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Q17.3-27PE

Expert-verified
Found in: Page 595

College Physics (Urone)

Book edition 1st Edition
Author(s) Paul Peter Urone
Pages 1272 pages
ISBN 9781938168000

(a) Ear trumpets were never very common, but they did aid people with hearing losses by gathering sound over a large area and concentrating it on the smaller area of the eardrum. What decibel increase does an ear trumpet produce if its sound gathering area is$$900\;{\rm{c}}{{\rm{m}}^{\rm{2}}}$$and the area of the eardrum is$$0.500\;{\rm{c}}{{\rm{m}}^{\rm{2}}}$$, but the trumpet only has an efficiency of $$5.00\%$$in transmitting the sound to the eardrum? (b) Comment on the usefulness of the decibel increase found in part (a)

(a) The sound increases by$$139.5\;{\rm{dB}}$$

(b) It is not sustainable in the longer term but it will amplify the sound.

See the step by step solution

Given Data

The area of the ear trumpet is$$900\;{\rm{c}}{{\rm{m}}^{\rm{2}}}$$.

The area of the ear is$$0.500\;{\rm{c}}{{\rm{m}}^{\rm{2}}}$$.

The efficiency of ear trumpet is$$5\%$$.

Loudness of Sound

The increase in intensity increases the loudness of a sound. The loudness is unsustainable if it is greater than the threshold level.

Calculation of the intensity of the first sound

(a)

The intensity of the unaided hearing is,

$${d_1} = 10\log \frac{{{I_0}}}{{{{10}^{12}}}}$$

The change in intensity of sound is,

$${Q_1} = {I_0} \times 900$$

$$\begin{array}{c}{I_{ear}} = efficiency \times \frac{{{Q_1}}}{{0.500}}\\{I_{ear}} = 5.00 \times {I_0} \times \frac{{900}}{{0.500}}\\{I_{ear}} = 90{I_0}\end{array}$$

The intensity of the sound is,

$$\begin{array}{c}d = 10\log \frac{{90{I_0}}}{{{{10}^{ - 12}}}}\\d = 10\log \frac{{{I_0}}}{{{{10}^{ - 12}}}} + 10\log \frac{{90}}{{{{10}^{ - 12}}}}\\d = {d_1} + 139.5\;{\rm{dB}}\end{array}$$

Hence, the intensity increases by$$139.5\;{\rm{dB}}$$

Usefulness of the situation

(b)

The maximum limit for the intensity of sound is about$$139.5\;{\rm{dB}}$$. The increase in the intensity level may damage the ear. The radius of the trumpet is close to $$16\;{\rm{cm}}$$ which is not practical for holding the sound to the ear. This situation is not sustainable for a short period.