Dynamics of Circular Motion

♦ Types of Circular Motion♦ Centripetal Force♦ Centripetal Acceleration♦ Horizontal Circle♦ Vertical Circle♦ Banking of Curves♦ Conical Pendulum

CIRCULAR MOTION- Motion of a body in a curved path

TYPES: Uniform Circular Motion, UCM – motion with

constant speed Ex. A car rounding an oval at 30 KPH

Non-Uniform circular Motion, NUCM – a roller coaster car that slows down and speeds up as it moves around a vertical loop.

DEFINITION OF UNIFORM CIRCULAR MOTION

Uniform circular motion is the motion of an object traveling at a constant speed on a circular path.

V

VS

r

Conversion:1 rev = 360° = 2π radc

Centripetal Acceleration

r

tv

v

v

r

v

t

v 2

r

vac

2

S

Concept in Uniform Circular Motion

A Body Moving In Circular Path on a Flat Horizontal Surface

Vertical Circular Motion

r

vmmgFN

21

1

r

vmmgFN

23

3

r

vmFN

22

2

r

vmFN

24

4

Definition of Terms

r vr

T

2

Period, T - time it takes for the object totravel once around the circle. (sec)

Frequency, f – no. of cycles/revolutions per unit time (rev/sec, cycles/sec, hertz)

𝑻 = 1/𝒇

aFm

mF

a

Recall Newton’s Second Law

When a net external force acts on an objectof mass m, the acceleration that results is directly proportional to the net force and hasa magnitude that is inversely proportional tothe mass. The direction of the acceleration isthe same as the direction of the net force.

Thus, in uniform circular motion there must be a netforce to produce the centripetal acceleration.

The centripetal force is the name given to the net force required to keep an object moving on a circular path.

The direction of the centripetal force always points towardthe center of the circle and continually changes direction as the object moves.

r

vmmaF cc

2

Banked CurvesOn an unbanked curve, the static frictional forceprovides the centripetal force.

Car Rounding a Flat Curve at Constant Speed

Banked CurvesOn a frictionless banked curve, the centripetal force is the horizontal component of the normal force. The vertical component of the normal force balances the car’s weight.

r

vmnFc

2

sin

mgn cos

Banked Curves ( Neglecting Friction)

rg

v2

tan

n

n cos θ

Banked Curves

Example : The Daytona 500

The turns at the Daytona International Speedway have a maximum radius of 316 m and are steely banked at 31degrees. Suppose these turns were frictionless. As what speed would the cars have to travel around them?

rg

v2

tan tanrgv

mph 96 sm4331tansm8.9m 316 2 v

Summary