Course Outline :

•Analysis Model: Particle in Uniform Circular Motion

•Tangential and Radial Acceleration

•Extending the Particle in Uniform Circular Model

Chapter 6 Circular Motion and Other

Applications’ of Newton’s Laws

6.1 Uniform circular motion

Uniform circular motion occurs when an object moves in a

circular path with a constant speed

An acceleration exists since the direction of the motion is

changing

This change in velocity is related to an acceleration

The velocity vector is always tangent to the path of the

object

Changing Velocity in Uniform Circular

Motion

The change in the velocity

vector is due to the

change in direction

The vector diagram

shows Dv = vf - vi

Centripetal Acceleration

The acceleration is always perpendicular to the path of the

motion

The acceleration always points toward the center of the circle

of motion

This acceleration is called the centripetal acceleration

Centripetal Acceleration, cont

The magnitude of the centripetal acceleration vector is given

by

The direction of the centripetal acceleration vector is always

changing, to stay directed toward the center of the circle of

motion

2

C

va

r

Period

The period, T, is the time required for one complete

revolution

The speed of the particle would be the circumference of the

circle of motion divided by the period

Therefore, the period is

2 rT

v

4.5 Tangential Acceleration

The magnitude of the velocity could also be changing

In this case, there would be a tangential acceleration

Total Acceleration The tangential

acceleration causes the

change in the speed of

the particle

The radial acceleration

comes from a change in

the direction of the

velocity vector

Total Acceleration, equations

The tangential acceleration:

The radial acceleration:

The total acceleration:

Magnitude

t

da

dt

v

2

r C

va a

r

2 2

r ta a a

Total Acceleration, In Terms of Unit

Vectors

Define the following unit

vectors

r lies along the radius vector

q is tangent to the circle

The total acceleration is

ˆˆ andr q

2

ˆ ˆt r

d v

dt rq v

a a a r

Analysis Model: Particle in Uniform Circular Motion

Based on Newton’s 2nd Law, there must have net force acting

on the particle to cause the acceleration. Other wise, the

particle will remain at rest or move with constant velocity.

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

Section 6.1

Example (6.1) : The Conical Pendulum

A small object of mass m is suspended from a string of length L. The object revolves with constant

speed v in a horizontal circle of radius r (Figure (6.3)). (Because the string sweeps out the surface of

a cone, the system is known as a conical pendulum). Find an expression for v.

Example (6.2) : How Fast Can it Spin?

A ball of mass 0.500 kg is attached to the end of a cord 1.50 m long. The ball is whirled in a horizontal

circle as was shown in Figure (6.1). If the cord can withstand a maximum tension of 50.0N, what is

the maximum speed the ball can attain before the cord breaks? Assume that the string remains

horizontal during the motion.

Example 1: A car of mass m round a curve on a flat, horizontal road of

radius R. If the coefficient of static friction between tires and the road is μs,

what is the maximum speed vmax at which the car can take the curve without

sliding?

The force that enables the car to move in curve is the static friction force

because no slipping occurs at the point of contact between road and tires. If

the car were on an icy road, it would only move in straight line

6.2 Non-Uniform Circular Motion

The acceleration and force have tangential components.

produces the centripetal acceleration

produces the tangential acceleration

The total force is

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.

Section 6.2

Example 3: A particle of mass m attached to the end of a cord of length l whirls in a

vertical circle. Find the tension in the cord and the tangential acceleration when the

speed of the particle is v and the cord makes an angle θ with the vertical.

Note : The tension, friction, normal, gravitational, electric, and

magnetic forces are examples of possible centripetal forces that

enable an object to move in circular path.