A block of mass m is released from a height of R. The block slides down along the path ABC. At point

C the block collides a spring of sti�ness k.
(a) If the circular arc AB is frictionless and the coe�cient of kinetic friction between the block and
surface BC is �k, then �nd the maximum compression in the spring.[3 points]
(b) If the coe�cient of friction through out the path ABC is �k, then �nd the maximum compression
in the spring.In both case (a) and (b) assume that the spring is initially undeformed. Hint - For case(b) you have
to �rst draw the FBD of the block to �nd the Normal force on the block as a function of the position
of the block on the circular arc. The work done by friction has to be obtained through an integration
B C
A O
R
R
0.5 R
Figure 1: Figure for question 1
2. A chain of length L is released from rest on a smooth incline with x = 0, as shown in �gure 3. Determine
the velocity v of the links in terms of x. (Hint - Use energy methods). [5 points].
L - x
x
Figure 2: Figure for question 2
1

To solve these problems, we can use principles of mechanics and energy conservation.

(a) In the first problem, we want to find the maximum compression in the spring when a block slides down along the path ABC.

To find the maximum compression in the spring, we can use the principle of mechanical energy conservation. The initial mechanical energy of the block at point A is equal to its final mechanical energy at point C.

The initial mechanical energy at point A is the potential energy due to its height, given by mgh, where m is the mass of the block, g is the acceleration due to gravity, and h is the height of point A.

The final mechanical energy at point C is the sum of the potential energy due to the spring compression and the kinetic energy of the block.

Assuming the block comes to rest momentarily at point C, the final kinetic energy is zero. Therefore, the final mechanical energy at point C is only the potential energy due to the spring compression, given by (1/2)kx^2, where k is the stiffness of the spring and x is the compression of the spring.

Equating the initial and final mechanical energies, we have:

mgh = (1/2)kx^2

Solving for x, we can find the maximum compression in the spring.

(b) In the second problem, we want to determine the velocity v of the links in terms of x, where a chain of length L is released from rest on a smooth incline.

To solve this problem, we can once again use the principle of mechanical energy conservation. The initial mechanical energy at the top of the incline is equal to the final mechanical energy at any point x along the incline.

The initial mechanical energy at the top of the incline is the potential energy due to the height of the chain, given by mgh, where m is the mass of the chain, g is the acceleration due to gravity, and h is the height of the top of the incline.

The final mechanical energy at point x is the sum of the potential energy due to the remaining height of the chain (L - x) and the kinetic energy of the chain.

Equating the initial and final mechanical energies, we have:

mgh = (1/2)mv^2 + mg(L - x)

Simplifying and solving for v, we can find the velocity of the links in terms of x.