Circular Motion

CIRCULAR MOTION- A type of motion exhibited by bodies following a circularcurved path

- Motion caused by centripetal force

Related Terms

Rotation ndash motion about an internal axis

Revolution ndash motion about an external axis

The earth revolves around the sun It takes earth a year to complete one revolution around the sun

To differentiate for example

The earth rotates on its own axis This is the reason why we are experiencing night and day

Basic Rotational Quantities The angular displacement is

defined by

For a circular path it follows that the angular velocity is

and the angular acceleration is

where the acceleration here is the tangential acceleration

Basic Rotational Quantities

Angular velocity can be considered to be a vector quantity with direction along the axis of rotation in the right-hand rule sense

Angular velocity For an object rotating about an axis every point on the

object has the same angular velocity The tangential velocity of any point is proportional to its distance from the axis of rotation Angular velocity has the units rads

Angular velocity is the rate of change of angular displacement and can be described by the relationship

1 What is the angular velocity in rads of the second hand of a watch

Sample Problem

Answer ω = 2πrad 60s = 010 rads

2 A bicycle travels 141 m along a circular track of radius 15 m What is the angular displacement in radians of the bicycle from its starting position1048712

Answer θ = 141m 15m = 94 rad

Centripetal Force

Any motion in a curved path represents accelerated motion and requires a force directed toward the center of curvature of the path This force is called the centripetal force which means center seeking force

where

m = mass of the object

v = tangential velocity

r = radius of the curved path

Centripetal Force

The straight line motion in the absence of the constraining force (tension) is an example of Newtonrsquos first law of motion

Centripetal Acceleration The centripetal acceleration can be derived for the

case of circular motion since the curved path at any point can be extended to a circle

Centripetal Acceleration

Problem 3

A 200g object is tied to the end of a cord and whirled in a horizontal circle of radius 120m at a constant 30 revs Assume that the cord is horizontal determine

(a) the acceleration of the object and

(b) the tension in the cord

222

)6)(201( sradmrrvac

Solution 3a The object is not accelerating tangentially to the circle but is undergoing a radial or centripetal acceleration

22

rrvac

2426 smac

where ω must be in rads then 30 revs = 60πrads

b To cause the acceleration found in (a) the cord must pull on the 020 kg mass with a centripetal force

)426)(200( 2smkgmaF cc

NFc 85

This is the tension in the cord

Centripetal force on banked highway curve The centripetal force is

proportional to the square of the velocity eg doubling of speed will require four times the centripetal force to keep the motion in a circle If centripetal force must be provided by friction alone on a curve an increase in speed could lead to an unexpected skid if friction is insufficient

Working Equations

Since centripetal force is provided by friction

If Ff = microFN and FN = mg then

The minimum speed required to make a turn

rmvFF fc

2

rmvmg

2

grv

Also the equation of micro = tanθ therefore

rmvmg

2

)(tan

grv 2

1tan

where θ is the banking angle

A car enters a horizontal curved roadbed of radius 50 m The coefficient of static friction between the tires and the roadbed is 020 What is the maximum speed with which the car can safely negotiate the unbanked curve

Problem 4

)50)(89)(200( grv

Solution

smv 899

Determine the minimum angle at which a roadbed should be banked so that a car traveling at 20ms can safely negotiate the curve if the radius of the curve is 200m

Problem 5

Solution

)200)(89()20(tantan

21

21 grv

511

Motion in a Vertical Circle

Motion in a Vertical Circle The motion of a mass on a string in a vertical

circle includes a number of mechanical concepts 1 It must satisfy the constraints of

centripetal force to remain in a circle2 It must satisfy the demands of

conservation of energy as gravitational potential energy is converted to kinetic energy when the mass moves downward

3 The velocity must increase as the mass moves downward from the top of the circle subject to the constraints stated

Motion in a Vertical Circle

Using the centrifugal force conditions the tension at the bottom can be related to the tension at the top

A 075-kg ball is attached to a 10-m rope and whirled in a vertical circle The rope will break when the tension exceeds 450 N What is the maximum speed the ball can have at the bottom of the circle without breaking the rope

Problem 6

Solution mgr

vmT bottombottom

2)(

smv 24750

)1()89)(750()450(

Circular Orbit

The force of gravity in keeping an object in circular motion is an example of centripetal force Since it acts always perpendicular to the motion gravity does not do work on the orbiting object if it is in a circular orbit

What is the acceleration due to gravity at an altitude of 100 times 106 m above the earths surface (Note the radius of the earth is 638 times 106 m)

Problem 7

Solution2

66

62

)10386101(10386)89(

xxx

rR

gg planetsurfaceorbit

2327 smgorbit

Assignment

1 In an amusement park ride a small child stands against the wall of a cylindrical room that is then made to rotate The floor drops downward and the child remains pinned against the wall If the radius of the device is 215 m and the relevant coefficient of friction between the child and the wall is 0400 with what minimum speed is the child moving if he is to remain pinned against the wall

2 An indoor track is to be designed such that each end is a banked semi-circle with a radius of 24 m What should the banking angle be for a person running at speed v = 60 ms

3 Outdoor Activity in ldquoMotionrdquo- tell your experience about motion in an amusement park- include your picture in ldquomotionrdquo- written output must be 1-2 pages only and must be compiled by group (lab group) in a clear book- deadline before midterm exam- other option experience motion within your vicinity