

Average angular velocity: angular displacement during a time interval

Instantenous angular velocity: limit as Δt → 0

4- Angular Acceleration (): If the angular velocity of a rotating body is not

constant, then the body has an angular acceleration.

Average angular acceleration:

angular velocity change during a

time interval

 If the body is rigid, these

equations hold for all points

on the body.

 Magnitude of angular

velocity = angular speed.

Intantenous angular acceleration: limit as Δt → 0

10.1_2 The Rotation & Rotational Variables

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Sample problem:

10.1_2 The Rotation & Rotational Variables

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Sample problem contd.:

10.1_2 The Rotation & Rotational Variables

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Sample problem contd.:

10.1_2 The Rotation & Rotational Variables

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Sample problem: Angular Velocity and Acceleration

10.1_2 The Rotation & Rotational Variables

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10.3 Are Angular Quantities Vectors?

 With right-hand rule to determine direction, angular velocity &

acceleration can be written as vectors.

 If the body rotates around the vector, then the vector points along the

axis of rotation,

 Angular displacements are not vectors, because the order of rotation

matters for rotations around different axes.

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10.4 Rotation with Constant Angular Acceleration

• Just as in the basic equations for constant linear acceleration, the basic

equations for constant angular acceleration can be derived in a

similar manner.

• The constant angular acceleration equations are similar to the constant

linear acceleration equations.

• We simply change linear quantities to angular ones.

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Sample problem: Constant Angular Acceleration

The angular acceleration is constant, so we can use the

rotation equation:

Substituting known values and setting 

=0 and  =5.0

rev =10p rad give us

Solving this quadratic equation for t, we find t =32 s.

(b) Describe the grindstone’s rotation between t =0 and

t =32 s.

Description: The wheel is initially rotating in the negative

(clockwise) direction with angular velocity w

=4.6 rad/s,

but its angular acceleration a is positive.

The initial opposite signs of angular velocity and angular

acceleration means that the wheel slows in its rotation in the

negative direction, stops, and then reverses to rotate in the

positive direction.

After the reference line comes back through its initial

orientation of q= 0, the wheel turns an additional 5.0 rev by

time t =32 s.

Calculation: With w =0, we solve for the corresponding time

10.4 Rotation with Constant Angular Acceleration

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Sample problem: Constant Angular Acceleration

10.4 Rotation with Constant Angular Acceleration

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10.5 Relating Linear and Angular Variables

• If a reference line rotates through an

angle , a point within the body at a

position r from the rotation axis moves a

distance s along a circular arc, where s is

given by:

Differentiating the above equation

with respect to time

Period of Revolution

Angular speed of particles are the same but

linear speed increases as going to outside of the

rotation axis!

r: radius of the circle travelled by

particle

 Linear and angular variables are

related by r (perpendicular

distance from the rotational axis)

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Differentiating with respect to

time

:The radial part of the acceleration is

the centripetal acceleration

: tangential acceleration

Tangential acceleration Radial acceleration

(in terms of angular velocity)

10.5 Relating Linear and Angular Variables

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Consider an induction roller coaster (which can be

accelerated by magnetic forces even on a horizontal track).

Each passenger is to leave the loading point with

acceleration g along the horizontal track.

That first section of track forms a circular arc (see Figure),

so that the passenger also experiences a centripetal

acceleration. As the passenger accelerates along the arc, the

magnitude of this centripetal acceleration increases

alarmingly. When the magnitude a of the net acceleration

reaches 4g at some point P and angle 

along the arc, the

passenger moves in a straight line, along a tangent to the

arc.

(a)What angle



should the arc subtend so that a is 4g at point P?

Calculations:

Substituting 

=0, and 

=0, and we find:

But

which gives:

Sample problem

This leads us to a total acceleration:

Substituting for a

, and solving for 

lead to:

When a reaches the design value of 4g, angle 

is the angle 

. Substituting a =4g, = 

, and

= g, we find:

10.5 Relating Linear and Angular Variables

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Sample problem cont.

(b) What is the magnitude a of the passenger’s net

acceleration at point P and after point P?

Reasoning: At P, a has the design value of 4g. Just

after P is reached, the passenger moves in a straight

line and no longer has centripetal acceleration.

Thus, the passenger has only the acceleration

magnitude g along the track.

