Physics CHAPTER 8

ROTATIONAL MOTION

The Radian The radian is a unit

of angular measure The radian can be

defined as the arc length s along a circle divided by the radius r

sr

57.3

More About Radians

Comparing degrees and radians

Converting from degrees to radians

3.572

360rad1

]rees[deg180

]rad[

Angular Displacement

Axis of rotation is the center of the disk

Need a fixed reference line

During time t, the reference line moves through angle θ

Angular Displacement, cont.

The angular displacement is defined as the angle the object rotates through during some time interval

The unit of angular displacement is

the radian Each point on the object undergoes

the same angular displacement

fi

Average Angular Speed

The average angular speed, ω, of a rotating rigid object is the ratio of the angular displacement to the time interval

fiav

fit t t

Angular Speed, cont.

The instantaneous angular speed is defined as the limit of the average speed as the time interval approaches zero

Units of angular speed are radians/sec rad/s

Speed will be positive if θ is increasing (counterclockwise)

Speed will be negative if θ is decreasing (clockwise)

Average Angular Acceleration

The average angular acceleration, , of an object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change:

fiav

fit t t

Angular Acceleration, cont

Units of angular acceleration are rad/s² Positive angular accelerations are in the

counterclockwise direction and negative accelerations are in the clockwise direction

When a rigid object rotates about a fixed axis, every portion of the object has the same angular speed and the same angular acceleration

Angular Acceleration, final

The sign of the acceleration does not have to be the same as the sign of the angular speed

The instantaneous angular acceleration is defined as the limit of the average acceleration as the time interval approaches zero

Angular AccelerationAngular acceleration α measures how rapidly the angular velocity is changing:

Slide 7-17

Linear and Circular Motion Compared

Slide 7-18

Linear and Circular Kinematics Compared

Slide 7-19

Sign of the Angular Acceleration

Slide 7-20

Relationship Between Angular and Linear Quantities Displacements

Speeds

Accelerations

Every point on the rotating object has the same angular motion

Every point on the rotating object does not have the same linear motion

rs

tv r

ta r

Centripetal Acceleration and Angular Velocity

The angular velocity and the linear velocity are related (v = ωr)

The centripetal acceleration can also be related to the angular velocity

ra 2C

Vector Nature of Angular Quantities

Angular displacement, velocity and acceleration are all vector quantities

Direction can be more completely defined by using the right hand rule Grasp the axis of rotation

with your right hand Wrap your fingers in the

direction of rotation Your thumb points in the

direction of ω

Velocity Directions, Example

In a, the disk rotates clockwise, the velocity is into the page

In b, the disk rotates counterclockwise, the velocity is out of the page

Righty-tighty

Lefty-loosey

Centripetal and Tangential Acceleration

Slide 7-22

Force vs. Torque

Forces cause accelerationsTorques cause angular

accelerationsForce and torque are related

Torque

The door is free to rotate about an axis through O There are three factors that determine the

effectiveness of the force in opening the door: The magnitude of the force The position of the application of the force The angle at which the force is applied

Torque, cont Torque, , is the tendency of a

force to rotate an object about some axis= Fr

is the torqueF is the force

symbol is the Greek taur is the length of the position

vector SI unit is N.m

Interpreting Torque

rF rF sin

Torque is due to the component of the force perpendicular to the radial line.

Slide 7-25

Torque is a vector quantityThe direction is perpendicular to the plane determined by the position vector and the force

A Second Interpretation of Torque

rF rF sin

Slide 7-26

FsinFsin

Signs and Strengths of the Torque

Slide 7-27

If the turning tendency of the force is counterclockwise, the torque will be positive

If the turning tendency is clockwise, the torque will be negative

Multiple Torques

When two or more torques are acting on an object, the torques are addedAs vectors

If the net torque is zero, the object’s rate of rotation doesn’t change

General Definition of Torque

The applied force is not always perpendicular to the position vector

The component of the force perpendicular to the object will cause it to rotate

When the force is parallel to the position vector, no rotation occurs

When the force is at some angle, the perpendicular component causes the rotation

General Definition of Torque, final

Taking the angle into account leads to a more general definition of torque: Fr sin

F is the forcer is the position vector is the angle between the force

and the position vector

Lever Arm

The lever arm, d, is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force

d = r sin

Net Torque

The net torque is the sum of all the torques produced by all the forces Remember to account for the

direction of the tendency for rotationCounterclockwise torques are

positiveClockwise torques are negative

Checking UnderstandingThe four forces shown have the same strength. Which force would be most effective in opening the door?

A. Force F1

B. Force F2

C. Force F3

D. Force F4

E. Either F1 or F3Slide 7-23

AnswerThe four forces shown have the same strength. Which force would be most effective in opening the door?

A. Force F1

B. Force F2

C. Force F3

D. Force F4

E. Either F1 or F3Slide 7-24

Moment of Inertia

The moment of inertia, I, of a point mass is equal to mass of the object times the square of the distance from the object’s axis of rotation.

SI units are kg m2

Applying Newton’s 2nd Law results in

2I m r

Moment of Inertia

The moment of inertia of an object is the rotational equivalent to the mass of the object in a linear motion.

