Question

A solid right circular cylinder of weight $10\text{kg}$ and cross - sectional area $100{\text{cm}}^2$ is suspended by a spring, where $K=1\text{kg-wt/cm}$, and hangs partially submerged in water of density $1000{\text{kg/m}}^3$. What is its period in seconds when it makes simple harmonic vertical oscillations? Answer upto 1 decimal place (Take $g=10{\text{m/s}}^2)$

Answer 1

Let the cylinder be depressed a distance x meters into water from the position of rest.
Increase in tension of spring $=100\text{x kg - wt}$
$(A=100{\text{cm}}^2={10}^{-2}{m}^2)$
Increase in buoyancy $=\frac{100}{{100}^2}x\times 1000\text{kg -wt}$
$K=1\text{kg - wt/cm}=100\text{kg - wt/m}$
Total unbalanced vertical force on the cylinder
$=100x+10x=110\text{x kg - wt}$
Acceleration of cylinder $=\frac{110x\times 10}{10}{\text{m/s}}^2=110{\text{x m/s}}^2$
Period of one oscillation $=\frac{2\pi }{\sqrt{110}}=0.6s$

Why this question? Tip- classic example of how time period can be found by just taking extra force into account, which saves us the effort of balancing forces in equilibrium position and some calculation!