A sphere of mass m 1 and a block of mass m 2 are connected by a light cord that

Question

A sphere of massm1and a block of massm2are connected by a light cord that passes over a pulley as shown in the figure. The radius of the pulley isR, and the mass of the thin rim isM. The spokes of the pulley have negligible mass. The block slides on a frictionless, horizontal surface. Find an expression for the linear acceleration of the two objects, using the concepts of angular momentum and torque.

Suppose the surface in the example is sloped as shown in the figure, but everything else remains the same. Determine the equation for the acceleration in this case using angular momentum and torque. Let a positive acceleration represent acceleration ofm2up the surface. (Use the following as necessary:m1,m2,M,?, andg.)

Explanation / Answer

m1*a = m1*g - T1

T2 = m2*a

(T1-T2)*R = I = MR^2*(a/R) = MaR

so, T1 - T2 = Ma

(m1*g - m1*a) - m2*a = Ma

a = m1*g/(M+m1+m2)

For slope,

m1*a = m1*g - T1

m2*a = T2 - m2*(g Cos)

T1 - T2 = Ma yields,

(m1*g - m1*a) - (m2*a + m2*g Cos) = Ma

a = (m1 + m2 cos)g/(M+m1+m2)

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