Given:
- Angular velocity of the disc, ω
- The disc is subjected to uniform angular retardation
- Angular velocity is decreased from ω to ω/2 during 120 rotations

To find:
- The number of rotations further made by the disc before coming to rest

Explanation:

To solve this problem, we need to use the concept of angular acceleration and kinematic equations of rotational motion.

1. Angular Acceleration:
Angular acceleration (α) is the rate of change of angular velocity (ω) with respect to time (t).
α = (ωf - ωi) / t
where,
ωf is the final angular velocity
ωi is the initial angular velocity
t is the time taken for the change in angular velocity

In this case, the angular acceleration is uniform as the disc is subjected to uniform angular retardation.

2. Rotational Kinematic Equation:
The rotational kinematic equation relates the angular displacement (θ), initial angular velocity (ωi), final angular velocity (ωf), and angular acceleration (α).
θ = ωi * t + (1/2) * α * t^2

3. Solving the Problem:
Given that the angular velocity (ω) is decreased from ω to ω/2 during 120 rotations, we can find the angular acceleration using the following formula:
α = (ωf - ωi) / t
α = (ω/2 - ω) / (120 * 2π)

Now, we can use the rotational kinematic equation to find the angular displacement (θ) when the disc comes to rest (ωf = 0).
0 = ω * t + (1/2) * α * t^2

Simplifying the equation, we get:
0 = ω * t + (1/2) * [(ω/2 - ω) / (120 * 2π)] * t^2

Solving this equation will give us the value of t, which represents the time taken for the disc to come to rest.

4. Number of Rotations:
Once we have the time taken (t) for the disc to come to rest, we can calculate the number of rotations made during this time.

Number of rotations = ω * t / 2π

This will give us the number of rotations made by the disc before coming to rest.

Summary:
- The angular acceleration is calculated using the formula α = (ωf - ωi) / t, where ωf is the final angular velocity, ωi is the initial angular velocity, and t is the time taken for the change in angular velocity.
- The rotational kinematic equation θ = ωi * t + (1/2) * α * t^2 relates the angular displacement, initial angular velocity, final angular velocity, and angular acceleration.
- By solving the equation ω * t + (1/2) * [(ω/2 - ω) / (120 * 2π)] * t^2 = 0, we can find the time taken for the disc to come to rest.
- The number of rotations made before coming to rest is calculated using the formula ω * t / 2π, where