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

$A$ body is suspended by a light string. The tensions in the string when the body is in air,when the body is totally immersed in water,and when the body is totally immersed in a liquid are respectively $40.2 \,N$,$28.4 \,N$,and $16.6 \,N$. The density of the liquid is

$A$ piece of steel has a weight $W$ in air,$W_1$ when completely immersed in water,and $W_2$ when completely immersed in an unknown liquid. The relative density (specific gravity) of the liquid is:

Equal masses of three liquids are kept in three identical cylindrical vessels $A$, $B$ and $C$. The densities are $\rho_A$, $\rho_B$ and $\rho_C$ with $\rho_A$ < $\rho_B$ < $\rho_C$. The force on the base will be maximum in vessel:

$A$ boat carrying a number of large stones is floating in a water tank. What would happen to the water level if a few stones are unloaded into the water?

Vedclass Test Series

Exam Paper Generator

Online Exam Module