Question

The bob of simple pendulum performs S.H.M. with a frequency \({}'n'\) in air and with frequency \({}'n_{1}'\) in water. If the density of the material of bob is \({}'x'\) g / cc and the ratio \(n/n_{1}\) is 3, then density of bob \({}'x'\) is ( density of water \(=1\) g /cc and neglect friction due to water)

• A. \(\frac{8}{9}\)
• B. \(\frac{9}{8}\)
• C. \(\frac{10}{9}\)
• D. \(\frac{9}{10}\)

Hint:

In water the buoyant force reduces the effective gravity, and frequency is proportional to the square root of the effective g.

Answer 1

The Correct Option is B

Step 1: Forces on the bob
Weight per unit volume is $x g$ and upthrust per unit volume is $1\times g$. Net downward force per unit volume $=(x-1)g$.

Step 2: Effective gravity
$g_{\text{eff}}=\frac{(x-1)g}{x}$. Since $n^{2}\propto g_{\text{eff}}$, $\frac{n^{2}}{n_{1}^{2}}=\frac{x}{x-1}=9$.

Step 3: Solve
$x=9x-9$, so $8x=9$ and $x=\frac{9}{8}$.

Final Answer:
The bob has density 9/8 g per cc. \[ \boxed{\text{(B) }\dfrac{9}{8}} \]