Hence, a =4g at P and a =g after P.

10.5 Relating Linear and Angular Variables

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10.6 Kinetic Energy of Rotation

• For an extended rotating rigid body,

• treat the body as a collection of particles with different linear

velocities (same angular velocity for all particles but possibly

different radii ),

• and add up the kinetic energies of all the particles to find the total

kinetic energy of the body:

is the mass of the i

particle and v

is its speed).



Rotational Inertia(or moment of inertia) I

(With respect to axis of rotation)

Kinetic Energy

of Rotation

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10.7 Calculating the Rotational Inertia

 I is a constant for a rigid object and given rotational axis.

 Caution: the axis for I must always be specified.

 Use these equations for a finite set of rotating particles.

 Rotational inertia corresponds to how difficult it is to change the state

of rotation (speed up, slow down or change the axis of rotation)

 If a rigid body consists of a great many adjacent particles (it is

continuous), we consider an integral and define the rotational inertia of

the body as

 In principle we can always use this equation.

 But there is a set of common shapes for which values have already

been calculated (see Table) for common axes.

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10.7 Calculating the Rotational Inertia

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• If a rigid body consists of a great many adjacent particles

Example:

10.7 Calculating the Rotational Inertia

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Parallel Axis Theorem:

If h is a perpendicular distance between a

given axis and the axis through the center of

mass (these two axes being parallel).Then

the rotational inertia I about the given axis is

• If we know the rotational inertia about an axis that extends through

the body’s center of mass (I

com

), then we can calculate rotational

inertia about an another axis parallel to the first one.

I = ?

com

h: perpendicular distance to com

 Note the axes must be parallel, and the first must go through the center

of mass.

 This does not relate the moment of inertia for two arbitrary axes.

10.7 Calculating the Rotational Inertia

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Sample problem: Rotational Inertia

10.7 Calculating the Rotational Inertia

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Sample problem: Rotational Inertia

10.7 Calculating the Rotational Inertia

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Sample problem: Rotational Inertia cont.

10.7 Calculating the Rotational Inertia

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10.8 Torque, t

• The ability of a force F to rotate the body depends on both the

magnitude of its tangential component F

and also on just how far from

O, the pivot point, the force is applied.

• To include both these factors, a quantity called torque t is defined as:

where is called the moment arm of F.



Again, torque is positive if it would cause a counterclockwise

rotation, otherwise negative.

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 Torque takes these factors into account:

1. A line extended through the applied force is called the line of action

of the force.

2. The perpendicular distance from the line of action to the axis is called

the moment arm.

 The unit of torque is the newton-meter, N m.

 Note that 1 J = 1 N m, but torques are never expressed in joules, torque

is not energy.

10.8 Torque, t

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10.9 Newton’s 2

Law of Rotation

For more than one force, we can generalize:

 Rewrite F = ma with rotational variables:

 It is torque that causes angular acceleration

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Sample problem: Newton’s 2

Law in Rotational Moition

Forces on block:

From the block’s freebody, we can write

Newton’s second law for components

along a vertical y axis as: T –mg= ma.

• The torque from the tension force, T, is -RT, negative

because the torque rotates the disk clockwise from

rest.

• The rotational inertia I of the disk is ½ MR

• But t

net

=I =-RT=1/2 MR

.

• Because the cord does not slip, the linear acceleration

a of the block and the (tangential) linear acceleration

of the rim of the disk are equal. We now have: T=-

1/2 Ma.

Combining

results:

We then find T:

The angular acceleration of the disk is:

Note that the acceleration a of the falling block is less

than g, and tension T in the cord (=6.0 N) is less than

the gravitational force on the hanging block ( mg =11.8

N).

10.9 Newton’s 2

Law of Rotation

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10.10 Work and Rotational Kinetic Energy

where t is the torque doing the

work W, and



and



are the

body’s angular positions before

and after the work is done,

respectively

When t is constant

The rate at which the work is done is the power

The rotational work-kinetic energy theorem states:

The work done in a rotation about a

fixed axis can be calculated by:

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10.10 Work and Rotational Kinetic Energy

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Sample problem: Work, Rotational KE,

Torque

10.10 Work and Rotational Kinetic Energy

Eq. (10-7)

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Angular Position

Measured around a rotation axis,

relative to a reference line:

Angular Displacement

A change in angular position

10 Summary

Eq. (10-1)

Eq. (10-5)

Angular Acceleration

Average and instantaneous values:

Eq. (10-4)

Angular Velocity and Speed

Average and instantaneous values:

Eq. (10-6)

Eq. (10-8)

Kinematic Equations

Given in Table 10-1 for constant

acceleration

Match the linear case

Rotational Kinetic Energy

and Rotational Inertia

Eq. (10-34)

Eq. (10-33)

Linear and Angular Variables

Linear and angular displacement,

velocity, and acceleration are

related by r

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10 Summary

The Parallel-Axis Theorem

Relate moment of inertia around

any parallel axis to value around

com axis

Eq. (10-36)

Torque

Force applied at distance from an

axis:

Moment arm: perpendicular

distance to the rotation axis

Newton's Second Law in

Angular Form

Work and Rotational Kinetic

Energy

Eq. (10-42)

Eq. (10-53)

Eq. (10-55)

Eq. (10-39)

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10 Solved Problems

1. A disk rotates about its central axis starting from rest and accelerates with constant

angular acceleration. At one time it is rotating at 10 rev/s; 60 revolutions later, its

angular speed is 15 rev/s. Calculate (a) the angular acceleration, (b) the time

required to complete the 60 revolutions, (c) the time required to reach the 10 rev/s

angular speed, and (d) the number of revolutions from rest until the time the disk

reaches the 10 rev/s angular speed.

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10 Solved Problems

2. If a 32.0 N·m torque on a wheel causes angular acceleration 25.0 rad/s

, what is

the wheel's rotational inertia?

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10 Solved Problems

3.In Figure, block 1 has mass m

=460 g, block 2 has mass

=500 g, and the pulley, which is mounted on a horizontal axle

with negligible friction, has radius R=5.00 cm. When released

from rest, block 2 falls 75.0 cm in 5.00 s without the cord

slipping on the pulley. (a) What is the magnitude of the

acceleration of the blocks? What are (b) tension T

and (c)

tension T

? (d) What is the magnitude of the pulley's angular

acceleration? (e) What is its rotational inertia?

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10 Solved Problems

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10 Solved Problems

4. A 32.0 kg wheel, essentially a thin hoop with radius 1.20 m, is rotating at

280 rev/min. It must be brought to a stop in 15.0 s. (a) How much work must be

done to stop it? (b) What is the required average power?

10 Solved Problems

5. In Figure, two 6.20 kg blocks are connected by a massless string over a pulley of

radius 2.40 cm and rotational inertia 7.40x10

−4

kg·m

. The string does not slip on

the pulley; it is not known whether there is friction between the table and the

sliding block; the pulley's axis is frictionless. When this system is released from

rest, the pulley turns through 0.650 rad in 91.0 ms and the acceleration of the

blocks is constant. What are (a) the magnitude of the pulley's angular acceleration,

(b) the magnitude of either block's acceleration, (c) string tension T

, and (d) string

tension T

10 Solved Problems

Additional Materials

10 Rotation

10.7 Calculating the Rotational Inertia

•Let O be the center of mass (and also the origin of the

coordinate system) of the arbitrarily shaped body shown

in cross section.

•Consider an axis through O perpendicular to the plane of

the figure, and another axis through point P parallel to the

first axis.

•Let the x and y coordinates of P be a and b.

•Let dm be a mass element with the general coordinates x

and y. The rotational inertia of the body about the axis

through P is:

•But x

+ y

, where R is the distance from O to dm,

the first integral is simply I

com

, the rotational inertia of the

body about an axis through its center of mass.

•The last term in is Mh

, where M is the body’s total mass.

10 Solved Problems

6.In Figure, a thin uniform rod (mass 3.0 kg, length 4.0 m) rotates freely

about a horizontal axis A that is perpendicular to the rod and passes through

a point at distance d=1.0 m from the end of the rod. The kinetic energy of

the rod as it passes through the vertical position is 20 J. (a) What is the

rotational inertia of the rod about axis ? (b) What is the (linear) speed of the

end of the rod as the rod passes through the vertical position? (c) At what

angle ϴ will the rod momentarily stop in its upward swing?

10-1_2 Rotational Variables

10.01 Identify that if all parts of a body rotate around a fixed axis locked

together, the body is a rigid body.

