physics question!!!!!!!?

Question

A cylinder with moment of inertia I about its center of mass, mass m, and radius r has a string wrapped around it which is tied to the ceiling. The cylinder's vertical position as a function of time is y(t).

At time t=0 the cylinder is released from rest at a height h above the ground.

Part A) The string constrains the rotational and translational motion of the cylinder. What is the relationship between the angular rotation rate w and v, the velocity of the center of mass of the cylinder?

Remember that upward motion corresponds to positive linear velocity, and counterclockwise rotation corresponds to positive angular velocity.

1) Express w in terms of vand other given quantities.

2) Suppose that at a certain instant the velocity of the cylinder is v. What is its total kinetic energy, Ktot, at that instant?
Express Ktotal in terms of m,r , I, and v.

3) Find vr , the cylinder's vertical velocity when it hits the ground.

Answer 1

The moment of inertia of a solid cylinder is .5*m*R^2
as the cylinder descends it gains translational kinetic energy
.5*m*v^2
and rotational kinetic energy
.5*I*w^2

since the cylinder is not slipping w/r/t the string, then w*R=v

Ignoring air and other frictional forces, the potential energy
m*g*h will get converted to kinetic energy as
m*g*h=.5*m*v^2+.5*I*w^2
Substituting in I=.5*m*R^2 and doing a bit of algebra
2*g*(h-y))=v^2+.5*R^2*w^2
substituting in w*R=v
2*g*(h-y)=w^2*R^2*1.5
or
w^2=4*g*(h-y)/(3*R^2)
b) ktot using .5*m*v^2+.5*I*w^2
and w=v/R
ktot=.5*v^2*(m+I/R^2)

c vr (will be negative)
using
w^2=4*g*(h-y)/(3*R^2)
and w=v/R, when y=0
v^2=4*g*h/3
v=-2*sqrt(g*h/3)
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