A uniform solid right circular cone of the 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 the 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, 2 r, 3 r, 3 r, 6 r

MARKS App · Accepted Answer

Volume of cone = 1 3 π r 2 h Mass of cone, m 1 = ρ × 1 3 π r 2 h Mass of hemisphere, m 2 = ρ × 1 2 × 4 3 π r 3 = ρ × 2 3 π r 3 Now, Y = m 1 y 1 + m 2 y 2 m 1 + m 2 ⇒ 0 = ρ × 1 3 π r 2 h × h 4 + 2 3 π r 3 ( − 3 r 8 ) ρ × 1 3 π r 2 h + ρ × 2 3 π r 2 ⇒ ρ × 1 3 π r 2 h 2 4 − 2 r × 3 r 8 = 0 h 2 4 − 3 r 2 4 = 0 or h = 3 r

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