Ex. For linear motion, the heavier the mass the more difficult it is to get it to move. In rotational motion, a high I, moment of inertia, means that it is difficult to get the object to rotate on an axis.

The size of the moment of inertia, depends on the radius of rotation from the center axis and the distribution of mass around the axis of rotation.

Moment of Inertia If the radius length is large, it

will be more difficult to get the mass to rotate which indicates a higher moment of inertia. So if we apply a torque closer to the axis of rotation, it will be easier to cause the rotation to occur.

If the mass is distributed closer to the axis of rotation, the rotation will be easier to start and the Moment of Inertia will be less

I1 > I2

Moments of Inertia for Various ObjectsObject Location of Axis Diagram Moment of Inertia

Equation

Thin hoop of radius r

Through central diameter

Solid uniform cylinder of radius r

Through center

Uniform Sphere of radius r

Through center

Long uniform rod of length l

Through center

Long uniform rod of length l

Through end

Thin rectangular plane of length l and width w

Through center

Newton’s Second Law for a Rotating Object

The angular acceleration is directly proportional to the net torque

The angular acceleration is inversely proportional to the moment of inertia of the object

I

More About Moment of Inertia

There is a major difference between moment of inertia and mass: the moment of inertia depends on the quantity of matter and its distribution in the rigid object.

The moment of inertia also depends upon the location of the axis of rotation

Moment of Inertia of a Uniform Ring Image the hoop is

divided into a number of small segments, m1 …

These segments are equidistant from the axis

2 2i iI m r M R

Other Moments of Inertia

Example, Newton’s Second Law for Rotation Draw free body

diagrams of each object Only the cylinder is

rotating, so apply = I

The bucket is falling, but not rotating, so apply F = m a

Remember that a = r and solve the resulting equations

Torque and Equilibrium

First Condition of Equilibrium The net external force must be zero

This is a necessary, but not sufficient, condition to ensure that an object is in complete mechanical equilibrium

This is a statement of translational equilibrium

0

0 0x y

or

and

F

F F

Torque and Equilibrium, cont

To ensure mechanical equilibrium, you need to ensure rotational equilibrium as well as translational

The Second Condition of Equilibrium states The net external torque must be zero

0

Equilibrium Example

The woman, mass m, sits on the left end of the see-saw

The man, mass M, sits where the see-saw will be balanced

Apply the Second Condition of Equilibrium and solve for the unknown distance, x

Axis of Rotation

If the object is in equilibrium, it does not matter where you put the axis of rotation for calculating the net torque The location of the axis of rotation is

completely arbitrary Often the nature of the problem will

suggest a convenient location for the axis When solving a problem, you must specify

an axis of rotation Once you have chosen an axis, you must

maintain that choice consistently throughout the problem

Notes About Equilibrium

A zero net torque does not mean the absence of rotational motion An object that rotates at uniform

angular velocity can be under the influence of a zero net torqueThis is analogous to the

translational situation where a zero net force does not mean the object is not in motion

Solving Equilibrium Problems

Draw a diagram of the system Include coordinates and choose a

rotation axis

Isolate the object being analyzed and draw a free body diagram showing all the external forces acting on the object For systems containing more than one

object, draw a separate free body diagram for each object

Problem Solving, cont.

Apply the Second Condition of Equilibrium This will yield a single equation, often with one

unknown which can be solved immediately

Apply the First Condition of Equilibrium This will give you two more equations

Solve the resulting simultaneous equations for all of the unknowns Solving by substitution is generally easiest

Example of a Free Body Diagram (Forearm)

Isolate the object to be analyzed Draw the free body diagram for that object

Include all the external forces acting on the object

Example of a Free Body Diagram (Beam) The free body

diagram includes the directions of the forces

The weights act through the centers of gravity of their objects

Fig 8.12, p.228

Slide 17

Example of a Free Body Diagram (Ladder)

The free body diagram shows the normal force and the force of static friction acting on the ladder at the ground

The last diagram shows the lever arms for the forces

Center of Gravity

The force of gravity acting on an object must be considered

In finding the torque produced by the force of gravity, all of the weight of the object can be considered to be concentrated at a single point

Calculating the Center of Gravity The object is

divided up into a large number of very small particles of weight (mg)

Each particle will have a set of coordinates indicating its location (x,y)

Calculating the Center of Gravity, cont.

We assume the object is free to rotate about its center

The torque produced by each particle about the axis of rotation is equal to its weight times its lever armFor example, m1 g x1

Calculating the Center of Gravity, cont.

We wish to locate the point of application of the single force whose magnitude is equal to the weight of the object, and whose effect on the rotation is the same as all the individual particles.

This point is called the center of gravity of the object

Coordinates of the Center of Gravity

The coordinates of the center of gravity can be found from the sum of the torques acting on the individual particles being set equal to the torque produced by the weight of the object

i i i icg cg

i i

m x m yx and y

m m

Center of Gravity of a Uniform Object

The center of gravity of a homogenous, symmetric body must lie on the axis of symmetry.

Often, the center of gravity of such an object is the geometric center of the object.