*Please show all intermediate steps with clarity Consider a pulley of radius r a

Question

*Please show all intermediate steps with clarity

Consider a pulley of radius r and of moment of inertia I, with a frictionless horizontal rotation axis. On the pulley is wound a light inextensible rope that supports a bucket of mass m. At some point the bucket is held stationary and the slack in the rope removed. The bucket is then released to fall under gravity. For the subsequent motion find (using terms defined above) algebraic expressions for; alpha, the angular acceleration of the pulley, a, the linear acceleration of the bucket, omega, the angular velocity of the bucket after time t s, T, the tension in the rope.

Explanation / Answer

By FBD of pulley we have-

I(alpha)=T(r) ..................(equ 1)

FBD of bucket-

mg-T=ma.....................(equ 2)

We also know-

a=r(alpha)...................(equ 3)

We have 3 equ and 3 variables(alpha,a,T)-

I(a)/(r*r)=T..................(equ 1)

mg-T=ma....................(equ 2)

equ 1 - equ 2

I(a)/(r*r)-mg=-ma

Therefore following are the results-

b) a=mg(r^2)/(I+mr^2)

a) alpha = a/r=mg(r)/(I+mr^2)

c) w=alpha(t)=mg(r)(t)/(I+mr^2)

d) From equ 1-

T=I(a)/(r*r)=mg(I)/(I+mr^2)

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