define the radian and express angular displacement in radians

Kinematics of Uniform Circular Motion

Objective

• Define the radian and express angular displacement in radians.
• Introduce angular speed (ω) and relate it to period (T) and frequency (f).
• Show the vector nature and direction of ω, centripetal acceleration a_c and centripetal force F_c.
• Connect the linear (tangential) speed v with ω (v = r ω).
• Derive and apply the centripetal‑acceleration formulas a_c = r ω² = v²/r.
• Relate centripetal force to mass and acceleration: F_c = m a_c, and discuss the physical interactions that can provide this force.
• Link the topic to vectors, Newton’s laws and work‑energy concepts.

1. The Radian

The radian (rad) is the SI unit for plane angles. One radian is the angle subtended at the centre of a circle by an arc whose length equals the radius of the circle.

2. Arc‑Length – Radius – Angle Relationship

When the angle θ is measured in radians, the arc length s is given by

$$s = r\,\theta$$

• s – arc length (m)
• r – radius (m)
• θ – angular displacement (rad)

3. Converting Between Radians and Degrees

The full circumference corresponds to an angle of 2π rad = 360°. Hence

$$1\;\text{rad}= \frac{180^\circ}{\pi}\approx57.3^\circ,\qquad 1^\circ = \frac{\pi}{180}\;\text{rad}\approx0.01745\;\text{rad}$$

4. Expressing Angular Displacement in Radians

For any motion along a circular path

$$\Delta\theta = \frac{\Delta s}{r}$$

This formula is valid only when Δθ is expressed in radians.

5. Angular Speed, Period and Frequency

• Angular speed (ω) – scalar magnitude of the angular‑velocity vector
  $$\omega = \frac{\Delta\theta}{\Delta t}\qquad[\text{rad s}^{-1}]$$
• Period (T) – time for one complete revolution
  $$T = \frac{2\pi}{\omega}\qquad[\text{s}]$$
• Frequency (f) – revolutions per second
  $$f = \frac{1}{T}= \frac{\omega}{2\pi}\qquad[\text{Hz}]$$
• Direction of ω – the angular‑velocity vector is perpendicular to the plane of motion and follows the right‑hand rule (curl fingers in the direction of motion; thumb points in the direction of ω).

6. Linear (Tangential) Speed

The speed of a point on the rim of a rotating object is linked to ω by

$$v = r\,\omega\qquad[\text{m s}^{-1}]$$

Because v is tangent to the circular path, its direction is always perpendicular to the radius.

7. Centripetal Acceleration

Uniform circular motion requires a continual change in the direction of the velocity vector, producing an inward (centripetal) acceleration.

$$a_c = r\,\omega^{2}= \frac{v^{2}}{r}\qquad[\text{m s}^{-2}]$$

• Both forms are equivalent because v = r ω.
• Direction: a_c is a vector pointing radially inward, i.e. towards the centre of the circular path.

Short Derivation

Consider two successive velocity vectors separated by a small angle Δθ. The magnitude of each vector is v, but the vector change Δv has magnitude v Δθ. Dividing by the time interval Δt gives

$$a_c = \frac{Δv}{Δt}= \frac{v\,Δθ}{Δt}=v\,\omega = r\,\omega^{2}= \frac{v^{2}}{r}$$

8. Centripetal Force

Newton’s second law applied to the centripetal acceleration yields the required inward force

$$F_c = m\,a_c = m\,r\,\omega^{2}= \frac{m\,v^{2}}{r}\qquad[\text{N}]$$

• Direction: F_c is a vector along the radius, pointing towards the centre.
• Typical physical interactions that can supply F_c:

• Tension in a string or rope (e.g., a stone tied to a string).
• Friction between tyres and road (e.g., a car rounding a curve).
• Normal reaction on a banked surface.
• Gravitational attraction (e.g., satellites, planets).
• Magnetic or electrostatic forces where applicable.

9. Links to Earlier Topics

• Vectors: ω, a_c and F_c are vector quantities; their directions are essential for solving problems.
• Newton’s Laws: The centripetal force is the net radial force required by Newton’s second law.
• Work‑Energy: For uniform circular motion the speed is constant, so the net work done by the centripetal force over one revolution is zero (force is always perpendicular to displacement).

10. Worked Examples

Example 1 – Rotating Wheel

A wheel of radius r = 0.40 m makes 15 revolutions in 6 s. Find ω, T, f and the linear speed of a point on the rim.

Solution:

$$\Delta\theta = 15\times2\pi = 30\pi\;\text{rad}$$ $$\omega = \frac{30\pi}{6}=5\pi\;\text{rad s}^{-1}\approx15.7\;\text{rad s}^{-1}$$ $$T = \frac{2\pi}{\omega}= \frac{2\pi}{5\pi}=0.40\;\text{s}$$ $$f = \frac{1}{T}=2.5\;\text{Hz}$$ $$v = r\omega =0.40\times5\pi\approx6.28\;\text{m s}^{-1}$$

Example 2 – Car on a Banked Curve

A car of mass m = 800 kg travels around a curve of radius r = 50 m banked at an angle β = 30°. The road is frictionless. Determine the speed for which the car can negotiate the curve without relying on friction.

Solution (using forces resolved perpendicular and parallel to the bank):

$$\tan\beta = \frac{v^{2}}{r\,g}\;\;\Longrightarrow\;\;v = \sqrt{r\,g\,\tan\beta} = \sqrt{50\times9.8\times\tan30^{\circ}} \approx 19.6\;\text{m s}^{-1}$$

The required centripetal force is supplied entirely by the horizontal component of the normal reaction.

Example 3 – Satellite in Circular Orbit

A satellite of mass m = 500 kg orbits Earth at an altitude where the orbital radius is r = 7.0×10⁶ m. Find the orbital speed and the magnitude of the gravitational (centripetal) force.

Solution (equating gravitational force to m v²/r):

$$\frac{G M_{\earth} m}{r^{2}} = \frac{m v^{2}}{r} \;\Longrightarrow\; v = \sqrt{\frac{G M_{\earth}}{r}} \approx 7.5\times10^{3}\;\text{m s}^{-1}$$ $$F_c = \frac{m v^{2}}{r} \approx 2.7\times10^{4}\;\text{N}$$

Here gravity provides the required centripetal force.

11. Common Angles

Angle (°)	Angle (rad)
0°	0
30°	\(\frac{\pi}{6}\)
45°	\(\frac{\pi}{4}\)
60°	\(\frac{\pi}{3}\)
90°	\(\frac{\pi}{2}\)
120°	\(\frac{2\pi}{3}\)
180°	\(\pi\)
270°	\(\frac{3\pi}{2}\)
360°	\(2\pi\)

12. Summary

• A radian is defined by an arc length equal to the radius; the relation s = rθ holds only for radian measures.
• Angular displacement: Δθ = Δs / r (rad).
• Angular speed ω = Δθ/Δt (rad s⁻¹) and ω = 2π/T = 2πf. The vector ω points perpendicular to the plane of motion (right‑hand rule).
• Linear speed and angular speed are linked by v = r ω (tangential direction).
• Centripetal acceleration: a_c = r ω² = v²/r, directed radially inward.
• Centripetal force: F_c = m a_c = m r ω² = m v²/r, also radially inward. It can be supplied by tension, friction, normal reaction, gravity, etc.
• Conversions: 2π rad = 360°, 1 rad ≈ 57.3°.
• Connections: the topic reinforces vector handling, Newton’s second law, and the work‑energy principle (centripetal force does no work in uniform circular motion).