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Radians and Wave Oscillation

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Radians and Wave Oscillation

Answer every question. Give exact answers in terms of π where appropriate, and show your working for calculations.

Radians and phase

A radian measures an angle using the radius and arc length of a circle. A complete turn is 2π radians. In wave equations, the phase angle describes progress through an oscillation cycle.

1.Which statement gives the definition of one radian?

The central angle that subtends an arc equal in length to the circle’s radius
The angle made by two perpendicular lines
The angle in half of a complete turn
The angle made by an arc twice as long as the radius

2.A rotating wheel turns through 120°. What is this angle in radians?

π/3
2π/3
3π/2
2π

3.In the expression for a sinusoidal oscillation, what does the phase angle tell you?

The greatest displacement from equilibrium
The position within the cycle at a particular time
The distance travelled in one second
The length of one complete wave

4.Select every quantity that represents one complete cycle of a travelling wave.

360° of phase
2π radians of phase
π radians of phase
One wavelength

5.Match each angle in degrees to its equivalent in radians.

30°
90°
180°
300°

π
π/6
5π/3
π/2

Calculating with radians

Use 180° = π radians when converting between degrees and radians. For an oscillation with frequency f, angular frequency is ω = 2πf, and phase after time t is θ = ωt when the starting phase is zero.

6.Convert 75° to radians. Give your answer in its simplest exact form and show your conversion.

7.Convert 240° to radians. Give your answer in its simplest exact form and show your conversion.

8.A vibration is modelled by y = 5 sin(2πft). Its frequency is 40 Hz. Find the phase angle when t = 0.0125 s. Give your answer in radians.

9.A sine wave is described by y = 3 sin(θ), where θ is the phase angle in radians. State its amplitude and period, and give the phase angle at which it first reaches its maximum positive displacement as θ increases from 0.

10.Explain how radians describe the phase of an oscillation. Include one reason radians are useful when analysing waves.

Applying angular frequency and phase

For a repeating oscillation, the period T is the time for one complete cycle. Angular frequency is ω = 2π/T.

11.A pendulum completes one full oscillation in 4.0 s. Calculate its angular frequency and its phase change after 1.0 s, assuming it starts at phase zero. Give both answers in radians or radians per second as appropriate.

12.Two oscillations are y₁ = 2 sin(5t) and y₂ = 2 sin(5t + π/3). Find their phase difference and state which oscillation leads.