A block of mass M is lying on a frictionless horizontal surface and it connected to a spring of spring constant k, whose other end is fixed to a wall, system executes simple harmonic motion of amplitude `A_(0)`, when the block is passing through its equilibrium positions, an object of mass m is put on it and the two move together with a new time period T and amplitude A. Find the value of T and A in terms of given constants.

Explanation

To solve the problem, we will derive the expressions for the new time period \( T \) and the new amplitude \( A \) after a mass \( m \) is placed on a block of mass \( M \) that is oscillating due to a spring with spring constant \( k \).

### Step 1: Determine the original time period \( T_0 \)

The time period \( T_0 \) of the simple harmonic motion (SHM) for the block of mass \( M \) connected to the spring is given by the formula:

\[
T_0 = 2\pi \sqrt{\frac{M}{k}}
...