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PART - 2 CH - 13 Oscillations — Physics STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 SciencePhysicsPART - 2 CH - 13 Oscillations3 Marks
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Answer

Let the volume of a chamber filled with a long neck of internal cross section $= V$.
Let $m= C$ mass of the bullet filled in the neck at position C.
Let $Pa =$ The pressure on both sides of the bullet $=$ Atmospheric pressure
$CD =y$ on increasing the pressure on the bullet, it comes down to the position D .
Where $CD =y$
Thus, the volume of air in the air chamber is less then V and the pressure become more than P .
If $\Delta V =$ reduction in volume in air inside the chamber $= A y$.
Volume distortion $=\frac{\text { Change in volume }}{\text { Volume }}=\frac{\Delta V }{ V }=\frac{ A y}{V}$
Volume modulus of elasticity (E)
$\begin{array}{l}E=\frac{\text { Stress }}{\text { Increase in volume }} \\E=\frac{-P}{\frac{\Delta V}{V}}=-\frac{P}{A y} \times V=-\frac{PV}{A y}\end{array}$
Here the negative sign indicates that an increase in pressure will result in a decrease in the volume of air in the chamber
$P=\frac{-E A}{V} y\ldots\ldots (1)$
Due to excess pressure, the restoring force acting on the bullet is
$F=P \times A=\frac{-EAy}{V} A=\frac{-EA^2}{V} y\ldots\ldots (2)$
Equation (2) with $F =-k y$
$k=\frac{E^2}{V}$
Putting the value in the formula of time period $k$
$\begin{array}{l} T =2 \pi \sqrt{\frac{m}{ k }} \\ T =2 \pi \sqrt{\frac{ mV }{ EA ^2}}\end{array}$

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