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## Homework Statement

In Ruchhardt’s method of measuring ##\gamma##, illustrated in Fig. 12.2, a ball of mass ##m## is placed snugly inside a tube (cross-sectional area ##A##) connected to a container of gas (volume ##V##). The pressure ##p## of the gas inside the container is slightly greater than atmospheric pressure ##p_0## because of the downward force of the ball, so that

##p = p_0 + \frac{mg}{A}##

Show that if the ball is given a slight downwards displacement, it will undergo simple harmonic motion with period ##T## given by

##T = 2\pi \sqrt{\frac{mV}{\gamma p A^2}}##

[You may neglect friction. As the oscillations are fairly rapid, the changes in ##p## and ##V## which occur can be treated as occurring adiabatically.]

In Rinkel’s 1929 modification of this experiment, the ball is held in position in the neck where the gas pressure ##p## in the container is exactly equal to air pressure, and then let drop, the distance ##L## that it falls before it starts to go up again is measured.

Show that this distance is given by

##mgL = \frac{\gamma pA^2 L^2}{2V}##

## Homework Equations

##pv^\gamma = const. ##

## The Attempt at a Solution

I have correctly derived the first equation of period of the SHM.

From this equation, I got,

##\omega ^2 = \frac{pA^2\gamma}{mV}##

As the initial velocity is 0, so

## L = 2 \cdot amplitude##

Applying energy conservation for the point A & B, we get:

##mgL = \frac{1}{2} m \omega^2 (\frac{L}{2})^2##

##mgL = \frac{pA^2 \gamma L^2}{8V}##