Use the law of conservation of mechanical energy to prove that the linear accele

Question

Use the law of conservation of mechanical energy to prove that the linear acceleration of a hoop rolling down an inclined plane (of angle ) is given by Equation A (as seen below). Refer to the image below (Figure 10.12) for the relevant moment of inertia. Show similar proofs for Equations B, C, and D.

Equation A (hoop): a = (1/2)*g*sin

Equation B (solid cylinder): a = (2/3)*g*sin

Equation C (spherical shell): a = (3/5)*g*sin

Equation D (solid sphere): a = (5/7)*g*sin

Ma Connections In statics, the net torque is zero, and there is no angular acceleration. In rotational motion, net torque is the cause of angular acceleration, exactly as in Newton's second law of motion for rotation. Axis Axis Hoop about cylinder axis Annular cylinder (or ring) about cylinder axis -MR Ax Axis Solid cylinder (or disk) about cylinder axis Solid cylinder (or disk) about central diameter 1=MR2 MF +M 4 12 Thin rod about axis through centerto Thin rod about axis through one end 1 to length Me M 2 12 Axis Solid sphere about any 2R 2R spherical shel about any 3 Slab about L axis through center Hoop about any diameter M (a2+ Figure 10 .12 Some rotational inertias.

Explanation / Answer

Applying energy conservation,

PEi + KEI = PEf + KEf

m g h + 0 = (m v^2 /2 ) + ( Iw^2 / 2)

2 m g (L sin(theta)) = m v^2 + ( m r^2)(v/r)^2

v^2 = g L sin(theta)

Now applying vf^2 - v0^2 = 2 a d

g L sin(theta) - 0 = 2 a L

a = (g sin(theta)) /2 ....Ans

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