Chapter 7

Rotational Motion
and
The Law of Gravity
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
s
  
r
More About Radians
 Comparing degrees and radians
360
1 rad   57.3
2
 Converting from degrees to
radians

 [rad]   [deg rees]
180
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
Rigid Body
 Every point on the object undergoes
circular motion about the point O
 All parts of the object of the body rotate
through the same angle during the
same time
 The object is considered to be a rigid
body
 This means that each part of the body is
fixed in position relative to all other parts of
the body
Angular Displacement,
cont.
 The angular displacement is defined as
the angle the object rotates through
during some time interval
    f   i
 The unit of angular displacement is the
radian
 Each point on the object undergoes the
same angular displacement
Average Angular Speed
 The average
angular speed, ω,
of a rotating rigid
object is the ratio
of the angular
displacement to
the time interval
f  i 
 av  
tf  ti 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:
f  i 
 av  
tf  ti 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
Analogies Between Linear
and Rotational Motion
Relationship Between Angular
and Linear Quantities
 Displacements  Every point on
s  r the rotating
object has the
 Speeds same angular
vt   r motion
 Accelerations  Every point on
at   r the rotating
object does not
have the same
linear motion
Centripetal Acceleration
 An object traveling in a circle,
even though it moves with a
constant speed, will have an
acceleration
 The centripetal acceleration is due
to the change in the direction of
the velocity
Centripetal Acceleration,
cont.
 Centripetal refers
to “center-
seeking”
 The direction of
the velocity
changes
 The acceleration
is directed toward
the center of the
circle of motion
Centripetal Acceleration,
final
 The magnitude of the centripetal
acceleration is given by
2
v
ac 
r
 This direction is toward the center of
the circle
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
v 2
rω 2 2
aC    rω 2

r r
Total Acceleration
 The tangential component of the
acceleration is due to changing
speed
 The centripetal component of the
acceleration is due to changing
direction
 Total acceleration can be found
from these components
a  a a
2
t
2
C
 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
Acceleration Directions
 If the angular acceleration and the
angular velocity are in the same
direction, the angular speed will
increase with time
 If the angular acceleration and the
angular velocity are in opposite
directions, the angular speed will
decrease with time
Forces Causing Centripetal
Acceleration
 Newton’s Second Law says that
the centripetal acceleration is
accompanied by a force
 FC = maC
 FC stands for any force that keeps an
object following a circular path
 Tension in a string
 Gravity
 Force of friction
Centripetal Force Example
 A ball of mass m
is attached to a
string
 Its weight is
supported by a
frictionless table
 The tension in the
string causes the
ball to move in a
circle
Centripetal Force
m v2
 General equation FC  m aC 
r
 If the force vanishes, the object will
move in a straight line tangent to the
circle of motion
 Centripetal force is a classification that
includes forces acting toward a central
point
 It is not a force in itself
Problem Solving Strategy
 Draw a free body diagram,
showing and labeling all the forces
acting on the object(s)
 Choose a coordinate system
that has one axis perpendicular to
the circular path and the other axis
tangent to the circular path
 The normal to the plane of motion is
also often needed
Problem Solving Strategy,
cont.
 Find the net force toward the
center of the circular path (this is the
force that causes the centripetal
acceleration, FC)
 Use Newton’s second law
 The directions will be radial, normal, and
tangential
 The acceleration in the radial direction will
be the centripetal acceleration
 Solve for the unknown(s)
Applications of Forces Causing
Centripetal Acceleration
 Many specific situations will use
forces that cause centripetal
acceleration
 Level curves
 Banked curves
 Horizontal circles
 Vertical circles
Level Curves
 Friction is the
force that
produces the
centripetal
acceleration
 Can find the
frictional force,
µ, or v
v  rg
Banked Curves
 A component of
the normal
force adds to
the frictional
force to allow
higher speeds
v2
tan  
rg
or ac  g tan 
Vertical Circle
 Look at the forces
at the top of the
circle
 The minimum
speed at the top
of the circle can
be found
v top  gR
Forces in Accelerating
Reference Frames
 Distinguish real forces from
fictitious forces
 “Centrifugal” force is a fictitious
force
 Real forces always represent
interactions between objects
Newton’s Law of Universal
Gravitation
 If two particles with masses m1
and m2 are separated by a
distance r, then a gravitational
force acts along a line joining
them, with magnitude given by
m1m2
FG 2
r
Universal Gravitation, 2
 G is the constant of universal
gravitational
 G = 6.673 x 10-11 N m² /kg²
 This is an example of an inverse
square law
 The gravitational force is always
attractive
Universal Gravitation, 3
 The force that
mass 1 exerts on
mass 2 is equal
and opposite to
the force mass 2
exerts on mass 1
 The forces form a
Newton’s third law
action-reaction
Universal Gravitation, 4
 The gravitational force exerted by
a uniform sphere on a particle
outside the sphere is the same as
the force exerted if the entire
mass of the sphere were
concentrated on its center
 This is called Gauss’ Law
Gravitation Constant
 Determined
experimentally
 Henry Cavendish
 1798
 The light beam
and mirror serve
to amplify the
motion
Applications of Universal
Gravitation
 Acceleration due
to gravity
 g will vary with
altitude
ME
gG 2
r
Gravitational Potential
Energy
 PE = mgy is valid only
near the earth’s
surface
 For objects high above
the earth’s surface, an
alternate expression is
needed
MEm
PE  G
r
 Zero reference level is
infinitely far from the
earth
Escape Speed
 The escape speed is the speed needed
for an object to soar off into space and
not return
2GME
v esc 
RE
 For the earth, vesc is about 11.2 km/s
 Note, v is independent of the mass of
the object
Various Escape Speeds
 The escape
speeds for
various members
of the solar
system
 Escape speed is
one factor that
determines a
planet’s
atmosphere
 Kepler’s Laws
 All planets move in elliptical orbits
with the Sun at one of the focal
points.
 A line drawn from the Sun to any
planet sweeps out equal areas in
equal time intervals.
 The square of the orbital period of
any planet is proportional to cube
of the average distance from the
Sun to the planet.
Kepler’s Laws, cont.
 Based on observations made by
Brahe
 Newton later demonstrated that
these laws were consequences of
the gravitational force between
any two objects together with
Newton’s laws of motion
 Kepler’s First Law
 All planets move
in elliptical orbits
with the Sun at
one focus.
 Any object bound
to another by an
inverse square law
will move in an
elliptical path
 Second focus is
empty
 Kepler’s Second Law
 A line drawn
from the Sun to
any planet will
sweep out
equal areas in
equal times
 Area from A to
B and C to D
are the same
 Kepler’s Third Law
 The square of the orbital period of any
planet is proportional to cube of the
average distance from the Sun to the
planet.
T  Kr
2 3

 For orbit around the Sun, K = KS =