DEC 1013ENGINEERING SCIENCES

13. CENTRIPETAL AND CENTRIFUGAL FORCE

NAZARIN B. NORDINnazarin@icam.edu.my

What you will learn:

• Centripetal force: acceleration, centrifugal force/ acceleration, mass-radius polygons

• Centrifugal force applied to wheel balancing/ clutches, governors

• Curved tracks: vehicles overturning/ sliding on level track, vehicles on banked track

• Angular displacement (q ) is usually expressed in radians, in degrees, or in revolutions.

ANGULAR DISPLACEMENT

57.30

One radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle.

1 rev = 3600 = 2p radians (rad)

1

2

3

4

56

6 segments getsto here.

2p segments gets completely around.

Thus the angle q in radians is given in terms of the arc length l it subtends on a circle of radius r by

rl

q

The radian measure of an angle is a dimensionless number.

THE ANGULAR SPEED

The angular speed (w ) of an object whose axis of rotation is fixed is the rate at which its angular coordinate, the angular displacement q, changes

with time. If q changes from qi to qf in a time t,

then the average angular speed is

tif qq

w

w = 2p f.

• f is the frequency in revolutions per second, rotations per second, or cycles per second.

• Accordingly, w is called the angular frequency. We can associate a direction with w and thereby create a vector quantity.

• The units of are exclusively rad/s. Since each complete turn or cycle of a revolving system carries it through 2p rad

w

THE ANGULAR ACCELERATION• The angular acceleration (a ) of an object whose

axis of rotation is fixed is the rate at which its angular speed changes with time.

• If the angular speed changes uniformly from wi

to wf in the time t, then the angular acceleration

is constant and

tif ww

a

The units of a are typically rad/s2, rev/min2, and such.

It is possible to associate a direction with w, and therefore with a, thereby specifying the angular acceleration vector a, but we will have no need to do so here.

tif ww

a

Equations for uniformly accelerated angular motion are exactly analogous to those for uniformly accelerated linear motion. In the usual notation we have:

t

tt

tt

if

if

i

if

fi

wwq

aqww

awq

awwwq

www

21

2

2

22

221

RELATIONS BETWEEN ANGULAR AND TANGENTIAL QUANTITIES:

• When a wheel of radius r rotates about an axis whose direction is fixed, a point on the rim of the wheel is described in terms of the circumferential distance s it has moved, its tangential speed v, and its tangential acceleration aT.

• These quantities are related to the angular quantities q, w, and a, which describe the rotation of the wheel, through the relations:

awq rarvrs T

• provided radian measure is used for q, w, and a. • By simple reasoning, s can be shown to be the

length of belt wound on the wheel or the distance the wheel would roll (without slipping) if free to do so.

• In such cases, v and aT refer to the tangential speed and acceleration of a point on the belt or of the center of the wheel.

awq rarvrs T

An object moving in a circle with constant speed, v, experiences a centripetal acceleration with: *a magnitude that is constant in time and

is equal to

*a direction that changes continuously in time and

always points toward thecenter of the circular path

Uniform Circular Motion

rva2

For uniform circular motion, the velocity is tangential to the circle and perpendicular to the acceleration

A circular motion is described in terms of the period T, which is the time for an object to complete one revolution.

Period and Frequency

2pr

f 1T

T 2prv

The distance traveled in one revolution is

The frequency, f, counts the number of revolutions per unit time.

r

The moon’s nearly circular orbit about the earth has a radius of about 384,000 km and a period T of 27.3 days. Determine the acceleration of the Moon towards the Earth.

Example of Uniform Circular Motion

T 2prv

v 2prT

a v2

r4p 2r2

T 2r4p 2rT 2

a 2.7210 3m / s2 g9.8m / s2

2.7810 4g

Uniform Circular MotionNewton’s 2nd Law: The net force on a body is equal to the product of the mass of the body and the acceleration of the body.

*The centripetal acceleration is caused by a centripetal force that is directed towards the center of the circle.

F ma mv2

r

ROTATIONAL INERTIA

• Law of inertia for rotating systemsAn object rotating about an axis tends to remain rotating at the same rate about the same axis unless interfered with by some external influence.

• Examples: bullet, arrow, and earth

• Demo – Football and spinning basketball• Demo - Whirly Tube (Zinger)• Demo – Whirly Shooter• Demo - Disc Gun• Demo - Rubber Bands

• Demo - Inertia Bars

• Moment of inertia (rotational inertia)

The sluggishness of an object to changes in its state of rotational motion

• Distribution of mass is the key.

• Example: Tightrope walker

CENTRIPETAL ACCELERATION• Centripetal acceleration (ac):• A point mass m moving with constant speed v around

a circle of radius r is undergoing acceleration. • The direction of the velocity is continually changing.• This gives rise to an acceleration ac of the mass,

directed toward the center of the circle. • We call this acceleration the centripetal acceleration;

its magnitude is given by

rvaC2

Because v = rw, we also have

rvac2

where w must be in rad/s.

rr 2)( w

2wrrr 22w

THE CENTRIPETAL FORCEThe centripetal force (Fc) is the force that must act on a mass m moving in a circular path of radius r to give it the centripetal acceleration v2/r. From F = ma, we have

2222

ww mrrrm

rvmmaFC

Where Fc is directed toward the center of the circular path.

• Centripetal force - center seeking force

• Examples: tin can and string, sling, moon and earth, car on circular path

CENTRIPETAL FORCE

• Demo - Coin on clothes hanger• Demo - String, ball, and tube• Demo - Loop the loop

CENTRIFUGAL FORCE

• Centrifugal force - center fleeing force

• Often confused with centripetal

• Examples: sling and bug in can

• Demo - Walk the Line

• Centrifugal force is attributed to inertia.

CENTRIFUGAL FORCE IN A ROTATING REFERENCE FRAME

• A frame of reference can influence our view of nature.

• For example: we observe a centrifugal force in a rotating frame of reference, yet it is a fictitious (pseudo) force.

• Centrifugal force stands alone (there is no action-reaction pair) - it is a fictitious force.

• Another pseudo force - Coriolis

THANK YOU