Rotational Motion

Angular Quantities

! = lr

In a purely rotational motion, all points on the object move in circles around the axis of rotation (“O”). The radius of the circle is r. All points on a straight line drawn through the axis move through the same angle in the same time. The angle ! in radians is defined:

where r is the radius of the circle, and " is the arc length subtended by the angle ! specified in radians. If " = r, then ! = 1 rad.

For a full circle, ! = "/r = 2!r/r = 2! rad. Thus 360°=2! rad.

One radian is then 360°/2! " 360°/6.28 " 57.3°. An object that makes one complete revolution (rev) has rotated through 360°, or 2! radians:

1!rev=360°=2!!rad.

Angular Quantities Angular displacement:

"! = !2 – !1

The average angular velocity is defined as the total angular displacement divided by time:

! = "#"t

! = lim"t#0

"$"t

The angular acceleration is the rate at which the angular velocity changes with time:

! = "2 !!1

"t= "!

"t

! = lim"t#0

"$"t

The instantaneous acceleration is:

The instantaneous angular velocity is:

(rad/s)

(rad/s2)

Angular Quantities

Or since "!/"t = #

Every point on a rotating body has an angular velocity # and a linear velocity v. They are related by "$ = r "!. Hence

v = !l!t

= r !"!t

v = r!Here, r is the distance of a point from the rotation axis and # is given in rad/s. Thus, although ! is the same for every point in the rotating object at any instant, the linear velocity v is greater for points farther from the axis. Therefore, objects farther from the axis of rotation will move faster.

NOTE: Since v = r# and rB > rA, then vB > vA

Angular Quantities If the angular velocity of a rotating object changes, the object as a whole—and each point in it—has an angular acceleration and a linear acceleration.

atan =!v!t

= r !"!t

# = !"!t

$

%&&

'&&

( atan = r#

The total acceleration vector at any point in the rotating object is the vector sum of the two components:

a = atan2 + aR

2 = (r! )2 + (r" 2 )2 = r ! 2 +" 4

aR =v2

r=r!( )2r

"#" aR = r!2

!a = !atan +

!aR

Angular Quantities

The correspondence between linear and rotational quantities can be summarized as:

Linear and Rotational Quantities

Angular Quantities (Reminder)

The frequency is the number of complete revolutions per second:

f = !2"

Frequencies are measured in hertz. 1 Hz = 1 s%1

The period is the time one revolution takes:

T = 1f

Example (1) A carousel is initially at rest. At t = 0 it is given a constant angular acceleration &=0.060 rad/s2, which increases its angular velocity for 8.0 s. At t = 8.0 s, determine (a)! the angular velocity of the carousel, (b)! the linear velocity of a child located 2.5 m from the center, (c)! child’s tangential (linear) acceleration, (d)! child’s centripetal acceleration, (e)! child’s total acceleration,

Constant Angular Acceleration

The equations of motion for constant angular acceleration are the same as those for linear motion, with the substitution of the angular quantities for the linear ones. [constant &, a]

Rolling Motion (Without Slipping)

Rolling without slipping involves both rotation and translation.

In (a), a wheel is rolling without slipping. The point P, touching the ground, is instantaneously at

rest, and the center moves with velocity v.

In (b) the same wheel is seen from a reference

frame where C is at rest. Now point P is moving

with velocity –v.

Relationship between linear and angular speeds is:

v = r!

Example (2) A bicycle slows down uniformly from v0 = 8.40 m/s to rest over a distance of 115 m. Each wheel and tire has an overall diameter of 68.0 cm. Determine (a)! the angular velocity of the wheels at the initial instant (t = 0); (b)! the total number of revolutions each wheel rotates before coming to rest; (c)! the angular acceleration of the wheel; and (d)! the time it took to come to a stop.

Torque To make an object start rotating about an axis requires a force. The direction of this force, and where it is applied, are important.

The overhead view of a door.

For example, the angular acceleration of a door is proportional to the magnitude of the force and is proportional to the perpendicular distance from the axis of rotation to the line along which the force acts. This distance is called the lever arm, or moment arm of the force.

For example, if rA = 3rB, then FB must be three times as large as FA to give the same angular acceleration.

So, a plumber can exert greater torque using a wrench with a long lever arm.

The lever arm is defined as the perpendicular distance from the axis of rotation to the line of action of the force.

