To solve this problem, we can use the equations of angular motion. The angular acceleration or retardation can be calculated using the formula:
θ = ω_i * t + (1/2) * α * t^2
where:
θ = angular displacement (in radians)
ω_i = initial angular velocity
α = angular acceleration/retardation
t = time (in seconds)
In this problem, the initial angular velocity ω_1 is ω, and the final angular velocity ω_2 is 0 (since the disc comes to rest). We need to find the time it takes to go from ω to ω_2 during 120 rotations.
First, we need to find the angular acceleration α. Using the formula:
ω_2 = ω_1 + α * t
0 = ω - α * t
α = ω / t
Now, we need to find the angular displacement θ for 120 rotations. Since one rotation corresponds to 2π radians, 120 rotations correspond to:
θ = 120 * 2π = 240π radians
Now, we can use the first equation of angular motion to find the time it takes to go from ω to 0 angular velocity:
θ = ω_i * t + (1/2) * α * t^2
240π = ω * t + (1/2) * (ω / t) * t^2
240π = ω * t + (1/2) * ω * t
240π = (3/2) * ω * t
t = (2/3) * (240π / ω)
Now, we can substitute the value of ω (initial angular velocity) and calculate t:
t = (2/3) * (240π / ω) = (2/3) * (240π / ω)
Now, we can find the number of rotations made before coming to rest using the formula:
Number of rotations = θ / (2π)
Number of rotations = (240π) / (2π) = 120
So, the number of rotations made by the disc before coming to rest is 120 rotations, which corresponds to option A.
