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Chapter 6
Circular Motion and Other Applications of Newton’s Laws
Circular Motion
 Objects traveling in circular paths
 Motion observed from an accelerating frame of reference
 Motion of an object through a viscous medium
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
2
v
 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.
Section 6.1
Conical Pendulum
The object is in e q u i l i b r i u m 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.
Tr
v
m
The maximum speed corresponds to the maximum tension the string can
withstand.
Section 6.1
Horizontal (Flat) Curve
The car as a particle in uniform circular motion in the
horizontal direction.
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.
The car as a particle in equilibrium in the vertical direction.
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.
T h e a n g l e of bank is found from
v2
tan 
rg
Section 6.1
Banked Curve, 2
T h e a n g l e is independent of the mass of the vehicle.
If the car rounds the curve a t l e s s than the design speed, friction is
necessary to keep it from sliding down the bank.
If the car rounds the curve a t m o r e than the design speed, friction is
necessary to keep it from sliding up the bank.
Section 6.1
Non-Uniform Circular Motion
The acceleration and
tangential components.
force
have
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
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 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 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 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
Example 1
Solution 1
Example 2
Solution 2
Solution 2
Example 3
Solution 3