A rectangular block of mass m and cross-secti | vedclass

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

Question

(A) In equilibrium,the weight of the block is balanced by the buoyant force:$mg = A l \rho g \Rightarrow m = A \rho l$where $l$ is the length of the p

Vedclass · Accepted Answer

$T^{2} \propto \frac{1}{\rho}$

Vedclass · Answer

(A) In equilibrium,the weight of the block is balanced by the buoyant force:$mg = A l ho g \Rightarrow m = A ho l$where $l$ is the length of the part immersed in the liquid.When the block is given a small downward displacement $y$,the additional buoyant force (restoring force) acting upwards is:$F = -[A(l+y) ho g - mg]$$F = -[A l ho g + A y ho g - mg]$Since $mg = A l ho g$,the equation simplifies to:$F = -A ho g y$This shows that $F \propto -y$,which is the condition for $SHM$. The effective spring constant $k$ is $A ho g$ and the inertia factor is $m$.The time period $T$ of $SHM$ is given by:$T = 2 \pi \sqrt{\frac{	ext{Inertia factor}}{	ext{Spring factor}}} = 2 \pi \sqrt{\frac{m}{A ho g}}$Squaring both sides,we get:$T^{2} = 4 \pi^{2} \frac{m}{A ho g}$Thus,$T^{2} \propto \frac{1}{ho}$.

$A$ rectangular block of mass $m$ and cross-sectional area $A$ floats in a liquid of density $\rho$. If we give it a small vertical displacement from equilibrium,it undergoes $SHM$ with a time period $T$. Then:

Similar Questions

$A$ bead of mass $m$ is attached to the mid-point of a taut,weightless string of length $l$ and placed on a frictionless horizontal table. Under a small transverse displacement $x$,as shown in the figure,if the tension in the string is $T$,then the frequency of oscillation is:

$A$ particle of mass $m$ is located in a one-dimensional potential field where the potential energy is given by: $V(x) = A(1 - \cos px)$, where $A$ and $p$ are constants. The period of small oscillations of the particle is

$A$ man of mass $M$ is swinging on a swing made of $2$ ropes of equal length $L$. The distance between the points of suspension is also $L$. The time period of the small oscillations about the mean position in a direction perpendicular to the plane of the swing is:

$A$ particle executes linear simple harmonic motion with an amplitude of $2 \ cm$. When the particle is at $1 \ cm$ from the mean position,the magnitude of its velocity is equal to that of its acceleration. Then its time period in seconds is

One end of a $U$-tube containing mercury is connected to a suction pump and the other end to the atmosphere. $A$ small pressure difference is maintained between the two columns. Show that, when the suction pump is removed, the column of mercury in the $U$-tube executes simple harmonic motion.

Vedclass Test Series

Exam Paper Generator

Online Exam Module