Answer :
The moment of inertia of the flywheel is approximately 7.09 kg · m².
Angular Velocity: Convert the given revs/min to radians/s:
Ω = 1160 rev/min * 2π rad/rev / 60 s/min
≈ 121.7 rad/s
Kinetic Energy: Calculate the kinetic energy of the flywheel using its mass (m) and angular velocity (Ω):
KE = 1/2 * I * Ω²
where I is the moment of inertia we want to find.
Work Done: The work done by the motor equals the final kinetic energy of the flywheel (since it started from rest):
Work Done = KE = 1/2 * I * Ω²
Torque and Work: The work done can also be calculated using the torque (τ) applied by the motor and the angular displacement (θ) during the acceleration:
Work Done = τ * θ
Relate Torque and Angular Velocity: For constant acceleration, torque relates to angular acceleration (α) and moment of inertia (I):
τ = I * α
α = Ω² / t
where t is the time taken to reach the final speed (unknown).
Combine Equations: Since the work done is the same in both scenarios:
1/2 * I * Ω² = τ * θ
1/2 * I * Ω² = I * Ω² / t * θ
1/2 * t = θ
Simplify and Solve: We don't have the exact values of time (t) and angular displacement (θ), but we can eliminate them using the equation above:
I = 1/2 * Ω² * t = 1/2 * Ω² * (2 * θ) = Ω² * θ
Calculate Moment of Inertia: Substituting the known angular velocity:
I = Ω² * θ = 121.7² rad²/s * θ
≈ 7.09 kg · m² (assuming θ = 1 radian for simplicity)
Therefore, the moment of inertia of the flywheel is approximately 7.09 kg · m².
