A heavy brass sphere is hung from a light spring and is set in vertical small oscillation with a period T. The sphere is now immersed in a non - viscous liquid with a density 1/10th the density of the sphere. If the system is now set in vertical S.H.M., its period will be

Engineering

Physics

SHM of Spring Block System

Basics of Simple Harmonic Motion

Buoyancy

Question

A heavy brass sphere is hung from a light spring and is set in vertical small oscillation with a period T. The sphere is now immersed in a non - viscous liquid with a density 1/10^th the density of the sphere. If the system is now set in vertical S.H.M., its period will be

(9/10)T

(9/10)²T

(10/9) T

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The period of a spring-mass system is given by $T = 2 π \sqrt{\frac{m}{k}}$ . When the sphere is immersed in a liquid, its effective mass changes due to buoyancy. The buoyant force reduces the effective weight. The effective mass, m_eff, is the actual mass minus the mass of the displaced liquid. Since the liquid density is ρ/10 and sphere density is ρ, the mass of displaced liquid is m/10. Therefore, the effective mass becomes m_eff = m - m/10 = (9m/10).

The new period T' is $T^{'} = 2 π \sqrt{\frac{\frac{9 m}{10}}{k}} = \sqrt{\frac{9}{10}} \times 2 π \sqrt{\frac{m}{k}} = \sqrt{\frac{9}{10}} T$ .

Final Answer: $\sqrt{\frac{9}{10}} T$