Radians and Wave Oscillation

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📚 Part 1: Understanding Radians and Waves

Key Information: A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. One complete revolution = 2π radians = 360°. In wave motion, radians help describe the phase of oscillation.

1. What is the definition of a radian?

The angle in a right triangle

The angle subtended by an arc equal to the radius

180° divided by π

Half a complete revolution

2. Convert 90° to radians:

π/4 radians

π/2 radians

π radians

2π radians

3. In wave motion, what does the phase angle represent?

The amplitude of the wave

The frequency of oscillation

The position in the wave cycle at a given time

The wavelength

4. Which of the following are equivalent to one complete wave cycle? (Select all that apply)

360°

2π radians

π radians

One wavelength

✏️ Part 2: Calculations and Problem Solving

5. Convert the following angles to radians (show your working):

a) 45° = _________ radians

b) 270° = _________ radians

6. A wave has the equation y = 3sin(2πft), where f = 50 Hz and t = 0.005 s.

Calculate the phase angle in radians: _________ radians

7. From the wave graph below, determine:

Sketch a sine wave with amplitude 2 units and period 4π radians

a) Amplitude = _________ units

b) Period = _________ radians

c) At what phase angle does the wave first reach maximum positive displacement? _________ radians

8. Explain the relationship between radians and wave oscillation. Include why radians are particularly useful in wave analysis.

🔬 Part 3: Applied Problems

9. A pendulum completes one full oscillation in 2 seconds. Calculate:

a) The angular frequency in rad/s: ω = _________ rad/s

b) The phase angle after 0.5 seconds: θ = _________ radians

10. Two waves are described by:

Wave A: y₁ = 4sin(3t)

Wave B: y₂ = 4sin(3t + π/2)

What is the phase difference between these waves? _________ radians

Describe what this phase difference means physically:

Answer Key

1. The angle subtended by an arc equal to the radius

2. π/2 radians

3. The position in the wave cycle at a given time

4. 360°, 2π radians, One wavelength

5a. π/4 radians (45° × π/180°)

5b. 3π/2 radians (270° × π/180°)

6. π/2 radians (2π × 50 × 0.005 = 0.5π)

7a. 2 units

7b. 4π radians

7c. π/2 radians

8. Radians provide a natural unit for measuring phase in oscillatory motion. Since 2π radians equals one complete cycle, calculations involving frequency, period, and phase relationships become more straightforward.

9a. π rad/s (ω = 2π/T = 2π/2)

9b. π/2 radians (θ = ωt = π × 0.5)

10. π/2 radians; Wave B leads Wave A by π/2 radians (90°), meaning B reaches its maximum 1/4 cycle before A.