Question

Speed of sound in rubber butyl 1830 m/s density is 1.35g/cm^3 elastic modulus

Answer 1

Given c= 1830 m/s

ρ=1.35 g/cm^3

We know that

c=√E/ρ

E=c²ρ

=1830² * 1.35

=4520000 Pa

Answer 2

Given info : the speed of sound in rubber butyl 1830m/s and density of it is 1.35 g/cm³.

To find : the elastic modulus of the rubber butyl is ..

solution : according to Laplace theory, the propagation of sound in any medium is an adiabatic process so the speed of sound is given by,

$v=\sqrt{\frac{\gamma RT}{M}}$ ...(1)

where $\gamma$ is adiabatic constant

and gas law of adiabatic process is given by, $PV^{\gamma}=\text{constant, C}$

⇒ $-V\frac{dP}{dV}=\frac{\gamma C}{V^{\gamma}}$  ...(2)

we know bulk modulus ,  $B=\frac{\Delta P}{-\frac{\Delta V}{V}}$  = $-V\frac{dP}{dV}$  ...(3)

from equations (2) and (3) we get, $B=\frac{\gamma C}{V^{\gamma}}$  ...(4)

from equations (1) and (4) we get, $v=\sqrt{\frac{BV^{\gamma}RT}{CM}}$

⇒ $v=\sqrt{\frac{BV^{\gamma}RT}{PV^{\gamma}M}}$

⇒ $v=\sqrt{\frac{BRT}{PM}}$

we know density of gas, $\rho=\frac{PM}{RT}$

⇒ $v=\sqrt{\frac{B}{\rho}}$ this the required equation of speed of sound.

for elastic medium , B is replaced by E ( elastic modulus )

here speed of sound, v = 1830 m/s

density of gas, $\rho$ = 1.35 g/cm³ = 1.35 × 10³ kg/m³

so the elastic modulus , E = v²p = 1830² × 1.35 × 10³ N/m² = 4.5 × 10⁹ N/m²

therefore the elastic modulus of the rubber butyl is  4.5 × 10⁹ N/m²