Please help! I can\'t figure this question out :( In the figure below, the hangi

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Please help! I can't figure this question out :(

In the figure below, the hanging object has a mass of m_1= 0.430 kg; the sliding block has a mass of m_2 = 0.870 kg; and the pulley is a hollow cylinder with a mass of M = 0.350 kg, an inner radius of R_1 = 0.020 0 m, and an outer radius of R_2 = 0.030 0 m. Assume the mass of the spokes is negligible. The coefficient of kinetic friction between the block and the horizontal surface is mu k = 0.250. The pulley turns without friction on its axle. The light cord does not stretch and does not slip on the pulley. The block has a velocity of v_1 = 0.820 m/s toward the pulley when it passes a reference point on the table. Use energy methods to predict its speed after it has moved to a second point, 0.700 m away, m/s Find the angular speed of the pulley at the same moment, rad/s

Explanation / Answer

m1 = 0.430 kg

m2 = 0.870 kg

M = mass of Pulley = 0.350 kg

R1 = inner radius = 0.0200 m

R2 = outer radius = 0.0300 m

h = 0.700m

uk = cofficient of friction = 0.250

Vi = initial velocity = 0.820 m/s

I = moment of interia of pulley = (1/2)*M*( R12+R22)

V = w*r

a) change in K.E of system ( block m1+m2+pulley) = net work done ( gravitation force on m1 and friction force on m2 has nonzero work)

(1/2)*m1*( Vf2 - Vi2) + (1/2)*m2*(  Vf2 - Vi2) + (1/2)* I * ( wf2 - wi2) = m1*g*h - uk*m2*g

( Vf2 - Vi2) ( 1/2)* ( m1+m2) + (1/2)*M*( R12+R22)*( Vf2 -Vi2 / R22) = m1*g*h - uk*m2*g

on rearranging the terms,

Vf = ( Vi2 + ( m1*g*h - uk*m2*g / (1/2)* (m1+m2) + (1/2)*M*( 1 + R12 / R22)) )1/2

on putting the values, Vf ( speed at 0,700m away) = 1.25 m/s

b) angular speed of pulley, wf = Vf / R2 ( V = w*R)

= 1.25 / 0.0300

= 41.66 rad/s

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