A spherical ball of radius r and relative den | vedclass

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

Question

(A) The volume of the ball is $V = \frac{4}{3}\pi r^3$. The density of the ball is $\sigma = 0.5\rho$.Initially,the ball floats with half its volume s

Vedclass · Accepted Answer

$\frac{5}{12}\pi r^4 \rho g$

Vedclass · Answer

(A) The volume of the ball is $V = \frac{4}{3}\pi r^3$. The density of the ball is $\sigma = 0.5\rho$.Initially,the ball floats with half its volume submerged. The buoyant force $F_B = V_{submerged} \rho g = (V/2)\rho g = (2/3)\pi r^3 \rho g$. Since it is in equilibrium,the weight of the ball $W = V \sigma g = V(0.5\rho)g = (2/3)\pi r^3 \rho g$. Thus,$F_B = W$.To push the ball down until it is fully submerged,we must apply an external force to overcome the net buoyant force. The work done is equal to the change in potential energy of the system.When the ball is pushed down by a distance $x$,the additional volume submerged is $A(x)dx$. The work done $W = \int_{0}^{r} F_{ext} dx$. The net force required at any depth $y$ (where $y$ is the depth of the center below the equilibrium position) is $F_{ext} = F_{buoyant} - Weight = (V_{submerged}(y) - V_{submerged, initial})\rho g$.Integrating the force over the displacement,the work done is $W = \frac{5}{12}\pi r^4 \rho g$.

$A$ spherical ball of radius $r$ and relative density $0.5$ is floating in equilibrium in water with half of it immersed in water. The work done in pushing the ball down so that the whole of it is just immersed in water is: (where $\rho$ is the density of water)

Similar Questions

An object suspended by a wire stretches it by $10 \,mm$. When the object is immersed in a liquid,the elongation in the wire reduces by $\frac{10}{3} \,mm$. The ratio of the relative densities of the object and the liquid is ............

$A$ beaker containing water is placed on the platform of a spring balance. The balance reads $1.5 \, kg$. $A$ stone of mass $0.5 \, kg$ and density $500 \, kg/m^3$ is immersed in water without touching the walls of the beaker. What will be the balance reading now?

When an air bubble rises in water,does its potential energy increase or decrease?

Vedclass Test Series

Exam Paper Generator

Online Exam Module