A wide-bottomed cylindrical massless plastic | vedclass

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

Question

(B) Let $m$ be the mass of each coin. The initial submerged depth is $h_0 = 3 \, cm$ for $40$ coins.By Archimedes' principle,the weight of the contain

Vedclass · Accepted Answer

$10$

Vedclass · Answer

(B) Let $m$ be the mass of each coin. The initial submerged depth is $h_0 = 3 \, cm$ for $40$ coins.By Archimedes' principle,the weight of the container equals the weight of the displaced water: $(40m)g = A \cdot h_0 \cdot \rho_w \cdot g$,where $A$ is the cross-sectional area and $\rho_w$ is the density of water.Thus,$A \rho_w = \frac{40m}{3}$.When $N$ additional coins are added,the total mass is $(40+N)m$. The new submerged depth $h'$ is given by $(40+N)m = A \cdot h' \cdot \rho_w$,so $h' = \frac{(40+N)m}{A \rho_w} = \frac{(40+N)m}{40m/3} = \frac{3(40+N)}{40} \, cm$.The centre of mass $(CM)$ of the system (measured from the base) is $CM = \frac{40m(0) + Nm(9)}{(40+N)m} = \frac{9N}{40+N} \, cm$.The geometric centre $(GC)$ of the submerged portion is at a distance $h'/2$ from the base: $GC = \frac{h'}{2} = \frac{3(40+N)}{80} \, cm$.For the transition from stable to unstable equilibrium,the $CM$ must coincide with the $GC$ $(CM = GC)$:$\frac{9N}{40+N} = \frac{3(40+N)}{80}$$720N = 3(40+N)^2$$240N = 1600 + 80N + N^2$$N^2 - 160N + 1600 = 0$Using the quadratic formula: $N = \frac{160 \pm \sqrt{160^2 - 4(1600)}}{2} = \frac{160 \pm \sqrt{25600 - 6400}}{2} = \frac{160 \pm \sqrt{19200}}{2} = 80 \pm 40\sqrt{3} \approx 80 \pm 69.28$.Since $N The value of $N$ is closest to $10$.

Similar Questions

$A$ small wooden ball of density $\rho$ is immersed in water of density $\sigma$ to a depth $h$ and then released. The height $H$ above the surface of water up to which the ball will jump out of water is

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?

$A$ boat carrying a number of stones is floating in a water tank. If the stones are unloaded into the water,the water level in the tank will ............

Vedclass Test Series

Exam Paper Generator

Online Exam Module