ROTATIONAL MOTION

CHAPTER 7 ROTATIONAL MOTION 7.1 Rotational Motion Parameters 7.2 Rotational Motion With Angular Acceleration 7.3 Relationship Between Linear and Rotational Motion Parameters

INTRODUCTION •Definition

“The act or process of turning around a centre or an axis”

•Uniform circular motion is the motion in which there is no change in speed, only a change in direction.

• There are many cases of objects moving in a curve or circular path about some point, such as bicycles or cars turning round corners.

7.1 Rotational Motion Parameters a) Angular Position/ Angular Displacement,  A particle P moves in a circle. At a particular constant, it has an angular position of  from its starting point. This is known as the angular displacement , .

Angular position, 

s

The definition:

 = the angle through which a rigid body/object rotates about a fixed axis. s r   360o , s  2  r

  , s  r

to complete 360o ,

s 2 r    360    2  rad r r unit of  : rad , revolution, or degree o

v Arc length, s  r  For complete circle : 1 rev  360 o  2  rad 1 rad  57.3o 3

b) Angular Velocity,  The definition of ,

 = the rate of the change of angular displacement 

   t t

2 2

 1 t1

if the angular velocity is constant,  will be a value, so; 

constant

 t

Unit : radian per second (rad s -1 ), -1

revolution per minutes (rev min )

r

Relation between linear velocity and angular velocity vr



Vector 

r



s t

 t

where  

s r 4

c) Angular Acceleration, The definition of ,

 = a rate of change of angular velocity of a revolving particle.  

2 1 t

2

t1

Unit : rad/s 2 , rev/s 2 , rev/min 2

Relation between linear acceleration, a and angular acceleration,  a  r

 in rad / s

SUM M ARY a ) s  r b) v  r c) a  r 

http://mriquestions.com/angular-frequency-omega.html

d) Period of Rotation The definition

T = the time taken to rotate through one round.

2  T

so

,T 

Period

2



unit : sec ond , minute, hour

e) Frequency or Revolution The definition

f = the number of rotation performed per frequency * let an object rotates n times in t seconds, then f 

n t

or t 

n f

* to complete one round, time taken is T seconds * to complete n rounds, time t taken is 

t  nT hence ,

nT 

n f

 f 

1 T

  2 f  f 2

SI unit : hertz Hz  or s 1

6

7.2 ROTATIONAL MOTION WITH CONSTANT ANGULAR ACCELERATION The equations of motion for constant angular acceleration are the same as

those for linear motion, with the substitution of the angular quantities for the linear ones.

LINEAR

ANGULAR

v  u  at

 f  i   t

1 u  v  t 2 1 s  ut  a t 2 2 v2  u 2  2 a s

 

s

1 i   f  t 2 1   i t   t 2 2  f 2  i2  2 

a)

s = r

b)

v = r

c)

a = r

7.3 RELATIONSHIP BETWEEN LINEAR & ROTATIONAL MOTION PARAMETERS If a reference line on a rigid body rotates through an angle q, a point within the body at a position r from the rotation axis moves a distance s along a circular arc, where s is given by: s  r (radian measure)

Differentiating the above equation with respect to time—with r held constant— leads to v  r  (radian measure)

The period of revolution T for the motion of each point and for the rigid body itself is given by T

2r v

Substituting for v we find also that

T

2



(radian measure)

Differentiating the velocity relation with respect to time : again with r held constant : leads to

a t   r (radian measure)

Here,



d dt

Note that dv / dt  at represents only the part of the linear acceleration that is responsible for changes in the magnitude v of the linear velocity. Like v, that part of the linear acceleration is tangent to the path of the point in question.

Also, the radial part of the acceleration is the centripetal acceleration given by

v2 ar    2 r r

Table 1 : Relation between linear and rotational quantities.

Example;

What is the angular size in radians as shown in the Figure.

Example;

s   r 6  10  0.6 rad

r = 10 m 

s=6m

A boy on a bicycle is traveling at 10 m/s. What is the angular speed of a point on the tire of the bicycle if the radius is 34 cm?

given v  10m / s , r  0.34 m ,   ? from v  r  v  r 10  0.34  29.4 rad / s

11

Example; A object undergoes circular motion with uniform angular speed 100 r.p.m.

Determine a) the period b) the frequency of revolution

given  = 100 rpm a)



100 rev 2  rad 1 min   min 1 rev 60 s

 10 .4 7 rads 1 from   T 

b)

2 T 2



 0.6 s

1 T  1.67 Hz

f 

Example;

Two lorries are going around two different circular paths at the same angular velocity. The speed of one lorry is 50 km/h on a track of radius R. What is the speed of the other lorry if its track has a radius of ¼ R?

given v1 13.87 m / s , r1  R , r2  1 R 4 v1  r1 

1  2 v1 v2  r1 r2 v2  r2

v1 r1

R  13.87 m / s    4 R   3.47 m / s 

v1  r1 

 

v1 13.87 m / s  R r1

R 4 R  13.87 m / s     R 4   3.47 m / s

v2  r2  , r2 

Example: A particular bird’s eye can just distinguish objects which subtended an angle not smaller than about 3 x 10-4 rad. How smaller an object can the bird just distinguish when flying at height of 100 m?

Given

From

r = 100 m  = 3 x 10-4 rad s = …..

r 

s

s  r  (100 m)(310  4 rad )  3 cm

Example: A car moves with tangential linear velocity of 50km/h in circle of radius 300m. Determine the angular velocity of the car as it goes round the circle.

Given r = 300m v = 50km/h = 13.9m/s 

v 13.9 m / s  r 300 m  0.016 rad / s

Youtube link for reference https://www.youtube.com/watch?v=fmXFWi-WfyU https://www.youtube.com/watch?v=grMWAI1RdVs

Example on youtube https://www.youtube.com/watch?v=6uHQP-axHM8 https://www.youtube.com/watch?v=0El-DqrCTZM https://www.youtube.com/watch?v=o8LvRDmozaA