Sound Intensity — Workshop Machine (Worked Example)

Given

• P = 0.5 W

• r = 2.0 m

Find

The intensity I.

Principle

I = P/(4πr²) — small power, short distance.

Formula

Solution

1. Write the formula: $I = P/(4\pi r^2)$

2. Substitute: $I = 0.5/(4\pi \times 2.0^2)$

3. Denominator: $4\pi \times 4 = 50.27$

4. Divide: $I = 0.5/50.27 = 0.00995\ \mathrm{W/m^2}$

Unit check

W ÷ m² = W/m². Units match intensity — power per unit area, as required.

What this means physically

0.00995 W/m² ≈ 100 dB — above the 85 dB threshold where prolonged exposure damages hearing. Ear protection is genuinely needed here.