Unit 6: Circular Motion

Angular Displacement, Velocity and Acceleration

When a particle moves along a circular path of radius r, its position can be described by the angle θ swept out from a reference line. The angular displacement is defined as the ratio of the arc length s travelled to the radius:

• θ = s / r (radians)

Angular velocity ω is the rate of change of angular displacement with time:

• ω = dθ/dt (rad s⁻¹)

Angular acceleration α is the rate of change of angular velocity:

• α = dω/dt (rad s⁻²)

For uniform circular motion the time period T (time for one revolution) relates to angular velocity as:

• T = 2π / ω

Diagram: A circle of radius r with a point P moving along the circumference. The arc length s subtends an angle θ at the centre.

Relation Between Angular and Linear Quantities

Linear motion along the tangent of the circle is directly linked to angular motion through the radius:

• Linear (tangential) velocity: v = ω r
• Tangential acceleration: a_t = α r
• Centripetal (radial) acceleration: a_c = ω² r = v² / r

These relations show that while ω and α describe the rotational state, v and a_t describe the instantaneous linear motion of the particle.

Diagram: Velocity vector v tangent to the circle, centripetal acceleration vector a_c pointing toward the centre, and radius r connecting the point to the centre.

Centripetal Acceleration

Centripetal acceleration is the inward‑directed acceleration required to keep an object moving in a circular path. It continuously changes the direction of the velocity vector without altering its magnitude (in uniform circular motion).

• Magnitude: a_c = v² / r = ω² r
• Direction: Always toward the centre of the circular path.

Because a_c is perpendicular to the velocity, it does no work on the object; it only redirects the motion.

Centripetal Force

According to Newton’s second law, the net force causing centripetal acceleration is the centripetal force:

• F_c = m a_c = m v² / r = m ω² r

This force can be supplied by various physical agents:

1. Tension in a string or rope.
2. Friction between tires and road.
3. Gravitational attraction (e.g., satellite orbit).
4. Electrostatic attraction (e.g., electron orbiting nucleus).

In a rotating reference frame, an observer perceives an outward centrifugal force, which is a pseudo‑force arising from the frame’s acceleration.

Diagram: Free‑body diagram of a mass m on a string showing tension T directed toward the centre providing F_c.

Conical Pendulum

A conical pendulum consists of a bob of mass m attached to a string of length L that sweeps out a horizontal circle, making the string trace a cone.

Let the string make an angle θ with the vertical. The radius of the circular path is r = L sin θ and the vertical height from the bob to the point of suspension is h = L cos θ.

Resolving forces:

• Vertical equilibrium: T cos θ = mg
• Horizontal (centripetal) requirement: T sin θ = m ω² r

Dividing the two equations eliminates tension:

• tan θ = ω² r / g

The time period of the pendulum depends only on the vertical height h:

• T = 2π √(h / g) = 2π √(L cos θ / g)

Diagram: Conical pendulum showing string length L, angle θ, radius r, and vertical height h.

Motion in a Vertical Circle

When a particle moves in a vertical circle of radius R, both speed and tension vary with position due to the interplay of gravity and centripetal requirement.

Minimum Speeds

• At the top of the circle, the minimum speed that maintains contact (tension can be zero) is obtained from mg = m v_top² / R: v_top = √(gR).
• At the bottom, using energy conservation between top and bottom: ½ m v_bottom² = ½ m v_top² + mg(2R). Substituting v_top² = gR yields: v_bottom = √(5gR).

Tension Variation

The tension T at any angle φ measured from the vertical upward direction is:

• T = m (v² / R + g cos φ)

Thus tension is maximum at the bottom (φ = 0) and minimum at the top (φ = π).

Energy Conservation

Throughout the motion, mechanical energy (kinetic + potential) remains constant if non‑conservative forces like friction are negligible:

• E = ½ m v² + m g y, where y is the vertical height measured from the lowest point.

Diagram: Vertical circle with labels for top, bottom, radius R, velocity vectors, and tension direction at various points.

Applications of Banking

Banking of roads or tracks raises the outer edge relative to the inner edge to provide a component of the normal reaction that acts as the required centripetal force, reducing reliance on friction.

Optimum Speed (No Friction)

For a banked curve of radius r and banking angle θ, the speed at which no lateral friction is needed satisfies:

• tan θ = v² / (r g) → v_opt = √(r g tan θ)

Maximum Safe Speed with Friction

When the coefficient of static friction between tyres and road is μ_s, the maximum speed before slipping outward is:

• v_max = √[ r g ( μ_s + tan θ ) / ( 1 – μ_s tan θ ) ]

If the denominator becomes zero or negative, the curve cannot be negotiated safely at any speed.

Diagram: Cross‑section of a banked road showing normal reaction N, its components N sinθ (providing centripetal force) and N cosθ (balancing weight), and friction force f acting down the slope when speed exceeds v_opt.

Summary

This chapter has linked angular and linear descriptions of motion, derived the essential expressions for centripetal acceleration and force, and applied them to diverse physical systems: the conical pendulum, vertical circular motion, and banked curves. Mastery of these concepts enables analysis of a wide range of real‑world phenomena, from amusement‑park rides to planetary orbits and vehicular dynamics on curved roads.