A cylindrical cork of uniform density floats | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let the cross-sectional area of the cork be $A$, its density be $\rho$, and its length be $L$. When the cork floats in a liquid of density $\rho_L

Vedclass · Accepted Answer

$1$/$4$

Vedclass · Answer

(D) Let the cross-sectional area of the cork be $A$, its density be $\rho$, and its length be $L$. When the cork floats in a liquid of density $\rho_L$, the submerged length $h$ is given by the condition of equilibrium: $A h \rho_L g = A L \rho g$, so $h = (\rho / \rho_L) L$.When the cork is depressed by a small distance $x$, the additional buoyant force acting on it is $F = -A x \rho_L g$. This force acts as a restoring force.The equation of motion is $m a = -A \rho_L g x$, where $m = A L \rho$ is the mass of the cork.Thus, $A L \rho (d^2x/dt^2) = -A \rho_L g x$, which gives $(d^2x/dt^2) + (\rho_L g / L \rho) x = 0$.This is the equation of simple harmonic motion with angular frequency $\omega = \sqrt{\rho_L g / L \rho}$.The time period is $T = 2\pi / \omega = 2\pi \sqrt{L \rho / \rho_L g}$.Since $T \propto 1/\sqrt{\rho_L}$, we have $T_1 / T_2 = \sqrt{\rho_2 / \rho_1}$.Given $T_1 = T$ and $T_2 = 2T$, we have $T / 2T = \sqrt{\rho_2 / \rho_1}$, which implies $1/2 = \sqrt{\rho_2 / \rho_1}$.Squaring both sides, we get $\rho_2 / \rho_1 = 1/4$.

Similar Questions

The metallic bob of a simple pendulum has the relative density $\rho$. The time period of this pendulum is $T$. If the metallic bob is immersed in water,then the new time period is given by

The time period of a simple pendulum is $T$. When the length is increased by $10 \ cm$, its period is $T_1$. When the length is decreased by $10 \ cm$, its period is $T_2$. Then, the relation between $T, T_1$, and $T_2$ is

Two simple pendulums are given: the first has a bob of mass $M_1$ and length $L_1$,and the second has a bob of mass $M_2$ and length $L_2$. Given $M_1 = M_2$ and $L_1 = 2L_2$. If the vibrational energy of both is the same,which of the following statements is correct?

$A$ simple pendulum keeps correct time at $0^{\circ}C$. At $25^{\circ}C$,it loses $12.5 \, s$ in a day. What is the coefficient of linear expansion of the metal of the pendulum?

Vedclass Test Series

Exam Paper Generator

Online Exam Module