Physics - SHM Question with Solution | TestHub

When the cylinder is displaced downwards by a small distance $x$ from its equilibrium position, the restoring force is due to the spring and the change in buoyant force. The change in spring force is $kx$. The change in buoyant force is $({\rho }_{L}Ag)x$. The total restoring force $F_{restoring}=-(kx+{\rho }_{L}Agx)=-(k+{\rho }_{L}Ag)x$. Thus, the effective spring constant is $k_{eff}=k+{\rho }_{L}Ag$. Substituting the given values: $k_{eff}=100+(1000)(0.1)(10)=100+1000=200$ N/m. The time period of oscillation is $T=2\pi \sqrt{\frac{M}{k_{eff}}}$.$$T=2\pi \sqrt{\frac{10}{200}}=2\pi \sqrt{\frac{1}{20}}=\frac{2\pi }{\sqrt{20}}=\frac{2\pi }{2\sqrt{5}}=\frac{\pi }{\sqrt{5}}$$ We are given $T=\pi \sqrt{\frac{X}{10}}$. So, $\frac{\pi }{\sqrt{5}}=\pi \sqrt{\frac{X}{10}}$. Squaring both sides: $\frac{1}{5}=\frac{X}{10}$. Therefore, $X=2$.