To solve this problem, we need to analyze the situation step by step, focusing on the principles of simple harmonic motion (SHM) and the effects of adding an additional mass to the system. Let's break it down.

Understanding the Initial Setup

Initially, we have a mass attached to a spring that is executing simple harmonic motion. The amplitude of this motion is given as A = √10. The spring constant is denoted by K, and the gravitational force acting on the mass is mg, which is provided as 2K. This means that the weight of the mass is twice the spring constant.

Calculating the Initial Mass

From the relationship mg = 2K, we can express the mass m in terms of K:

• m = 2K/g

Here, g is the acceleration due to gravity. This relationship will help us understand how the system behaves when another mass is added.

Introducing the Additional Mass

When the stationary mass m gently sticks to the mass already attached to the spring, the total mass of the system becomes:

• m_total = m + m = 2m

Now, substituting the expression for m we derived earlier:

• m_total = 2(2K/g) = 4K/g

Effect on the Amplitude

In SHM, the amplitude is related to the energy of the system. The total mechanical energy in SHM is given by:

• E = (1/2)K A²

Initially, the energy of the system with amplitude A = √10 is:

• E_initial = (1/2)K (√10)² = (1/2)K (10) = 5K

After the additional mass is added, the new system has a total mass of 2m, which we calculated as 4K/g. The new spring constant remains K, but the total mass affects the frequency of oscillation. The new amplitude A' can be found using the conservation of energy principle:

• E_initial = E_final
• 5K = (1/2)K (A')²

Solving for the New Amplitude

Now, we can set the energies equal to each other:

• 5K = (1/2)K (A')²

Dividing both sides by K (assuming K is not zero) gives:

• 5 = (1/2) (A')²

Multiplying both sides by 2 results in:

• 10 = (A')²

Taking the square root of both sides, we find:

• A' = √10

Final Result

Interestingly, the new amplitude A' after the additional mass sticks to the original mass remains the same as the initial amplitude, which is √10. This outcome occurs because while the mass increased, the energy of the system adjusted accordingly, keeping the amplitude unchanged.

In summary, when the stationary mass is added to the system, the new amplitude of the oscillation remains at √10, despite the increase in mass. This illustrates the balance between mass and energy in simple harmonic motion.