CBSE Class 9 Physics Uniform Circular Motion Notes Set 13

Angular Displacement and Angular Velocity

When a particle moves on a circle, the radius joining it to the centre sweeps an angle called the angular displacement. It equals arc length divided by radius, \( \theta = \dfrac{s}{r} \), and is measured in radians (\( \pi \) rad = 180°, 1 rad = 57.3°).
1. Angular velocity \( \omega = \dfrac{\theta}{t} \) in rad/s.
2. Linear speed and angular velocity are related by \( v = r\omega \).
3. The period T is the time for one revolution and the frequency n is the number of revolutions per second: \( T = \dfrac{2\pi r}{v} = \dfrac{2\pi}{\omega} = \dfrac{1}{n} \) and \( \omega = 2\pi n \).
4. Particles at different distances on the same rotating body have the same angular velocity but different linear speeds.

Centripetal Acceleration and Force

In uniform circular motion the speed is constant, but the direction of velocity changes continuously, so the body is accelerating. The acceleration is always perpendicular to the velocity and directed towards the centre.
1. Centripetal acceleration: \( a = \dfrac{v^2}{r} = r\omega^2 \).
2. Centripetal force: \( F = \dfrac{mv^2}{r} = mr\omega^2 \), directed towards the centre. It does no work, so the kinetic energy stays constant.
3. It is supplied by tension in a string, friction on a turn, gravity for a satellite and so on.
4. The centrifugal force is a fictitious (pseudo) force that appears in the rotating frame, acting away from the centre with the same magnitude \( \dfrac{mv^2}{r} \).

Motion in a Vertical Circle

1. For a body to stay on the track at the top of a vertical circle, \( \dfrac{mv^2}{r} \geq mg \), so the minimum speed at the top is \( v = \sqrt{rg} \).
2. For a stone whirled on a string, the tension is greatest at the bottom, \( T = mg + \dfrac{mv^2}{r} \), and least at the top, \( T = \dfrac{mv^2}{r} - mg \).
3. A bucket of water whirled in a vertical circle does not spill if its angular speed at the top is at least \( \sqrt{g/r} \).

Everyday Applications

1. A cream separator or washing machine drier throws the heavier particles away from the axis.
2. A cyclist leans inwards on a turn, and a banked road with the outer edge raised gives the extra centripetal force needed for safe turning.
3. The Earth bulges at the equator because the effect of rotation is greatest there.

Worked Examples

Example 1. A bicycle wheel of diameter 0.7 m moves at 18 km/h. Find its angular velocity.
v = 5 m/s and r = 0.35 m, so \( \omega = \dfrac{v}{r} = 14.3 \) rad/s, which is \( \dfrac{14.3}{2\pi} \approx 2.27 \) revolutions per second.

Example 2. A stone of mass 0.2 kg is whirled in a horizontal circle of radius 0.5 m at 4 revolutions per second. Find the tension.
\( \omega = 2\pi \times 4 = 25.13 \) rad/s, so \( T = m\omega^2 r = 0.2 \times 631.7 \times 0.5 = 63.2 \) N.