Two buckets are connected by a massless rope passing over a frictionless pulley

Question

Two buckets are connected by a massless rope passing over a frictionless pulley of negligible mass and friction. If the system is released from rest with the bucket of mass M_1 at a height h_0 and bucket of mass M_2 on the floor ( for the pulley I = 1/2 M_pR_p ^2) find the speed of bucket M_1 when it strikes the floor. If the Pulley had mass M_p and radius R_p what speed would the mass m_1 strike the floor with? If the Pulley had mass M_p and radius R_p what would the acceleration of mass m_1 be?

Explanation / Answer

a) using energy conservation ,

initial PE + Ke = final PE + KE

( 0 + M1gh) + (0 + 0) = (M2gh + 0) + (M1 +M2)v^2 / 2

v = sqrt[ 2(M1 - M2)gh / (M1 + M2) ]

b) when pulleey also have mass then it will get rotational KE.

using energy conservation,

( 0 + M1gh) + (0 + 0) = (M2gh + 0) + (M1 +M2)v^2 / 2 + ((Mp Rp^2 /2) ( v / Rp)^2)

M1gh - M2gh = (2M1 + 2M2 + Mp) v^2 / 4

v = sqrt [ 4(M1 - M2)gh / (2M1 + 2M2 + mp) ]

c) using v^2 - u^2 = 2ah

4(M1 - M2)gh / (2M1 + 2M2 + mp) = 2 x a x h

a = 2(M1 - M2)g / (2M1 + 2M2 +Mp )

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