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

The centripetal acceleration, , is directed toward the center of the circle.

The centripetal acceleration is always perpendicular to the velocity.

2

c

va

r

ca

Section 6.1

Uniform Circular Motion, Force

A force, , 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

c

vF ma m

r

rF

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 theedge 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 msin tanv Lg

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.

The maximum speed corresponds to the maximum tension the string can withstand.

Trv

m

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:

Note, this does not depend on the mass of the car.

sv gr

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

tanv

rg

2

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.

2

2

1

bot

bot

mvF n mg

r

vn mg

rg

Section 6.1

Ferris Wheel, final

At the top of the circle, the force exerted on the object is less than its weight.

2

2

1

top

top

mvF n mg

r

vn mg

rg

Section 6.1

Non-Uniform Circular Motion

The acceleration and force have tangential components.

produces the centripetal acceleration

produces the tangential acceleration

The total force is

r t F F F

tF

rF

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.2

cosv

T mgRg

Section 6.2

Top and Bottom of Circle

The tension at the bottom is a maximum.

The tension at the top is a minimum.

If Ttop = 0, then

2

1topvT mg

Rg

2

1botvT mg

Rg

topv 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

The non-inertial observer (b) sees

These are equivalent if Ffictiitous = ma

sin

cos 0

x

y

F T ma

F T mg

' sin

' cos 0

x fictitious

y

F T F ma

F T mg

Section 6.3