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Chapter 6
Circular Motion and
Other Applications of Newton’s Laws
Circular Motion and Other Applications of Newton’s Laws
Circular Motion
Two analysis models using Newton’s Laws of Motion have been developed.
The models have been applied to linear motion.
Newton’s Laws can be applied to other situations:
 Objects traveling in circular paths
 Motion observed from an accelerating frame of reference
 Motion of an object through a viscous medium
Many examples will be used to illustrate the application of Newton’s Laws to a
variety of new circumstances.
Introduction
Uniform Circular Motion, Acceleration
A particle moves with a constant speed in a circular path of radius r with an
acceleration.
The magnitude of the acceleration is given by
v2
ac 
r
 The centripetal acceleration, a c , is directed toward the center of the circle.
The centripetal acceleration is always perpendicular to the velocity.
Section 6.1
Uniform Circular Motion, Force
A force, Fr , is associated with the
centripetal acceleration.
The force is also directed toward the
center of the circle.
Applying Newton’s Second Law along
the radial direction gives
v2
 F  mac  m r
Section 6.1
Uniform Circular Motion, cont.
A force causing a centripetal
acceleration acts toward the center of
the circle.
It causes a change in the direction of
the velocity vector.
If the force vanishes, the object would
move in a straight-line path tangent to
the circle.
 See various release points in the
active figure
Section 6.1
Centripetal Force
The Force causing ac is sometimes called Centripetal Force (Center directed
force).
So far we know forces in nature:
 Friction, Gravity, Normal, Tension.
Should we add Centripetal Force to this List?
NO!!!!!
Force causing ac is NOT a new kind of force!
 It is a New Role for force!!!
It is simply one or more of the forces we know acting in the role of a force that
causes a circular motion.
Book will not use the term Centripetal Force!!!
Centripetal Force as New Role
Earth-Sun Motion:
 Centripetal Force ≡ Gravity
Object sitting on a rotating turntable:
 Centripetal Force ≡ Friction
Rock-String (horizontal plane):
 Centripetal Force ≡ Tension
Wall-Person (rotating circular room):
 Centripetal Force ≡ Normal
Ferris Wheel (lowest point):
 Centripetal Force ≡ Normal – Gravity
Ferris Wheel (highest point):
 Centripetal Force ≡ Normal – Gravity
Rock-String (vertical plane):
 Centripetal Force ≡ Tension ± Gravity
Centrifugal Force
Centrifugal Force (Outward) is
Another Misconception
Force on the ball is
NEVER outward ≡ Centrifugal Force
Force is ALWAYS inward
If Centrifugal Force existed, the ball would
Fly Off as in (a) when released.
Ball Flies off as in figure (b).
Similar to sparks flying in straight line from the
edge of rotating grinding wheel.
(a)
(b)
Conical Pendulum
The object is in equilibrium in the
vertical direction .
It undergoes uniform circular motion in
the horizontal direction.
 ∑Fy = 0 → T cos θ = mg
 ∑Fx = T sin θ = m ac
v is independent of m
v  Lg sin tan
Section 6.1
Motion in a Horizontal Circle
The speed at which the object moves depends on the mass of the object and the
tension in the cord.
The centripetal force is supplied by the tension.
v
Tr
m
The maximum speed corresponds to the maximum tension the string can
withstand.
Section 6.1
Horizontal (Flat) Curve
Model the car as a particle in
uniform circular motion in the
horizontal direction.
Model the car as a particle in
equilibrium in the vertical
direction.
The force of static friction
supplies the centripetal force.
The maximum speed at which
the car can negotiate the curve
is:
v  s gr
 Note, this does not depend on
the mass of the car.
Section 6.1
Banked Curve
These are designed with friction
equaling zero.
Model the car as a particle in
equilibrium in the vertical direction.
Model the car as a particle in uniform
circular motion in the horizontal
direction.
There is a component of the normal
force that supplies the centripetal force.
The angle of bank is found from
v2
tan 
rg
Section 6.1
Banked Curve, 2
The banking angle is independent of the mass of the vehicle.
If the car rounds the curve at less than the design speed, friction is necessary to
keep it from sliding down the bank.
If the car rounds the curve at more than the design speed, friction is necessary to
keep it from sliding up the bank.
Section 6.1
Ferris Wheel
The normal and gravitational forces act
in opposite direction at the top and
bottom of the path.
Categorize the problem as uniform
circular motion with the addition of
gravity.
 The child is the particle.
Section 6.1
Ferris Wheel, cont.
At the bottom of the loop, the upward
force (the normal) experienced by the
object is greater than its weight.
mv 2
 F  nbot  mg  r
 v2 
nbot  mg  1  
rg 

Section 6.1
Ferris Wheel, final
At the top of the circle, the force
exerted on the object is less than its
weight.
mv 2
 F  ntop  mg  r
 v2

ntop  mg   1
 rg

Section 6.1
Non-Uniform Circular Motion
The acceleration and force have
tangential components.
Fr produces the centripetal
acceleration
Ft produces the tangential acceleration
The total force is
F  F  F
r
t
Section 6.2
Vertical Circle with Non-Uniform Speed
The gravitational force exerts a
tangential force on the object.
 Look at the components of Fg
Model the sphere as a particle under a
net force and moving in a circular path.
 Not uniform circular motion
The tension at any point can be found.
 v2

T  mg 
 cos 
 Rg

Section 6.2
Top and Bottom of Circle
The tension at the bottom is a maximum.
2
 v bot

T  mg 
 1
Rg


The tension at the top is a minimum.
2
 v top

T  mg 
 1
 Rg



If Ttop = 0, then
v top  gR
Section 6.2
Motion in Accelerated Frames
A fictitious force results from an accelerated frame of reference.
 The fictitious force is due to observations made in an accelerated frame.
 A fictitious force appears to act on an object in the same way as a real force,
but you cannot identify a second object for the fictitious force.
 Remember that real forces are always interactions between two objects.
 Simple fictitious forces appear to act in the direction opposite that of the
acceleration of the non-inertial frame.
Section 6.3
“Centrifugal” Force
From the frame of the passenger (b), a force
appears to push her toward the door.
From the frame of the Earth, the car applies
a leftward force on the passenger.
The outward force is often called a
centrifugal force.
 It is a fictitious force due to the
centripetal acceleration associated with
the car’s change in direction.
In actuality, friction supplies the force to
allow the passenger to move with the car.
 If the frictional force is not large enough,
the passenger continues on her initial
path according to Newton’s First Law.
Section 6.3
“Coriolis Force”
This is an apparent force caused by changing the radial position of an object in a
rotating coordinate system.
The result of the rotation is the curved path of the thrown ball.
From the catcher’s point of view, a sideways force caused the ball to follow a
curved path.
Section 6.3
Fictitious Forces, examples
Although fictitious forces are not real forces, they can have real effects.
Examples:
 Objects in the car do slide
 You feel pushed to the outside of a rotating platform
 The Coriolis force is responsible for the rotation of weather systems,
including hurricanes, and ocean currents.
Section 6.3
Fictitious Forces in Linear Systems
The inertial observer models the sphere
as a particle under a net force in the
horizontal direction and a particle in
equilibrium in the vertical direction.
The non-inertial observer models the
sphere as a particle in equilibrium in
both directions.
The inertial observer (a) at rest sees
 Fx  T sin  ma
F
y
 T cos   mg  0
The non-inertial observer (b) sees
F '
F '
x
 T sin  Ffictitious  ma
y
 T cos   mg  0
These are equivalent if Ffictiitous = ma
Section 6.3