Name	Symbol	Formula
One rotation		2π  radians.
Angular displacement	θ	θ =  d/ r.
Angular velocity	ω	ω =   θ / t
Angular acceleration	α	α   =    (  ω f – ω i )   /  t.
Torque	τ	τ =  F x r.
Inertia	I	I = τ /

Fig. 1  Rotating points

In translational (linear) motion  or circular motion an object has a definite velocity. For example a ball tied to a string and turned around.  In rotation, look at the diagram, different coloured objects rotate at different speeds counterclock  wise making 5 revolution per second. Observe the objects V,B and R . Are they moving at the same velocity? No. R does a large circle in 1/5 th of a second while B and V go along shorter and shorter paths.

Radian Measure

The method for measuring rotation is to measure the change of angle in a radial line as shown by ‘theta’ θ .

Fig.2 Radian defined.

This change in rotation θ can be measured in two ways. It could be measured in degrees or it can be measured in Radians. What is a Radian?

This is a varying length depending on the size of the circle. In a large circle the radian is longer as the radius is longer.

One Radian is the distance at the circumference equal to its radius.

It is useful to know the total number of radians on the whole circumference.

The circumference of a circle =  2 π r.

One radian = r

  Total number of radians in one turn of the circle is =  2 π r. /  r   = 2 π .

Angular  Velocity.

If the disc does 5 revolutions per second,  it will amount to  = 5 x 2 π

If the time taken is 5 seconds  the Velocity of rotation =  5 x 2 π   /  5 Rad s-1.

This is usually denoted by the simple letter for omega .   ω   ( a curly w )

      . ω  = θ / time.

Angular Acceleration

As acceleration is the rate of change of velocity

. α   =    (  ω f – ω i )   /  time

Example   1.

63 m of wire is pulled off a reel attached to an axle in 20 s.  The reel had a  radius of 0.10 m.

Find the following:-

a.      Angular displacement.

b.      The number of rotations of the reel.

c.      Angular velocity of the drum.

d.     Average linear speed of the wire.

ANSWERS

a. Linear displacement of wire =63 m

       Converting to  radians  = 63/ radius

                                  = 63/ 0.10  =  630 rad.

b.      Number of rotations =  displacement / circumferemce.

                                            =   63  /  2 π r =  63/ 6.28 x 0.1  = 100. 32.

c.      Angular velocity  =  Angular displacement / time

                         =   630 / 20   = 3.1 rad. S-1.

Example 2.

The blade of a  ceiling fan is 0.5m long. At a low speed it does 15 turns a second .  When switched to high in 30 second it does   45 turns per second.

Find the Angular acceleration.

   ω I   =   2 π x 15 =  94.2

  ω f    = 2 π x 45   =  282.6

. α  =        (  ω f – ω i )   /  time

      =  282.6 – 94,2  /  t     =   188.4 / 30  = 6.28 rad s-2

Torque

Fig. 3 Forces acting on a fixed body.

The diagram shows forces, in coloured arrows acting on a body attached to an axle. What can you say about them shown in different colours?

Blue -  can rotate the body in a clockwise direction.

Red – Capable of rotating in the opposite direction.

Black – they will not rotate the body as they are passing through the axle.

Blue and red forces will have torque effect while the black will have zero torque.  A torque is a force that will have a turning effect on a body.

Torque is the product of force and the distance to axle . ( radius of the turning circle)

Fig. 4 What numbered position will provide the maximum torque?.

There is a piece of iron placed on a float.  The magnet, marked as NS exerts an attractive force on the iron. The body gets pulled to the magnet without a twist.

Explain what may happen when shifted to numbered spots.

From this simple experiment we realise it is not only the magnitude and direction of a force that affects the motion of an object but the distance from force to the axle or the centre of mass is also crucial.

Torque considers all these 3  factors, namely direction, magnitude and the distance from the axle.

Thus the formula   Torque =  Force x radius. ( Radius being the distance to the rotating axis.    

   . τ  = F r

Answer=  Number 1 position. ( Reason- highest r value.)

Rotational Inertia

Inertia is the reluctance to accelerate. For linear motion, mass is the only factor that determines inertia.  In rotatory motion it is the unbalanced torque that causes acceleration and inertia is what offers reluctance.

In rotation inertia depends not only on mass but , where the mass of an object is more.

If a rotating balet dancer stretches the hands out the speed will reduce showing inertia change. If she folds the hands again she will speed up.

Even in regular objects there is no single formula to detrmine Inertia.

Object shape	Axis of Rotation	Formula for I.
Hollow cylinder	at the centre.	I = mr2
Solid wheel	Axis at the centre	I = 1/2 mr2
Thin rod	At end points.	I = 1/3 mr2
Sphere (Hollow)	Axis at the centre	I = 2/3 mr2

Rotational Inertia = Torque / angular acceleration.

Example 3.

The mass of 1.0 kg is accelerating down at 2.0 ms-2  turning the green wheel. The radius of the wheel is 0.16 m.

Find the following:-

                          .a.  Tension force in the string.

                           .b. The torque on the wheel.

                            ,c. The angular acceleration of the wheel.

.d. Rotational Inertia of the wheel.

.a.

Effective down force =  weight – tension. Fu = mg – Ft.

As the downward acceleration is due to downward force  Fu = ma.

                                             Fu = 1.0 x 2

                                   Ft = mg – Fu

                                   Ft = 1 x 9.8  -1  x2

                                        =  9.8 -2 = 7.8 N.  

.b.   The force turning the wheel is the tension force.

    τ  = F r

  τ  =  7.8 x 0.16  = 1.25 Nm.

.d.  Rotational Inertia  =   Torque/ angular acceleration.

                                   I   =  τ  / α

                              I  = 1.25 / 12.5 =   0.01 kgm².

 

 1.1 What is a Radian ?     .a – An angle.   .b. A distance.  .c – A force  .d. Energy.

1.2 How many radians are there in a single rotation?

.a.  2 π.      .b -  2 π r.    .c -  360 ˚ .    .d -  57.

1.3  How much is 1 radian in degrees?

.a – 45 ˚        .b -  2 π x 360 ˚     .c -  6.28 x r ˚.   .d – 360/2 π ˚.

1.4 .a,b,c and d are equal forces acting in the horizontal plane on the oval shaped object. The ob

ject is a flat disc fixed to an axle on one side. In your opinion which force will produce the maximum torque?

1.5  Referring to diagram in

 question 1,4  which force will produce the minimum torque?

2.0 Fig. Measuring distance.

 A wheel attached to a handle is sometimes used to measure a distance. . It had a radius of 0.25 m..   In 2 minutes  the wheel rotated 30 times while pushing

Find the following.

2.1 Time taken for one complete revolution,

2.2 Angular velocity.

2.3 Average velocity of the man.

 

3.0

 

Fig. 

Wheels  Bicycle .

According to the specifications given find the following: Neglect friction.

Force on pedal  =  25 N.

The pedal arm = 0.16 m.     .

Radius large cog wheel. ˚0.08 m,   

Radius small cog wheel = 0.04 m

3.1. Torque on the pedal.

3.2 Tension force on  chain.

3.3   How many times will the rare wheel turn for 1 turn of the pedal ?

3.4  The distance cycle willtravel.

 

4.0

 

The graph indicates how an electric fan started to rotate when switched on. Find the following.

4.1  Average angular acceleration  for the first 4 seconds.

4.2  Angular acceleration for last two seconds.

4.3 The number of revolutions for the first 5 seconds.

4.4 The number of revolutions for the last 2 seconds.

4.5 If the fan had a blade 0.24 m long  the linear speed of its outer most edge.

Answers visit Answer page - physics