Answer:
Consider an air chamber of volume V with a long neck of
uniform area of cross-section A, and a frictionless ball of mass m fitted
smoothly in the neck at position C, Fig. 10 (NCT). 7. The pressure of air below
the ball inside the chamber is equal to the atmospheric pressure. Increase the pressure
on the ball by a little amount p, so that the ball is depressed to position D,
where CD = y.
There
will be decrease in volume and hence increase in pressure of air inside the
chamber. The decrease in volume of the air inside the chamber.
\[{{\text{a}}_{\text{c}}}\text{=}\frac{{{\text{
}\!\!\upsilon\!\!\text{ }}^{\text{2}}}}{\text{R}}\] Volumetic strain
\[\text{g }\!\!'\!\!\text{
=}\sqrt{{{\text{g}}^{\text{2}}}\text{+}\frac{{{\text{ }\!\!\upsilon\!\!\text{
}}^{\text{4}}}}{{{\text{R}}^{\text{2}}}}}\]\[\therefore \]
\[\text{T=2 }\!\!\pi\!\!\text{
}\sqrt{\frac{\text{l}}{\text{g }\!\!'\!\!\text{ }}}\text{=2 }\!\!\pi\!\!\text{
}\sqrt{\frac{\text{l}}{{{\text{g}}^{\text{2}}}\text{+}{{\text{
}\!\!\upsilon\!\!\text{ }}^{\text{4}}}\text{/}{{\text{R}}^{\text{2}}}}}\] Bulk
Modulus of elasticity E, will be
\[{{\text{ }\!\!\rho\!\!\text{
}}_{\text{1}}}\]
\[\text{T=2 }\!\!\pi\!\!\text{
}\sqrt{\frac{\text{h }\!\!\rho\!\!\text{ }}{{{\text{p}}_{\text{1}}}\text{g}}}\]
Here,
negative sign shows that the increase in pressure will decrease the volume of
air in the chamber.
\[\text{
}\!\!\rho\!\!\text{ }\] \[\vartriangle \text{V=Ay}\]
Due
to this excess pressure, the restoring force acting on the ball is
\[\text{=}\frac{\text{change
in volume}}{\text{original volume}}\]
Clearly,
\[=\frac{\vartriangle V}{V}=\frac{\text{Ay}}{\text{V}}\] Negative sign shows
that the force is directed towards equilibrium position. If the applied increased
pressure is removed from the ball, the ball will start executing linear SHM in
the neck of chamber with C as mean position.
In S.H.M., the restoring force, \[F=-ky\]..
(ii)
Comparing (i) and (ii), we have
\[\therefore \], which is the spring factor.
Now, inertia factor = mass of ball = m.
As, \[\text{E=}\frac{\text{stress}\left( \text{orincrease
in pressure} \right)}{\text{volumetric strain}}\]
\[\text{=}\frac{\text{-p}}{\text{Ay/V}}\text{=}\frac{\text{-pV}}{\text{Ay}}\]
\[\therefore \] Frequency, \[\text{p=}\frac{\text{-EAy}}{\text{V}}\]
Note. If the ball oscillates in the neck of chamber under
isothermal conditions, then E = P = pressure of air inside the chamber, when
ball is at equilibrium position. If the ball oscillates in the neck of chamber
under adiabatic conditions, then \[\text{F=p }\!\!\times\!\!\text{
A=}\frac{\text{-eaY}}{\text{v}}\text{.A=}\frac{\text{-E}{{\text{A}}^{\text{2}}}}{\text{V}}\text{y
}......\text{(i)}\] where\[F\alpha y;\]
You need to login to perform this action.
You will be redirected in
3 sec
