Question

A solid right circular cylinder of weight $10kg$ and cross sectional area $100c{m}^2$ is suspended by a spring, where $k=1kg.wt/cm$, and hangs partially submerged in water of density $1000kg/m^3$ as shown in the given figure. What is its period when it makes simple harmonic vertical oscillations? (Take $g=10m/s^2)$.

• A. $0.6s$
• B. $1s$
• C. $1.4s$
• D. $2.2s$

Answer 1

The correct option is A $0.6s$

Given:

$m=10kg,A=100c{m}^2=\frac{100}{100\times 100}{m}^2=\frac{1}{100}{m}^2$,

$k=1kg.wt/cm=100\frac{kg.wt}{m}=1000\frac{N}{m}$,

$\varrho =1000kg/m^3,g=10m/s^2$

Restoring force $=$ spring force $+$ buoyant force

$\therefore ma=-kx-A.x.\varrho .g$

where $x=$ displacement

$\therefore 10a=-1000x-\frac{10x\times 1000}{100}$

$10a=-1000x-100x=-1100x$

$a=-110x$ ...(1)

For oscillatory motion,

$a=-w^{2}x$ ....(2)

comparing (1) & (2),

$w^2=110$

$\therefore {\left(\frac{2\pi }{T}\right)}^2=110$

$T=\frac{2\pi }{\sqrt{110}}=\frac{2\times 3.142}{\sqrt{110}}$

$\therefore T=\frac{6.284}{10.488}$

$\therefore T=0.599s$

$\therefore T\approx 0.6s$