Question

The time period of a spring-mass system is $T$ in air. When the mass is partially immersed in water, the time period of oscillation is $T^{\prime }$. Then:


[Assume the mass is cubical]

• A. $T^{\prime }=T$
• B. $T^{\prime }<T$
• C. $T^{\prime }>T$
• D. $T^{\prime }\le T$

Answer 1

The correct option is B $T^{\prime }<T$

Let $A$ be the cross-sectional area of the cubical body and $h$ be the length of the cubical body inside the water. [Assuming the shape of mass is cubical]
The time period of the spring mass system in air is
$T=2\pi \sqrt{\frac{m}{k}}$ ..............(1)
When the body is immersed in water partially to a height $h$,
Buoyant force $(=\rho Ahg)$ and the spring force $(=k{x}_0)$ will act.
At equilibrium, $k{x}_0+F_b=mg$


When the body is displaced through a small distance $\Delta x$,
The restoring force $\left(F\right)$,
$F=-m{\omega }^2\Delta x=-(k+A\rho g)\Delta x$
$\Rightarrow \omega =\sqrt{\frac{k+A\rho g}{m}}$
$\Rightarrow T^{\prime }=2\pi \sqrt{\frac{m}{k+A\rho g}}$ .....(2)
From Eq.(1) and (2),
$\frac{T^{\prime }}{T}=\sqrt{\frac{k}{k+A\rho g}}<\perp$
$\Rightarrow T^{\prime }<T$
Hence, option $\left(b\right)$ is the correct answer.