Understanding the Problem:
We are given that a flywheel rotates about an axis due to friction, and this friction is proportional to its angular velocity. The problem states that when the angular velocity of the flywheel falls to one fourth of its initial value, the flywheel will come to rest. We need to determine the number of additional rotations the flywheel will make before coming to rest, given that it initially made 'n' rotations.

Analysis:
Let's assume the initial angular velocity of the flywheel is ω₀ and the final angular velocity (when it comes to rest) is ωf. We are given that ωf = ω₀/4.

We can use the concept of angular velocity to find the number of rotations. The angular velocity (ω) is defined as the rate of change of angle with respect to time. It is given by the equation:

ω = θ/t

where:
ω = angular velocity (in radians per second)
θ = angle rotated (in radians)
t = time taken (in seconds)

We know that the time taken for 'n' rotations is equal to the time taken for one rotation multiplied by 'n'. So, we can write:

θ = ω₀ * t₁ * n

where:
t₁ = time taken for one rotation

Similarly, for the final angular velocity, we have:

θ = ωf * t₂

where:
t₂ = time taken for the flywheel to come to rest

Solution:
Step 1: Finding the initial angle rotated:
We can rearrange the equation for θ to solve for t₁:
t₁ = θ / (ω₀ * n)
Substituting the values, we have:
t₁ = 2πn / (ω₀ * n)
t₁ = 2π / ω₀

Now, we can find the initial angle rotated (θ₀) using the equation:
θ₀ = ω₀ * t₁ * n
θ₀ = ω₀ * (2π / ω₀) * n
θ₀ = 2πn

Step 2: Finding the time taken for the flywheel to come to rest:
Using the equation for θ and ωf, we have:
θ = ωf * t₂
θ = (ω₀/4) * t₂
θ = ω₀ * (t₂/4)

Since the flywheel comes to rest when θ = 2π, we can solve for t₂:
2π = ω₀ * (t₂/4)
8π = ω₀ * t₂
t₂ = 8π / ω₀

Step 3: Finding the additional rotations:
The additional rotations (n') can be calculated by dividing the time taken for the flywheel to come to rest by the time taken for one rotation:
n' = t₂ / t₁
n' = (8π / ω₀) / (2π / ω₀)
n' = 4

Therefore, the flywheel will make an additional 4 rotations before coming to rest.

Conclusion:
In summary, when the angular velocity of the flywheel falls to one fourth of its initial value, it will make an additional 4 rotations before coming to