Rotational motion physics question?

Question

A frictionless pulley (a solid cylinder) of unknown mass M and radius r = 0.41 m is used to draw water from a well. A bucket of mass m = 1.42 kg is attached to a cord wrapped around the cylinder. A bucket starts from rest at the top of the well and falls for t = 2.82 s before hitting the water h = 7.85 m below. Neglect the mass of the cord.
a) What is the linear acceleration a of the falling bucket?
b) What is the angular acceleration,a, of the rotating pulley?
c) What is the tension T force for the cord?
d) What is the value of the torque,T, that is applied to the pulley by the bucket hanging on the card?
e) Using the value of the torque and the angular acceleration, find the moment of inertia I of the rotating pulley
f) Using the moment of inertia, calculate the mass of the pulley, M
g) What is the linear velocity, v, of the falling bucket just before it hits the water?
h) What is the angular velocity, w, of the rotating pulley just before the falling bucket hits the water?

Answer 1

Potential energy of the bucket
Pe(b)=mgh

Kinetic energy of the bucket
Ke(b)=(1/2)mV^2

Kinetic energy of the cylinder
Ke(c)=(1/2)I w^2

I=(1/2) m r^2
w= V/r
Ke(c)=(1/2) (1/2) M r^2 V^2/r^2=
Ke(c)= (1/4)M V^2

conservation of energy equation
Pe(b)= Ke(b)+Ke(c)

Force equation

Wb - T/r=F(b)
mg -T/r = ma and since T= I A= I a/r =(1/2)Mr^2 a/r
mg - (1/2)M a r^2/r^2=ma

mg= ma + (1/2)Ma

a) So a= mg/(m+ 0.5 M)
b) A=a/r= mg/(m+ 0.5 M)/r
c) Ft=ma
d) T=I A or T=r Ft
e) I=T/A
f) since I=0.5 Mr^2 then
M=2I/r^2
g) v=at
h) w=v/r