A uniform solid right circular cone of base radius $r$ is joined to a uniform solid hemisphere of radius $r$ and of the same density, so as to have a common face. The centre of mass of the composite solid lies on the common face. The height of the cone is

Question

A uniform solid right circular cone of base radius

r

is joined to a uniform solid hemisphere of radius

r

and of the same density, so as to have a common face. The centre of mass of the composite solid lies on the common face. The height of the cone is

A uniform solid right circular cone of base radius r is joined to a uniform solid hemisphere of radius r and of the

A uniform solid right circular cone of base radius r is joined to a uniform solid hemisphere of radius r and of the

Quantrex Academy · Accepted Answer

Correct answer: B. 3 r

Volume of cone = (1)/(3) πr ^ 2 h Mass of cone, m _ 1 = ρ × (1)/(3) πr ^ 2 h Mass of hemisphere, m _ 2 = ρ × (1)/(2) × (4)/(3) πr ^ 3 & & & & & & & & & & & & & & & & & & = ρ × (2)/(3) πr ^ 3 Now, Y = m _ 1 y _ 1 + m _ 2 y _ 2 m _ 1 + m _ 2 ⇒ 0 = ρ × (1)/(3) πr ^ 2 h × (h)/(4) + (2)/(3) πr ^ 3 ( - (3r)/(8) ) ρ × (1)/(3) πr ^ 2 h + ρ × (2)/(3) πr ^ 2 ⇒ ρ × (1)/(3) πr ^ 3 ((h2)/(4) - 2r × (3r)/(8)) = 0 (h2)/(4) - 3r^ 2 4 = 0 or h = 3 r

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