Question

The metallic bob of simple pendulum has the relative density 5. The time period of this pendulum is 10 s. If the metallic bob is immersed in water, then the new time period becomes 5√x s. The value of x will be _____.

Answer 1

Correct Answer: 5

The time period \(T\) of a simple pendulum is determined by the formula: 

\( T = 2\pi \sqrt{\frac{L}{g_{\text{eff}}}} \)

where \(L\) is the length of the pendulum and \(g_{\text{eff}}\) is the effective acceleration due to gravity. Initially, with the metallic bob in air, the time period is given as 10 seconds. Upon immersing the bob in water, its apparent weight changes due to the buoyant force, thus changing \(g_{\text{eff}}\).

The relative density (specific gravity) of the metallic bob is 5, meaning the bob is 5 times as dense as water. When immersed, the effective acceleration due to gravity \(g_{\text{eff}}\) becomes:

\( g_{\text{eff}} = g - \frac{\text{Buoyant force}}{\text{Mass of bob}} = g - \frac{\rho_{\text{water}}g}{\rho_{\text{metal}}} \)

Given that the relative density is 5, or \(\rho_{\text{metal}} = 5\rho_{\text{water}}\), then:

\( g_{\text{eff}} = g \left( 1 - \frac{1}{5} \right) = \frac{4g}{5} \)

The new time period \(T'\) when the bob is immersed in water can be calculated as:

\( T' = 2\pi \sqrt{\frac{L}{g_{\text{eff}}}} = 2\pi \sqrt{\frac{L}{\frac{4g}{5}}} = 2\pi \sqrt{\frac{5L}{4g}} = 5\sqrt{x} \text{ seconds} \)

Comparing with the original time period in air, for the immersed time period we can set:

\( 5\sqrt{x} = 2\pi \sqrt{\frac{5L}{4g}} \)

Since \(T = 2\pi \sqrt{\frac{L}{g}}\) was initially 10 s, we equate:

\( \sqrt{x} = \sqrt{\frac{5}{4}} \)

Solving for \(x\):

\( x = \frac{5}{4} = 1.25 \)

Thus, the value of \(x\) is 1.25. The task specifies a range of 5 to 5, which appears to be an oversight since the value of \(x\) logically derives as 1.25 from the calculations.

Answer 2

\(T=2π\sqrt{\frac{ℓ}{g}}=10\)
\(T=2π\sqrt{\frac{ℓℓ}{g(1−\frac{1}{ρ})}}\)
\(=2π\frac{ℓ}{g}×\frac{5}{4}\)
\(=10\sqrt{\frac{5}{4}}\)
=5√5
Given that, new time period becomes 5√x s.
On comparing, x = 5
So, the answer is 5.