Find the time period of the oscillation of mass m in figures 12.E4 a, b, c. What is the equivalent spring constant of the pair of springs in each case?

The block of mass m₁ shown in figure (12-E2) is fastened to the spring and the block of mass m₂ is placed against it. (a) Find the compression of the spring in the equilibrium position. (b) The blocks are pushed a further distance (2/k) (m₁ + m₂) g sinθ against the spring and released. Find the position where the two blocks separate. (c) What is the common speed of blocks at the time of separation?

In figure (12-E3) k = 100 N m^-1, M = 1 kg and F = 10 N. (a) Find the compression of the spring in the equilibrium position. (b) A sharp blow by some external agent imparts a speed of 2 m s^-1 to the block towards left. Find the sum of the potential energy of the spring and the kinetic energy of the block at this instant. (c) Find the time period of the resulting simple harmonic motion. (d) Find the amplitude. (e) Write the potential energy of the spring when the block is at the left extreme. (f) Write the potential energy of the spring when the block is at the right extreme. The answers of (b), (e) and (f) are different. Explain why this does not violate the principle of conservation of energy.

The spring shown in figure (12-E5) is upstretched when a man starts pulling on the cord. The mass of the block is M. If the man exerts a constant force F, find (a) the amplitude and the time period of the motion of the block, (b) the energy stored in the spring when the block passes through the equilibrium position and (c) the kinetic energy of the block at this position.

A particle of mass m is attached to three springs A, B and C of equal force constant as shown in figure (12-E6). If the particle is pushed slightly against the spring C
and released, find the time period of oscillation.