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Chapters 7 & 8
The Law of Gravity
and
Rotational Motion
Newton’s Law of Universal
Gravitation

Every particle in the Universe
attracts every other particle with a
force that is directly proportional
to the product of the masses and
inversely proportional to the
square of the distance between
them.
m1m 2
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
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 actionreaction
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
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 =
2.97x10-19 s2/m3
K is independent of the mass of the
planet
Kepler’s Third Law

Period of a Satellite in a
Circular Orbit

Speed of a Satellite in
Circular Orbit


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
 av
 f   i 


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:
 av
 f   i 


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 Linear
and Angular Measures
Quantity
Linear
Angular
Relationship
Displacement
x (m)
θ (rad)
d=rθ
Velocity
v (m/s)
ω (rad/s)
v=rω
Acceleration
a (m/s2)
α (rad/s2)
a=rα
Centripetal Force
Key Question:
Why does a roller coaster stay on a track
upside down on a loop?
Centripetal Force


We usually think of acceleration as a change in
speed.
Because velocity includes both speed and
direction, acceleration can also be a change in
the direction of motion.
Centripetal Force


Any force that causes an object to move in a
circle is called a centripetal force.
A centripetal force is always perpendicular to an
object’s motion, toward the center of the circle.
Centripetal Force
Mass (kg)
Centripetal
force (N)
Fc = mv2
r
Linear speed
(m/sec)
Radius of path
(m)
Calculate centripetal force


A 50-kilogram passenger on an amusement park
ride stands with his back against the wall of a
cylindrical room with radius of 3 m.
What is the centripetal force of the wall pressing
into his back when the room spins and he is
moving at 6 m/sec?
Centripetal Acceleration


Acceleration is the rate at which an object’s
velocity changes as the result of a force.
Centripetal acceleration is the acceleration of an
object moving in a circle due to the centripetal
force.
Centripetal Acceleration
Centripetal
acceleration (m/sec2)
ac = v2
r
Speed
(m/sec)
Radius of path
(m)
Calculate centripetal
acceleration


A motorcycle drives around a bend with a 50meter radius at 10 m/sec.
Find the motor cycle’s centripetal acceleration
and compare it with g, the acceleration of
gravity.
Centrifugal Force
 We call an object’s tendency to
resist a change in its motion its
inertia.
 An object moving in a circle is
constantly changing its direction
of motion.


Although the centripetal force pushes you toward
the center of the circular path...
...it seems as if there also is a force pushing you
to the outside. This apparent outward force is
called centrifugal force.
Centrifugal Force
 Centrifugal force is not a true
force exerted on your body.
 It is simply your tendency to
move in a straight line due to
inertia.


This is easy to observe by twirling a small object
at the end of a string.
When the string is released, the object flies off in
a straight line tangent to the circle.
Force vs. Torque

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Forces cause accelerations
Torques cause angular
accelerations
Force 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, t, is the tendency of a
force to rotate an object about
some axis

t= r F


t is the torque
F is the force



symbol is the Greek tau
r is the length of the position vector
SI unit is N.m
Direction of Torque

Torque is a vector quantity



The direction is perpendicular to the
plane determined by the position
vector and the force
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 added


As 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
General Definition of
Torque, cont


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:

t  r F sin 

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
F is the force
r 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 
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