10.02 Identify that the angular position of a rotating rigid body is the

angle that an internal reference line makes with a fixed, external

reference line.

10.03 Apply the relationship between angular displacement and the initial

and final angular positions.

10.04 Apply the relationship between average angular velocity, angular

displacement, and the time interval for that displacement.

10.05 Apply the relationship between average angular acceleration,

change in angular velocity, and the time interval for that change.

10.06 Identify that counterclockwise motion is in the positive direction

and clockwise motion is in the negative direction.

10.07 Given angular position as a function of time, calculate the

instantaneous angular velocity at any particular time and the average

angular velocity between any two particular times.

Learning Objectives

10-1_2 Rotational Variables

10.08 Given a graph of angular position versus time, determine the

instantaneous angular velocity at a particular time and the average

angular velocity between any two particular times.

10.09 Identify instantaneous angular speed as the magnitude of the

instantaneous angular velocity.

10.10 Given angular velocity as a function of time, calculate the

instantaneous angular acceleration at any particular time and the

average angular acceleration between any two particular times.

10.11 Given a graph of angular velocity versus time, determine the

instantaneous angular acceleration at any particular time and the

average angular acceleration between any two particular times.

10.12 Calculate a body’s change in angular velocity by integrating its

angular acceleration function with respect to time.

10.13 Calculate a body’s change in angular position by integrating its

angular velocity function with respect to time.

10-3_4 Rotation with Constant Angular Acceleration

10.14 For constant angular acceleration, apply the relationships between

angular position, angular displacement, angular velocity, angular

acceleration, and elapsed time (Table 10-1).

Learning Objectives

10-5 Relating the Linear and Angular Variables

10.15 For a rigid body rotating

about a fixed axis, relate the

angular variables of the body

(angular position, angular

velocity, and angular

acceleration) and the linear

variables of a particle on the

body (position, velocity, and

acceleration) at any given

radius.

10.16 Distinguish between

tangential acceleration and

radial acceleration, and draw

a vector for each in a sketch

of a particle on a body

rotating about an axis, for

both an increase in angular

speed and a decrease.

Learning Objectives

10-6 Kinetic Energy of Rotation

10.17 Find the rotational

inertia of a particle about a

point.

10.18 Find the total rotational

inertia of many particles

moving around the same

fixed axis.

10.19 Calculate the rotational

kinetic energy of a body in

terms of its rotational inertia

and its angular speed.

Learning Objectives

10-7 Calculating the Rotational Inertia

10.20 Determine the rotational

inertia of a body if it is given

in Table 10-2.

10.21 Calculate the rotational

inertia of body by integration

over the mass elements of the

body.

10.22 Apply the parallel-axis

theorem for a rotation axis

that is displaced from a

parallel axis through the

center of mass of a body.

Learning Objectives

10-8 Torque

10.23 Identify that a torque on a body

involves a force and a position

vector, which extends from a

rotation axis to the point where the

force is applied.

10.24 Calculate the torque by using

(a) the angle between the position

vector and the force vector, (b) the

line of action and the moment arm

of the force, and (c) the force

component perpendicular to the

position vector.

10.25 Identify that a rotation axis

must always be specified to

calculate a torque.

10.26 Identify that a torque is

assigned a positive or negative

sign depending on the direction

it tends to make the body rotate

about a specified rotation axis:

“clocks are negative.”

10.27 When more than one torque

acts on a body about a rotation

axis, calculate the net torque.

Learning Objectives

10-9 Newton's Second Law for Rotation

10.28 Apply Newton's second law for rotation to relate the net

torque on a body to the body's rotational inertia and rotational

acceleration, all calculated relative to a specified rotation axis.

Learning Objectives

10-10 Work and Rotational Kinetic Energy

10.29 Calculate the work done

by a torque acting on a

rotating body by integrating

the torque with respect to the

angle of rotation.

10.30 Apply the work-kinetic

energy theorem to relate the

work done by a torque to the

resulting change in the

rotational kinetic energy of

the body.

10.31 Calculate the work done

by a constant torque by

relating the work to the angle

through which the body

rotates.

10.32 Calculate the power of a

torque by finding the rate at

which work is done.

10.33 Calculate the power of a

torque at any given instant by

relating it to the torque and

the angular velocity at that

instant.

Learning Objectives