Rotation and Linear motion..
There are two kinds of circular motion. Can you recognise them on the minute hand of this clock?
Any body will at once say the hour hand does a complete round in one hour. This is rotation or spinning.
What is the other type of motion?
Consider the blue and red paper markers on the minute hand. They also move with the hand. Do they have the same speed?
The blue dot covers a larger circle than the red dot during the same time. So the blue dot has to be faster. This is the other type of motion, which is similar to linear motion. This is sometimes referred to as the orbital motion.
Translational Motion considered for a quarter turn which was in 15 minutes. (900 s)
Orbital motion of red dot. Radius = 3. cm. | Orbital motion of Blue dot. Radius = 4.cm. |
Distance moved = 2π r / 4 = 2x3.14 x 3 ÷ 4 = 4.71 cm. | Distance moved = 2π r / 4 = 2x3.14x 4 ÷ 4 = 6.28 cm. |
Translational velocity = d/t = 4.71 ÷ 900. = 5.2x 10-3 cm.s-1 | Translational velocity = d/t = 6.28 ÷ 900 = 6.98 x10 -3 cm.s-1 |
Rotation, can be measured by considering the turning angle.This is angular measurement.
When the angle is considered, there is no difference between the red and blue points.
Angular motion.
When the minute hand does a round it has to cover 360° As we know the time it takes, the velocity can be determined.
Let us take an example where the hand moved for 15 minutes.
Using Degrees. 
The angular displacement = 90°. The symbol for angular displacement - Theta ‘θ’. Time taken for this will be 15 minutes. That is 900 seconds. ( 15 x60).
Angular velocity . ω = Angular displacement / time. ω.= θ / t . ω . = 90° / 900 = 0.1° s-1 (degree per second). | Using Radians. 
Radius of the circle is marked by a red line. A distance on the circumference, equal to the radius subtends an angle equal to one radian. In this figure, the angle θ is equal to 1 radian. The circumference = 2πr. Angular velocity ω = Angular displacement / time. Converting degrees to radians:- Any complete circle = 360°. This is the circumference angle of 2πr. length divided by the radius gives the radians. Therefore a turn of a circle = 2π radians. Fifteen minutes take a ¼ turn. Then θ = 2π / 4……..= 1.57 radians. ω. = θ / 900 radians per second. = 1.57 / 900 .= 1.74x10 -3 . rad s-1 |
Angular acceleration.
Angular acceleration ,α = change in velocity ÷ time.
α = ω f - ω i ÷ t. rad.s-2
Example.
Muralitharan spins a cricket ball making an angular displacement of 4 rad. in 0.08 seconds. The radius of the bowler’s wrist is 3 cm. Assume that the starting angular velocity is zero and the acceleration was uniform. Find the following.
The linear distance his fore finger moved during the time. b) the average angular velocity.
c) the maximum angular velocity. d) Angular acceleration.
e) Spinning rate of the ball leaving the hand.
One radian (radius of wrist) = 3 cm. Therefor 4 rad = 12 cm.
b) Average Angular velocity.
ω av = θ / t. = 4 / 0.08 rad.S-1 = 50 rad.S-1 | c) Maximum angular velocity.
ω f =ωav x 2 = 50x2 radS-1 =100 radS-1
| d) Angular acceleration. α . = Δ ω / t α . = 100 / 0.08 = 1250 radS-2
|
e) Final angular velocity = 100 rsd S-1.
One turn = 2π radians. Turninings = 100 / 6.28 …….= 16 turns per second.
Example 2
The graph shows the angular acceleration of a rotating disc against time.
Find the following.
Maximum angular velocity and the time it lasted.
30 rad S-1 lasted for 5 seconds. | b) Acceleration and the time it lasted.. α = The gradient. Gr. = 30 radS-1/ 5 s……. = 6 radS-2 Lasted for seconds 0 to 5 . |
c) Angular displacement during acceleration Angular displacement = Area below graph. Θ = ½ x30 x 5 =75 rad. | d) Total angular displacement. Θ = 75 + 150 = 225 rs |
Rotational Inertia
In translational motion, mass is the property that inhibits acceleration. Greater the mass slower the acceleration when a force is used. Acceleration is inversely proportional to mass. F = ma.
In rotation the factor that inhibits acceleration, when a torque is used is INERTIA. (I) A torque is a force used to rotate and object. 
= F x lever arm.
Considering inertia = I α,
A and B are two wooden wheels that can rotate in a vertical plane. They are identical except for the way nails are embedded. In A the metal nails are closer to the axis while in B. they are closer to the circumference. Same torque is used on both. The figure shows that A had a higher acceleration than B. The difference is due to the difference in inertia of the two wheels. The Inertia of a body depends on how the mass is distributed. If there is more mass closer to the centre of rotation, the inertia will be lower than when the mass accumulates near the periphery.
The inertia of some regular bodies can be found using these formulae.
The unit for inertia is kg. m2 (This is also called "Moment of Inertia".)