JEE Physics Circular Motion Notes Set 03

Kinematics of Circular Motion

1. Angular displacement \( \theta \) is measured in radians. Angular velocity \( \omega = d\theta/dt \) and angular acceleration \( \alpha = d\omega/dt \). Frequency f and period T are linked by \( \omega = 2\pi f = 2\pi/T \).
2. Linear and angular quantities are connected by \( v = \omega r \), \( a_t = \alpha r \) and \( a_c = \omega^2 r = v^2/r = v\omega \).
3. The velocity is always tangential. The centripetal acceleration points to the centre. The tangential acceleration is along the velocity and exists only when the speed changes.
4. Total acceleration \( a = \sqrt{a_c^2 + a_t^2} \). Its angle with the velocity satisfies \( \tan\phi = a_c/a_t \).
5. Rotational equations for constant \( \alpha \): \( \omega = \omega_0 + \alpha t \), \( \theta = \omega_0 t + \tfrac{1}{2}\alpha t^2 \), \( \omega^2 = \omega_0^2 + 2\alpha\theta \). The angle in the n-th second is \( \omega_0 + \tfrac{\alpha}{2}(2n-1) \).
6. For uniform circular motion the speed and kinetic energy are constant, \( \alpha = 0 \), and the centripetal force does no work.

Relative Angular Velocity

The angular velocity of a body A as seen from a body B is the component of the relative velocity perpendicular to the line AB divided by the separation: \( \omega_{AB} = \dfrac{v_{rel,\perp}}{d} \). Components along the line of sight change the separation, not the direction.

Dynamics on a Horizontal Circle

Vertical Circular Motion

1. At angle \( \theta \) from the lowest point, \( T = \dfrac{mv^2}{r} + mg\cos\theta \) and \( v^2 = u^2 - 2gr(1-\cos\theta) \), where u is the speed at the bottom.
2. For a string, the minimum speed at the top is \( \sqrt{gr} \) and at the bottom \( \sqrt{5gr} \). The tension at the bottom exceeds that at the top by 6mg. The speed at the horizontal position is \( \sqrt{3gr} \) when the top speed is critical.
3. For a rod the top speed may be zero, so the minimum bottom speed is \( \sqrt{4gr} \).
4. A body sliding from rest down a frictionless track into a loop of radius R just completes it from height \( h = 2.5R \).
5. Tension is greatest at the lowest point, so a string tends to break there. At the highest point of a string-driven circle the tension can be zero only at the critical speed.
6. A car on a convex bridge presses on the road with \( mg - mv^2/r \); on a concave dip the thrust is \( mg + mv^2/r \).

Worked Examples

1. A stone of mass 0.5 kg moves on a horizontal circle of radius 0.4 m at 2 rev/s. \( \omega = 4\pi \), \( T = m\omega^2 r = 0.5\times 157.9\times 0.4 \approx 31.6 \) N.
2. A pendulum bob swings from the horizontal position on a string of 1 m. At the lowest point \( v = \sqrt{2gL} = \sqrt{20} \approx 4.5 \) m/s and \( T = 3mg \).
3. A road is banked at \( \tan\theta = 0.25 \) with radius 40 m. The no-friction speed is \( \sqrt{40\times 10\times 0.25} = 10 \) m/s.