Torque

In general, the magnitude of the torque about a given axis can be written as ! = r!Fwhere r! is the lever arm, and the perpendicular symbol (!) reminds us that we must use the distance from the axis of rotation that is perpendicular to the line of action of the force.

We have, F! = F sin ! and r! = r sin !. Thus

! = rF"

The component F|| exerts no torque since it is directed at the rotation axis. Hence the torque will be equal to F! times the distance r from the axis to the point of application of the force:

! = rF sin"

In example of the door, the lever arm for FA is the distance from the knob to the hinge (rA); the lever arm for FD is zero; and the lever arm for FC is rC .

Rotational dynamics; Torque & Rotational Inertia

Knowing that F = ma, we see that

This is for a single point mass; what about an extended object?

As the angular acceleration is the same for the whole object, we can write:

! = rF = rma = rm(r" ) = mr2"

! = mr2"( )#"The quantity mr2 represents the rotational inertia of the particle and is called its moment of inertia.

! = mr2"( )#"I = mr2" = m1r1

2 +m2r22 +!

$%&

'&()( ! = I#"

This is the rotational equivalent of Newton's second law. It is valid for the rotation of a rigid object about a fixed axis.

The rotational inertia of an object depends not only on its mass distribution but also the location of the axis of rotation.

For example, compare (f) and (g).

Rotational Inertia

Example (3)

A 15.0-N force is applied to a cord wrapped around a pulley of mass M = 4.00 kg and radius R = 33.0 cm. The pulley accelerates uniformly from rest to an angular speed of 30.0 rad/s in 3.00 s. If there is a frictional torque 'fr = 1.10 m.N at the axle, determine the moment of inertia of the pulley. The pulley rotates about its center.

Rotational Kinetic Energy The kinetic energy of a rotating object is given by

By substituting the rotational quantities, we find that the rotational kinetic energy can be written:

KE = 12 mv

2( )! = 12 mr

2" 2( )! = 12

mr2!( )" 2

Rotational KE = 12I! 2

An object that has both translational and rotational motion also has both translational and rotational kinetic energy:

KE = 12MvCM

2 + 12ICM!

2

The torque does work as it moves the wheel through an angle !:

W = F!l = Fr!"# = rF

$%&' W = #!"

Angular Momentum & Its Conservation

In analogy with linear momentum, we can define angular momentum L:

We can then write the total torque as being the rate of change of angular momentum. L = I!

I! = I0!0 = constant

!" = #L#t

= I$ % I$0

#t=I $ %$0( )

#t= I #$

#t= I&

If the net torque on an object is zero, the total angular momentum is constant.

Therefore, systems that can change their rotational inertia through internal forces will also change their rate of rotation.

Example (4) A clutch is a mechanism for connecting and disconnecting a vehicle’s engine from its transmission system. A simple clutch consists of two cylindrical plates that can be pressed together to connect two sections of an axle, as needed. Assume that the two plates have masses MA = 6.0 kg and MB = 9.0 kg, with equal radii R = 0.60 m. They are initially separated. Plate MA is accelerated from rest to an angular velocity #1 = 7.2 rad/s in time "t = 2.0 s. Calculate (a)! the angular momentum of MA, (b)! the torque required to accelerate MA from rest to #1. (c) Next, plate MB, initially at rest but free to rotate without friction, is placed in firm contact with freely rotating plate MA, and the two plates then both rotate at a constant angular velocity #2, which is considerably less than #1. What is #2?

Summary !! Angles are measured in radians; a whole circle is 2! radians.

!! Angular velocity is the rate of change of angular position.

!! Angular acceleration is the rate of change of angular velocity.

!! The angular velocity and acceleration can be related to the linear velocity and acceleration.

!! The frequency is the number of full revolutions per second; the period is the inverse of the frequency.

!! The equations for rotational motion with constant angular acceleration have the same form as those for linear motion with constant acceleration.

!! Torque is the product of force and lever arm.

!! The rotational inertia depends not only on the mass of an object but also on the way its mass is distributed around the axis of rotation.

!! The angular acceleration is proportional to the torque and inversely proportional to the rotational inertia.

Summary !! An object that is rotating has rotational kinetic energy. If it is translating as

well, the translational kinetic energy must be added to the rotational to find the total kinetic energy.

!! Angular momentum is L = I#

!! If the net torque on an object is zero, its angular momentum does not change.