Basics of Rotation

Why rotation feels different from straight-line motion

When a car moves on a road, every part of it shifts forward together. But when a wheel rotates, different points on it keep changing direction even if the wheel stays in one place. That is rotational motion: motion of a body about a fixed axis.

Think of a classroom globe spinning about its stand. The globe does not travel across the room, but it turns around an axis. In the same way, a fan blade, a potter's wheel, and a merry-go-round all rotate.

To describe rotation, we use angular quantities instead of linear ones:

• Angular displacement b8 tells how much the body has turned.
• Angular velocity c9 tells how fast it is turning.
• Angular acceleration b1 tells how quickly that turning speed changes.

These are rotational versions of displacement, velocity, and acceleration in straight-line motion.

Angular displacement: how much has it turned?

Angular displacement is the angle through which a body rotates about its axis. It is represented by b8 and measured in radian or degree. In physics, radian is preferred.

If a wheel turns from one spoke position to another by a quarter turn, the angular displacement is:

• 90�b0 or
• c0/2 rad

Useful facts:

• One full rotation = 360�b0 = 2c0 rad
• Half rotation = 180�b0 = c0 rad
• Quarter rotation = 90�b0 = c0/2 rad

Worked example

A disc rotates through 3 complete revolutions. Find angular displacement.

1 revolution = 2c0 rad

So, b8 = 3 �d7 2c0 = 6c0 rad

Just as distance tells "how far" in linear motion, angular displacement tells how much turning happened.

Angular velocity: how fast the turning happens

If angular displacement changes with time, the body has angular velocity. It is denoted by c9.

The average angular velocity is:

c9 = b8 / t

where b8 is angular displacement and t is time. Its SI unit is rad/s.

Worked example

A wheel turns through 4c0 rad in 2 s.

c9 = b8 / t = 4c0 / 2 = 2c0 rad/s

That means the wheel completes 1 full revolution every 0.5 s.

A fast mixer in a kitchen has large angular velocity; a slowly turning clock hand has small angular velocity. Even though both rotate, their turning rates are very different. So angular velocity answers the question: how quickly is the angle changing?

Angular acceleration: when rotation speeds up or slows down

When angular velocity changes with time, we get angular acceleration, represented by b1.

b1 = (c9_2 - c9_1) / t

Its SI unit is rad/s�b2.

Worked example

A motor increases its angular velocity from 2 rad/s to 10 rad/s in 4 s.

b1 = (10 - 2)/4 = 8/4 = 2 rad/s�b2

So the rotation is speeding up steadily.

If a spinning top slows due to friction, angular acceleration is negative. That is called angular retardation.

You can think of it like this:

• Angular displacement  how much it turned
• Angular velocity  how fast it turned
• Angular acceleration  how fast the turning speed changed

These three ideas form the language of rotational motion.