general energy problems

1. An object is attached to a vertical spring and slowly lowered to its equilibrium position. This stretches the spring by an amount d. If the same object is attached to the same vertical spring but permitted to fall instead, through what maximum distance does it stretch the spring?

2. A small block of mass m slides along the frictionless loop-the-loop track (of radius r) shown in the diagram. What would the minimum height be such that the block goes all the way around the loop?

3. A point mass m starts from rest and slides down the surface of a frictionless solid sphere of radius r. Find the angle at which the mass flies off the sphere.

4. A mass m is dropped from rest above a relaxed spring of stiffness k a distance D. Find the position from where it was released where the mass attains its maximum velocity and find that maximum velocity.

5. A mass attached to a string of length L is released from rest from an initial horizontal position (although the diagram implies it is not that high, please ignore that part of the diargram - sorry). From veritcally below where the string is attached at its top, there is a peg a distance x below the top attachment of the string. As the mass swings down to its lowest position, the string then touches the peg and the mass swings up and around from this new point (i.e. the peg's position). The mass continues to swing upward. Find the tension in the string when the mass is at its highest vertical point.

6. Find the potential energy function for the three mass, earth, and pulley system as shown. The potential enegy will be as a function of the vertical position downward as shown in the diagram. Also, find the equlibrium position of this system. The two outside masses are equal.

7. A rope with a length L and total mass M lies coiled on a table. How much work would it take an applied force to lift the rope completely off the table at constant speed? At constant kinetic energy?