Explanation:

To understand why the time period of oscillation for a mass 4M is 2 seconds, we need to first understand the relationship between the time period of oscillation and the mass of the object.

Time Period of Oscillation:
The time period of oscillation is the time taken for one complete cycle of oscillation. It is denoted by the symbol T and is measured in seconds.

Hooke's Law and the Time Period:
The time period of oscillation for a mass-spring system can be determined using Hooke's law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position.

Mathematically, Hooke's law can be expressed as:

F = -kx

Where F is the force exerted by the spring, k is the spring constant, and x is the displacement from the equilibrium position.

The force exerted by the spring can also be expressed as:

F = ma

Where m is the mass of the object and a is the acceleration.

Equating the two expressions for F, we get:

ma = -kx

This equation can be rearranged to:

a = -(k/m)x

This equation represents the acceleration of the object as a function of its displacement.

Simple Harmonic Motion:
When an object is subjected to a restoring force that is proportional to its displacement and acts in the opposite direction, it undergoes simple harmonic motion.

In the case of a mass-spring system, the object oscillates back and forth around its equilibrium position with a constant amplitude.

The equation of motion for simple harmonic motion is given by:

x(t) = A cos(ωt + φ)

Where x(t) is the displacement of the object at time t, A is the amplitude of the motion, ω is the angular frequency, and φ is the phase angle.

Angular Frequency and Time Period:
The angular frequency of the oscillation is given by:

ω = √(k/m)

And the time period is given by:

T = 2π/ω

Substituting the value of ω in terms of k and m, we get:

T = 2π√(m/k)

From the above equation, we can see that the time period is inversely proportional to the square root of the mass.

Relation between Mass and Time Period:
If the time period of oscillation of a mass M is one second, then we can write:

1 = 2π√(M/k)

Squaring both sides of the equation, we get:

1 = 4π^2(M/k)

Simplifying the equation further, we get:

k = 4π^2M

Now, we need to find the time period for a mass 4M. Let's denote it as T'.

Using the equation for time period, we have:

T' = 2π√(4M/k)

Substituting the value of k, we get:

T' = 2π√(4M/4π^2M)

Simplifying the equation further, we get:

T' = 2

Therefore, the time period of oscillation for a mass 4M is 2 seconds.