Question

The speed of sound in mercury at a certain temperature is 1450m/s. Given the density of mercury as $13.6 \times {10^3}\dfrac{{kg}}{{{m^3}}}$ . The bulk modulus for mercury is?
(a) $2.86 \times {10^{10}}\dfrac{N}{{{m^2}}}$
(b) $3.86 \times {10^{10}}\dfrac{N}{{{m^2}}}$
(c) $4.86 \times {10^{10}}\dfrac{N}{{{m^2}}}$
(d) $5.86 \times {10^{10}}\dfrac{N}{{{m^2}}}$

Answer 1

Hint
We will use the formula for speed of sound in solids which relates Bulk Modulus, Density and sound speed to calculate the bulk modulus for mercury. The formula is given by: -
 $v = \sqrt {\dfrac{B}{\rho }} $
Where, v = velocity of sound in the body.
B = Bulk Modulus of the body.
$\rho = $ density of the body.

Complete step by step answer
We will apply the formula for speed of sound in solids. It is given by the following expression: -
 $v = \sqrt {\dfrac{B}{\rho }} $
Where, v = velocity of sound in the body.
B = Bulk Modulus of the body.
 $\rho = $ density of the body.
According to question: -
Speed of sound (v) = 1450m/s
Density $(\rho ) = 13.6 \times {10^3}\dfrac{{kg}}{{{m^3}}}$
Bulk Modulus (B) = Required (Unit for Bulk Modulus will be $\dfrac{N}{{{m^2}}}$ as all other quantities are expressed in S.I. units as well)
Plugging in these values in the formula we get: -
 $1450 = \sqrt {\dfrac{B}{{13.6 \times {{10}^3}}}} $
Squaring both sides, we get: -
 ${(1450)^{^2}} = \dfrac{B}{{13.6 \times {{10}^3}}}$
$B = {(1450)^{^2}} \times 13.6 \times {10^3}$
 $B = 28594000 \times {10^3}$
 $ \Rightarrow B = 2.86 \times {10^{10}}\dfrac{N}{{{m^2}}}$
The bulk modulus for mercury is $2.86 \times {10^{10}}\dfrac{N}{{{m^2}}}$ .
Hence, Option (A) is the correct solution.

Note
We all have seen Mercury in liquid state, yet the formula for speed of sound in solids is used here. It is because mercury is the only metal that is liquid in room temperature but the temperature at which the speed of sound is calculated must not lie within the range of temperature at which mercury stays a liquid. That’s why it is justified to utilize this formula here.