Calculate $T_A, T_B, T_C$ for the following diagram: ($B$ is the mid-point of rope)
 

A body of mass 'm' is suspended from a vertical spring of length 'L'. It execute S.H.M. with a time period T. Find the time period if

(a) Length of spring is made half.

(b) Mass of body is made half.

A block of mass m placed on a smooth floor is connected to a fixed support with the help of a spring of force constant k. It is pulled by a rope as shown in the figure. Tension T of the rope is increased gradually without changing its direction, until the block losses contact the floor. The increase in rope tension T is so gradual that acceleration in the block can be neglected. What is the necessary tension in the rope so that the block looses contact with the floor?

Determine the acceleration of the masses w.r.t. lift and tension in the string if the whole system is moving vertically upwards with uniform acceleration a₀. (m₁ > m₂)

A cart of mass $M$ has a block of mass $m$ in contact with it as shown in figure. The coefficient of friction between the block and the cart is $\mathrm{\mu }$. What should be the minimum acceleration of the cart so that the block of mass m does not fall ?