A solid sphere A of mass 5m and a thin spherical shell B of mass m have the same radius. Each is placed on a rough horizontal surface. A horizontal force of the same magnitude F is applied tangentially at the highest point of each body. Both bodies start rolling without slipping with linear accelerations $a_{A}$ and $a_{B}$ , respectively. What is the ratio $a_{\char"391 }$ : $a_{B}$ ?
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Question

A solid sphere A of mass 5m and a thin spherical shell B of mass m have the same radius. Each is placed on a rough horizontal surface. A horizontal force of the same magnitude F is applied tangentially at the highest point of each body. Both bodies start rolling without slipping with linear accelerations $a_{A}$ and $a_{B}$ , respectively. What is the ratio $a_{\char"391 }$ : $a_{B}$ ?

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G1 Physics · Accepted Answer

For the solid sphere A, I(A) = (2 ÷ 5)(5m)R² = 2mR². Let friction be f(A). Translational motion gives F+f(A) = 5m a(A). Torque about the centre gives FR-f(A)R = I(A)α. For rolling without slipping, α = a(A) ÷ R. Hence, F-f(A) = 2m a(A). Adding the translational and rotational equations: 2F = 7m a(A), so a(A) = 2F ÷ (7m). For the spherical shell B, I(B) = (2 ÷ 3)mR². Similarly, F+f(B) = m a(B) and F-f(B) = (2 ÷ 3)m a(B). Adding: 2F = (5 ÷ 3)m a(B), so a(B) = 6F ÷ (5m). Therefore, a(A) ÷ a(B) = […