Question

Sound with intensity larger than 120 dB appears painful to a person. A small speaker delivers 2.0 W of audio output. How close can the person get to the speaker without hurting his ears?

Answer 1

Answer ⇒ 40 cm.

Explanation ⇒

The intensity of the sound at a distance r from the speaker,

I = 2.0/4πr²

I = 1/2πr²

Since,

Sound level = 10 log [I/I₀]

∴ 120 = 10 log [(1/2πr²)/10⁻¹²]

Since, I₀ = 10⁻¹² W/m², thus,

Thus,  log [10¹²/2πr²] = 12

log 10¹² -log [2πr²] = 12

12 - log[2π] - log r² = 12

log r² = -log [2π] = log [2π]⁻¹

r² = [2π]⁻¹

r² = 1/2π

r² = 0.16

r = 0.40 m

or r = 40 cm

Hope it helps.

Answer 2

Answer:

The intensity of the sound at a distance r from the speaker,

I = 2.0/4πr²

I = 1/2πr²

Since,

Sound level = 10 log [I/I₀]

∴ 120 = 10 log [(1/2πr²)/10⁻¹²]

Since, I₀ = 10⁻¹² W/m², thus,

Thus,  log [10¹²/2πr²] = 12

log 10¹² -log [2πr²] = 12

12 - log[2π] - log r² = 12

log r² = -log [2π] = log [2π]⁻¹

r² = [2π]⁻¹

r² = 1/2π

r² = 0.16

r = 0.40 m

or r = 40 cm

Explanation: