14.20 An air chamber of volume V has a neck area of cross section a into which a ball of mass m just fits and can move up and down without any friction (Fig.14.27). Show that when the ball is pressed down a little and released , it executes SHM. Obtain an expression for the time period of oscillations assuming pressure-volume variations of air to be isothermal [see Fig. 14.27].

<p>Volume of the air chamber = V</p><p>Area of cross-section of the neck = a</p><p>Mass of the ball = m</p><p>The pressure inside the chamber is equal to atmospheric pressure.</p><p>Let the ball be depressed by x units. As a result of depression, there would be decrease in volume and an increase of pressure inside the cylinder.</p><p>Decrease in the volume, &nbsp;<span title="Click to copy mathml"><math><mo>? </mo><mi>V</mi></math></span>&nbsp;= ax</p><p>Volumetric strain =&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mo>? </mo><mi>V</mi></mrow></mrow><mrow><mrow><mi>V</mi></mrow></mrow></mfrac></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mi>a</mi><mi>x</mi></mrow></mrow><mrow><mrow><mi>V</mi></mrow></mrow></mfrac></math></span></p><p>Bulk modulus of air, B =&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mi>S</mi><mi>t</mi><mi>r</mi><mi>e</mi><mi>s</mi><mi>s</mi></mrow></mrow><mrow><mrow><mi>S</mi><mi>t</mi><mi>r</mi><mi>a</mi><mi>i</mi><mi>n</mi></mrow></mrow></mfrac></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mo>-</mo><mi>p</mi></mrow></mrow><mrow><mrow><mi mathvariant="normal"></mi><mfrac><mrow><mrow><mi>a</mi><mi>x</mi></mrow></mrow><mrow><mrow><mi>V</mi></mrow></mrow></mfrac></mrow></mrow></mfrac></math></span>&nbsp;: here stress is the increase in pressure. The negative sign indicates that pressure increases with a decrease in volume.</p><p>So p =&nbsp;<span title="Click to copy mathml"><math><mo>-</mo><mfrac><mrow><mrow><mi>B</mi><mi>a</mi><mi>x</mi></mrow></mrow><mrow><mrow><mi>V</mi></mrow></mrow></mfrac></math></span></p><p>The restoring force acting on the ball, F = p&nbsp;<span title="Click to copy mathml"><math><mo>×</mo><mi>a</mi></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><mo>-</mo><mfrac><mrow><mrow><mi>B</mi><msup><mrow><mrow><mi>a</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup><mi>x</mi></mrow></mrow><mrow><mrow><mi>V</mi></mrow></mrow></mfrac></math></span>&nbsp;…. (i)</p><p>In SHM, the equation of restoring force, F = -kx …. (ii)</p><p>Combining equation (i) and (ii), we get k =&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mi>B</mi><msup><mrow><mrow><mi>a</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></mrow></mrow><mrow><mrow><mi>V</mi></mrow></mrow></mfrac></math></span></p><p>Time period, T = 2&nbsp;<span title="Click to copy mathml"><math><mi>π</mi><msqrt><mrow><mfrac><mrow><mrow><mi>m</mi></mrow></mrow><mrow><mrow><mi>k</mi></mrow></mrow></mfrac></mrow></msqrt></math></span>&nbsp;= 2&nbsp;<span title="Click to copy mathml"><math><mi>π</mi><msqrt><mrow><mfrac><mrow><mrow><mi>V</mi><mi>m</mi></mrow></mrow><mrow><mrow><mi>B</mi><msup><mrow><mrow><mi>a</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></mrow></mrow></mfrac></mrow></msqrt></math></span></p>

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