A solid cone of mass m (density 32/9 g cm⁻³) fits within a thin massless spherical shell of radius R = 6 m. Apex angle = 60°, the immersed system has a horizontal cone axis. Net torque immediately after release is 10π(12x)³ N m. Find x (g = 10 m s⁻²). [See Image.]
Physics · Mixed Chapter
JEE Main Practice

Question

A solid cone of mass m (density 32/9 g cm⁻³) fits within a thin massless spherical shell of radius R = 6 m. Apex angle = 60°, the immersed system has a horizontal cone axis. Net torque immediately after release is 10π(12x)³ N m. Find x (g = 10 m s⁻²). [See Image.]

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1 — In water, buoyancy acts at the centroid of the displaced volume, whereas the cone weight acts at cone centre of mass. Step 2 — Identify the displacement between buoyancy centre and cone centre of mass from the cone inscribed in the shell. Step 3 — Since the axis is initially horizontal, buoyancy and gravity create a couple τ = |B x₍B₎ − mg x₍G₎| with B = ρwater gV. Step 4 — Insert R = 6 m, density 32/9 g cm⁻³ and apex angle 60°, keeping all dimensions in SI. Printed source reduces to 1…