A cone of mass m is tightly fitted inside a thin massless spherical shell of radius R = 6 m. The apex angle of the cone is 60° and density is 32/9 g cm⁻³. The system is kept immersed inside water so that the axis of the cone lies in a horizontal plane as shown. If it is released from this position, net torque about centre of mass of the system immediately after the release is 10π(12x)³ N m. Find the value of x. [g = 10 m s⁻²]
Physics · Mixed Chapter
JEE Mains

Question

A cone of mass m is tightly fitted inside a thin massless spherical shell of radius R = 6 m. The apex angle of the cone is 60° and density is 32/9 g cm⁻³. The system is kept immersed inside water so that the axis of the cone lies in a horizontal plane as shown. If it is released from this position, net torque about centre of mass of the system immediately after the release is 10π(12x)³ N m. Find the value of x. [g = 10 m s⁻²]

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: The cone is inscribed in the sphere. For apex angle 60°, its height is h=3R/2. Its centre of mass is 3h/4 from the apex, i.e. at a distance R/8 from the sphere centre. Step 2: The gravitational force passes through the cone’s centre of mass; the buoyancy force acts vertically at the sphere centre. The net torque about the mass centre is τ=F_b(R/8). Step 3: F_b=ρ_w g(4πR³/3); hence τ=ρ_w gπR⁴/6. Step 4: With ρ_w=1000 kg m⁻³ and R=6 m, τ=2,160,000π N m. Equate 10π(12x)³=2,160,000π, giving…