Answer :
The angular displacement of the top accelerating from 12 rpm to 100 rpm in 13 seconds is approximately (b) 76.2 radians.
Calculating Angular Displacement
The angular displacement can be found using the formula for angular displacement under constant angular acceleration:
θ = ω₀t + 0.5αt²
First, we convert the given rates from rpm to rad/s: ω₀ = 12 rpm × (2π / 60) = 1.26 rad/s and ω_f = 100 rpm × (2π / 60) = 10.47 rad/s.
Next, the angular acceleration is calculated using the formula: α = (ω_f - ω₀) / t
So, α = (10.47 - 1.26) / 13 = 0.71 rad/s².
Finally, substituting the values into the angular displacement formula:
θ = 1.26 × 13 + 0.5 × 0.71 × 13² = 16.38 + 0.5 × 0.71 × 169 = 16.38 + 59.95 = 76.33 radians ≈ 76.2 radians.