Question

A cylindrical object of outer diameter 10 cm, height 20 cm and density 8000 kg/m³ is supported by a vertical spring and is half dipped in water as shown in figure. (a) Find the elongation of the spring in equilibrium condition. (b) If the object is slightly depressed and released, find the time period of resulting oscillations of the object. The spring constant = 500 N/m. 

Answer 1

Given:
Diameter of the cylindrical object, d = 10 cm
∴ Radius, r = 5 cm
Height of the object, h = 20 cm
Density of the object, ρ_b = 8000 kg/m³ = 8 gm/cc
Density of water, ρ_w = 1000 kg/m³
Spring constant, k =  500 N/m = 500 × 10³ dyn/cm

(a) Now,
F + U = mg          [F = kx]
Here, U is the upward thrust and x is the small displacement from the equilibrium position.
kx + Vρ_wg = mg 

\[\Rightarrow 500 \times {10}^3 \times (x) + \left( \pi r^2 \right) \times \left( \frac{h}{2} \right) \times 1 \times 1000 = \pi r^2 \times h \times \rho_b \times 1000\]

\[ \Rightarrow 500 \times {10}^3 \times (x) = \pi r^2 \times \text{h} \times 1000\left( \rho_b - \frac{1}{2} \right)\]

\[ = \pi \times (5 )^2 \times 20 \times 1000\left( \rho_b - \frac{1}{2} \right)\]

\[ \Rightarrow 50x = \pi \times 25 \times 2 \times \left( \rho_b - \frac{1}{2} \right)\]

\[x = \pi(8 - 0 . 5)\]

\[\text{or, x }= \pi \times 7 . 5 = 23 . 5 \text{ cm} \]

(b) We know that X is the displacement of the block from the equilibrium position.
Now,
Driving force : \[\text{F = kX + V  }\rho_\text{w} \times \text{g}\]

\[ \Rightarrow\text{ ma = kx }+ \pi \text{r}^2 \times (\text{X}) \times \rho_\text{w} \times\text{ g}\]

\[ = (\text{k} + \pi \text{r}^2 \times \rho_\text{w} \times g)\text{x}\]

\[ \Rightarrow \omega^2 \times (X) = \frac{\left( k + \pi r^2 \times \rho_\text{w} \times g \right)}{\text{m}} \times (X)\]

\[\text{In SHM }\left( a = \omega^2 \text{X} \right), \text{ time period }\left( T \right) \text{is}\]

\[ T = 2\pi\frac{m}{k + \pi r^2 \times \rho_w \times g}\]

\[ =2\pi\frac{\pi \times 25 \times 20 \times 8}{500 + {10}^3 + \pi \times 25 \times 1 \times 1000}\]

 = 0.935 s