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Chapter
8
Rotational
Motion
Circular Motion


Specific type of
2-dimensional
motion.
In discussing
circular motion:



Angular speed
Angular
acceleration
Centripetal
acceleration
Linear Vs. Angular
LINEAR
ANGULAR
Displacement
∆x = xf - xi
∆θ = θf – θi
Speed
v = ∆x/∆t
ω = ∆θ/∆t
Acceleration
a = ∆v/∆t
α = ∆ω/∆t
Kinematic Equations
vf = vo + at
ωf = ωo + αt
∆x = vot + 1/2at2
∆θ = ωot + 1/2αt2
vf2 = vo2 + 2a∆x
ωf 2 = ωo2 + 2α∆θ
Angular Displacement: ∆θ



Axis of rotation is
the center of the
disk.
Need a fixed
reference line.
During time t, the
reference line
moves through
angle θ.
Average Angular Speed: ω

The average
angular speed, ω,
of a rotating rigid
object is the ratio
of the angular
___________ to
the __________.
 av
f  i



tf  ti
t
Average Angular
Acceleration: α

The average angular acceleration 
of an object is defined as the ratio
of the change in the __________
to the _____ it takes for the object
to undergo the change:
 av 
f  i
tf  ti


t
Sample Problem
If a truck has a linear acceleration of
1.85m/s2 and the wheels have an
angular acceleration of 5.23 rad/s2,
what is the diameter of the truck’s
wheels?
Linear Vs. Angular – Part 2

When a rigid object rotates about a fixed axis every
portion of the object has the _____________and
__________. This is not true with the linear speed
and acceleration.
Relationship Between Angular
and Linear Quantities

Displacements

s  r

Speeds
vt   r

Accelerations
at   r

Every point on
the rotating
object has the
same angular
motion.
Every point on
the rotating
object does not
have the same
linear motion.
Sample Problem
A wheel rotates with a constant
angular acceleration of 3.50 rad/s2. If
the angular speed of the wheel is
2.00 rad/s at t= 0s, through what
angle does the wheel rotate between
t =0 and t=2.00 s?
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
Force vs. Torque




Forces cause ___________
accelerations.
Torques cause ____________
accelerations.
Force and torque are related.
Think of torque as the rotational
equivalent to force.
Torque and
Newton’s 2nd Law

Translational Motion

Rotational Motion


I = moment of inertia
= angular acceleration
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

Torque, t, is the ________ of a force
to _______ an object about some
____.

t= r F

t is the torque




Symbol is the Greek tau
F is the force
r is the length of the position vector
SI unit is N.m
Torque

For a given force, torque increases
with increasing distance from the axis
of rotation (r).

t= r F
Torque

For a given force,
torque is at a
________ when the
angle between the
force and r is at ___
and at a ______
when the angle is
__________.

t= r F
Direction of Torque

Torque is a vector quantity



The direction is ___________ to the
plane determined by the position
vector and the force.
If the turning tendency of the force is
______________, the torque will be
__________.
If the turning tendency is ________,
the torque will be ________.
Multiple Torques

When two or more torques are
acting on an object, the torques
are ________.


This is a vector addition.
If the net torque is zero, the
object’s rate of rotation doesn’t
change.
Sample Problem
A packing crate is being pushed by two
forces of equal magnitude acting in
opposite directions. Find the net torque
exerted on the crate by these two
forces have a magnitude of 500 N and
the width of the crate is 1.0 m.
Assume the axis of rotation is through
the center of the crate.
Lever Arm



The lever arm, d, is the ___________ distance
from the axis of rotation to a line drawn along
the direction of the force
d = r sin 
This also gives t = F r sin 
Sample Problem
Find the torque
produced by
the 300 N force
applied at an
angle of 60o to
the door of the
figure at the
right.
Right Hand Rule



Point the fingers
in the direction of
the position
vector
Curl the fingers
toward the force
vector
The thumb points
in the direction of
the torque
Torque and Equilibrium

First Condition of Equilibrium

The net external force must be zero:
F  0 or
Fx  0 and Fy  0


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.
Torque and Equilibrium


To ensure ____________ equilibrium,
you need to ensure _________
equilibrium as well as ___________.
The Second Condition of Equilibrium
states:

The ___________________must be zero.
t  0
If an object has the ability to rotate, but is not moving at all,
i.e.
F = 0
and
t = 0
Then the object is in STATIC EQUILIBRIUM.
Applications:
Seesaws:
Consider a seesaw that is not moving, has 2 people on it
and is supported off centre.
Sample Problem
Little Johnny (40kg) and Silly Sally
(30kg) are playing on a 3.0m long
seesaw (60kg) that is supported in the
middle. If Silly Sally is sitting on the
very edge, how far from the centre
should little Johnny sit in order for the
seesaw to balance horizontally?
Notes About Equilibrium

A zero net torque does not mean
the _________ of rotational motion.

An object that rotates at uniform
angular velocity can be under the
influence of a zero net torque.

This is analogous to the translational
situation where a zero net force does not
mean the object is not in motion.
Chapter
8
THE
END
Rotational
Motion