09_Slide_Handout_1(3) Circular and Rotational Motion

*Property of STI H0055

Circular Motion and Rotational Motion Angular Quantities

Uniform Circular Motion

Rotational Motion

Center of Gravity

Angular Momentum

Torque

H0055

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Angular QuantitiesAngular Position/Displacement

1. Angular Position (θ)

Denotes the orientation of object in space (rotation)

Direction of the displacement vector drawn from the axis to the location of the object (revolution)

2. Angular Displacement (Δθ)

Change in angular position

Difference between initial (θi) and

final (θf) angular positions

Δθ = θf -θi

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Angular QuantitiesAngular Velocity

Rate of change in object’s angular position

Angular displacement per unit of time t

t

Angular Acceleration

Rate at which the angular velocity changes:

where ωf and ωi are the final and initial angular velocities

respectively

t

if

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Find the angular

velocity of a sprinter

that makes two laps

around an oval in

10.0 minutes.

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A compact disc makes 500

revolutions per minute. If

the CD accelerates

uniformly in 4.0 seconds

from rest, what is its

angular acceleration in

radians per second?

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Angular quantities as vectors

Angular velocity and angular acceleration are

vectors

↑ ω = same direction as α

↓ ω = opposite direction of α

Right-hand rule

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Angular displacement is not a vector quantity

Addition of angular displacements is not

commutative

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Uniform Circular Motion

Movement of an object along a circular path

with constant speed and constant angular

velocity

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Tangential Speed (v)

Speed of an object along a circular path

where v is the tangential speed, r is the radius of

the circular path and t is the time

t

Δθrv

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Tangential Speed (v)

Other equations:

ωrv

fT

v r2r2

Tf

1

T = period of revolution

f = frequency

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Centripetal Acceleration (ac)

Acceleration of object moving in a circular path

Always directed towards the center

Not denoted as a vector because its direction is

never constant

r

v 2

= a c

r2= a c

2

24= a c

T

r

Formulas:

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Centripetal Force (Fc)

Force that causes the object to move in circular path

Always have the same direction as ac

It is not denoted as vector because its direction is never constant

r

vm =Fc

2

r =mFc

2

2

24

T

r =mFc

Fc = mac

Formulas:

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A 40 kg child makes 10 revolutions around the

Cliffhanger in 29.3 seconds. The radius of the barrel

is 2.9 meters, what is the centripetal force acting on

the child?

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A 0.035 kg pebble is made to whirl in a circular

path at the end of a 1.00 m string. What is the

tangential velocity if the pebble makes one

revolution in 1.50 s? What is the centripetal

acceleration? What is the tension in the string?

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Centrifugal Force

Not a real force, but experienced as if they are

actual force

Arises due to inertia

Objects tend to resist centripetal force and

move in straight path

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Rotational Motion Rotation

Movement of an object along its axis causing it

to change its orientation in space

Translational Motion

Movement of an object along a path causing it

to change its location in space

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Rotational Inertia

Tendency to resist any change in rotational motion

Depends on the distribution of mass in relation to the

axis

The farther is the bulk of mass from the axis, the

greater is the rotational inertia

Expressed using Newton’s Second Law for Rotational

Motion

t = I a

I = rotational inertia

t = the torque

a = the angular acceleration

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Center of Gravity

A point within the object which when used as a

pivot point will result in balanced torque and

hence will not cause any rotation

May lie at a point within the object where there

is no mass

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Angular Momentum

The product of the moment of inertia I

and the angular velocity

L = I

“An object or system of objects will maintain

its angular momentum unless acted upon

by an unbalanced external torque.”

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Law of Conservation of Angular Momentum

If no unbalanced external torque acts on a rotating system, the angular momentum of that system is constant

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Cross Multiplication

Type of vector multiplication where the product of two vectors a and b is another vector c, such that c lies perpendicular to the plane of a and b

c = a × b

Magnitude of c is determined by the formula:

where θ is the angle between a and b

Direction is determined by right-hand rule

Not commutative

sinabc

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Torque (τ) Force acting at a certain distance away from the axis;

this distance is called lever arm

The cross-product of force and lever arm:

τ = F × r τ = Fr sinθF = force

r = lever arm

θ = the angle between them

unit of torque is Newton meter (N m)

F

r

θ

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You pull the door open with a

perpendicular force of 10.0 N.

Your hand is 0.400 m away

from the door hinge. How

much torque is produced?

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Solve for the torque due to a force of 15 N on a

body if the lever of arm is 0.75 m in length.

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Rotational Equilibrium

An object is in rotational equilibrium if the sum

of all torques acting on it is zero

0321 NET

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At what distance from the center of the seesaw

should a man weighing 75.0 kg sit in order to

balance the seesaw while a 30.0 kg child is sitting

on the other side 2.0 m away from the center?

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09_Slide_Handout_1(3) Circular and Rotational Motion

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