CIRCULAR MOTION

Course: Diploma

Subject: Applied Science Physics

Unit: IV

Chapter: II

Circular motion

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

Let T be the time it takes for the object to travel once around the circle.

T=period of the circular motion

2 r

T

Angular velocity

the angular velocity is defined as the rate of change of angular displacement and is a vector quantity which specifies the angular speed (rotational speed) of an object and the axis about which the object is rotating.

Angular velocity can be expressed as :ω = θ / twhere

ω= angular velocity (rad/s)

θ = angular distance (rad)

t = time (s)

rad=radians

Frequency and PeriodIf we assume an object is continuously rotating, then

another way to look at rotational motion is to examine the period of rotation, T.  

Measurable in units of time (milliseconds, second, hours, years, eons…) the period is how much time is takes to make one complete rotation.  

The frequency, f, of an object is actually the inverse of the period of rotation.

Frequency and PeriodT=1/f

and

f =1/TThe metric unit for frequency is Hertz (Hz), where 1

Hertz = 1 cycle/second.  You are probably familiar with the term Hertz.

Relation between angular velocity, period and frequency

Angular frequency, f, is defined as the number of circular revolutions in a given time interval. It is commonly measured in units of Hertz (Hz), where 1 Hz = 1 s–1. For example, the second hand on a clock completes one revolution every 60 seconds and therefore has an angular frequency of 1 /60 Hz.

The relationship between frequency and angular velocity is:

Where,ω=angular velocity

f=angular frequency

f2

Now, Angular period,

Therefore, relation between angular velocity & angular period is,

1T

f

1f &

T

f2

1

T 22

T

Angular Acceleration

Angular acceleration is the rate of change in angular velocity. (Radians per sec per sec.)

The angular acceleration can also be found from the change in frequency, as follows:

2 ( ) 2

fSince f

t

2 Angular acceleration (rad/s )t

Centripetal Acceleration and Angular Velocity

The angular velocity and the linear velocity are related (v = ωr)

The centripetal acceleration can also be related to the angular velocity

2 2 2

2

2c

2 r 1v & f

T Tv 2 fr & 2 f

v r

v r

v 1a & f

t t 2

v ra

2 2

a 2 a r

Centripetal Force

We should remember from that motion is always caused by some kind of force.

The same is true for circular motion

Centripetal Force is any force that causes an object to follow a circular path.

Centripetal Force

Centripetal force can be thought of as a “center seeking” force.

Centripetal force is always directed toward the center

Measuring Centripetal Force

We know that Force is always F=m(a)Centripetal force is always measured as

follows---->

Try it!

A car is rounding a curve at 65 m/s. The mass of the car is 1500 kg. The centripetal force of the car going around the curve is 28 N. What is the radius of the curve?

Centrifugal Force

Just like we have an inward seeking force with circular motion, we can also have an outward seeking motion.

This outward seeking motion is called Centrifugal force.

Centrifugal ForceCentrifugal force can be seen if we tie a string to

a bucket and fill the bucket with water.The force holding the water in the bucket and

keeping it from falling out is Centrifugal Force. Centrifugal force is measured the same way we

measure centripetal forceThe only difference is the fact that our direction

is outward and not inward.

Example 1: A rope is wrapped many times around a drum of radius 50 cm. How many revolutions of the drum are required to raise a bucket to a height of 20 m?

h = 20 m

RƟ = 40 rad

Now, 1 rev = 2π rad

Ɵ = 6.37 revƟ = 6.37 rev

1 rev40 rad

2 rad

20 m

0.50 m

s

R

Example 2: A bicycle tire has a radius of 25 cm. If the wheel makes 400 rev, how far will the bike have traveled?

Ɵ = 2513 rad

s = Ɵ R = 2513 rad (0.25 m)

s = 628 ms = 628 m

2 rad400 rev

1 rev

Example 3: A rope is wrapped many times around a drum of radius 20 cm. What is the angular velocity of the drum if it lifts the bucket to 10 m in 5 s?

h = 10 m

R

ω = 10.0 rad/sω = 10.0 rad/s

ω = = Ɵt

50 rad5 s

Ɵ = 50 rad10 m

0.20 m

s

R

Example 4: In the previous example, what is the frequency of revolution for the drum? Recall that ω = 10.0 rad/s.

h = 10 m

R

f = 95.5 rpmf = 95.5 rpm

2 or 2

f f

10.0 rad/s1.59 rev/s

2 rad/revf

Or, since 60 s = 1 min:

rev 60 s rev1.59 95.5

1 min minf

s

Example 5: The block is lifted from rest until the angular velocity of the drum is 16 rad/s after a time of 4 s. What is the average angular acceleration?

h = 20 m

R

a = 4.00 rad/s2a = 4.00 rad/s2

f o fort t

2

16 rad/s rad4.00

4 s s

REFERENCE BOOKS AUTHOR/PUBLICATION

ENGINEERING PHYSICSB.L.THERAJA, S. CHAND

PUBLISHERS

References

1. http://www.engineeringtoolbox.com/motion-formulas-d_941.html

2. http://kleins-klasses.wikispaces.com/file/view/Centripetal+and+Centrifugal+Forces.ppt