A large grinding wheel in the shape of a solid cylinder of radius 0.330 m is free to rotate on a frictionless, vertical axle. A constant tangential force of 280 N applied to its edge causes the wheel to have an angular acceleration of 0.960 rad/s2.
(a) What is the moment of inertia of the wheel?
kg · m2
(b) What is the mass of the wheel?
kg
(c) If the wheel starts from rest, what is its angular velocity after 4.50 s have elapsed, assuming the force is acting during that time?
rad/s
To find the moment of inertia of the wheel, you can use the formula:
Moment of Inertia = Torque / Angular Acceleration
In this case, the torque can be calculated using the formula:
Torque = Force * Radius
where the force is given as 280 N and the radius is given as 0.330 m.
Therefore, the torque is Torque = 280 N * 0.330 m = 92.4 N·m.
Next, substitute the torque and the given angular acceleration (0.960 rad/s^2) into the formula for moment of inertia:
Moment of Inertia = 92.4 N·m / 0.960 rad/s^2 = 96.25 kg·m^2 (a)
To find the mass of the wheel, you can use the formula:
Moment of Inertia = Mass * Radius^2
Rearranging the formula, you get:
Mass = Moment of Inertia / Radius^2
Substitute the given moment of inertia (96.25 kg·m^2) and the given radius (0.330 m) into the formula to find the mass:
Mass = 96.25 kg·m^2 / (0.330 m)^2 = 883.58 kg (b)
To find the angular velocity of the wheel after 4.50 s have elapsed, you can use the formula:
Angular Velocity = Initial Angular Velocity + (Angular Acceleration * Time)
The initial angular velocity is zero since the wheel starts from rest. The given angular acceleration is 0.960 rad/s^2, and the given time is 4.50 s.
Therefore, the angular velocity is:
Angular Velocity = 0 + (0.960 rad/s^2 * 4.50 s) = 4.32 rad/s (c)
So, after 4.50 s have elapsed, the angular velocity of the wheel is 4.32 rad/s.