A hydrometer executes simple harmonic motion | vedclass

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

Question

(A) When the hydrometer is floating in equilibrium,its weight is balanced by the buoyant force.When it is pushed down by a small distance $x$,the addi

Vedclass · Accepted Answer

$T = 2 \pi \sqrt{\frac{m}{\pi r^2 \rho g}}$

Vedclass · Answer

(A) When the hydrometer is floating in equilibrium,its weight is balanced by the buoyant force.When it is pushed down by a small distance $x$,the additional buoyant force acts as the restoring force.The additional volume submerged is $V = \pi r^2 x$.The additional buoyant force is $F = \rho V g = \rho (\pi r^2 x) g$.Since this force acts in the direction opposite to the displacement,the restoring force is $F_{restoring} = -\rho \pi r^2 g x$.Using Newton's second law,$m a = -\rho \pi r^2 g x$,which gives $a = -(\frac{\rho \pi r^2 g}{m}) x$.Comparing this with the standard $SHM$ equation $a = -\omega^2 x$,we get $\omega^2 = \frac{\rho \pi r^2 g}{m}$.The time period $T$ is given by $T = \frac{2 \pi}{\omega} = 2 \pi \sqrt{\frac{m}{\rho \pi r^2 g}}$.

$A$ hydrometer executes simple harmonic motion when it is pushed down vertically in a liquid of density $\rho$. If the mass of the hydrometer is $m$ and the radius of the hydrometer tube is $r$,then the time period of oscillation is

Similar Questions

$A$ simple pendulum consists of a small sphere of mass $m$ suspended by a thread of length $l$. The sphere carries a positive charge $q$. The pendulum is allowed to do small oscillations in a uniform electric field $E$ with direction vertically upwards. The time period of oscillation is

The time period of oscillations of a simple pendulum is $1$ minute. If its length is increased by $44 \%$,then its new time period of oscillation will be ......... $s$.

Two simple pendulums $A$ and $B$ are made to oscillate simultaneously. It is found that $A$ completes $10$ oscillations in $20 \ s$ and $B$ completes $8$ oscillations in $10 \ s$. The ratio of the lengths of $A$ and $B$ is

The time period of a simple pendulum,on a satellite orbiting the Earth,is

Vedclass Test Series

Exam Paper Generator

Online Exam Module