Question

A high-speed flywheel in a motor is spinning at 500 rpm when a power failure suddenly occurs. the flywheel has mass 41.0 kg and diameter 71.0 cm . the power is off for 26.0 s , and during this time the flywheel slows due to friction in its axle bearings. during the time the power is off, the flywheel makes 180 complete revolutions.

(a) At what rate is the flywheel spinning when the power comes back on?
(b) How long after the beginning of the power failure would it have taken the flywheel to stop if the power had not come back on?
(c) How many revolutions would the wheel have made during this time?

Answer 1

To solve this problem, we can use the principle of conservation of angular momentum and the equation for rotational kinetic energy. We can find the rate at which the flywheel is spinning when the power comes back on using the conservation of angular momentum. By calculating the angular deceleration, we can determine how long it would have taken for the flywheel to stop if the power had not come back on. Finally, we can find the number of revolutions the wheel would have made during this time using the formula for angular displacement.

Step-by-step explanation:

To solve this problem, we can use the principle of conservation of angular momentum and the equation for rotational kinetic energy.

(a) When the power comes back on, the flywheel is still spinning. We can use the conservation of angular momentum to find the final angular velocity. Since angular momentum is conserved, we can write:

L_initial = L_final

I_initial * w_initial = I_final * w_final

where L_initial is the initial angular momentum, L_final is the final angular momentum, I_initial is the initial moment of inertia, I_final is the final moment of inertia, w_initial is the initial angular velocity, and w_final is the final angular velocity.

In this case, the initial moment of inertia and angular velocity are known, and the final moment of inertia can be calculated using the formula for the moment of inertia of a flywheel:

I = 0.5 * m * r^2

where I is the moment of inertia, m is the mass of the flywheel, and r is the radius of the flywheel.

(b) To find how long it would have taken for the flywheel to stop if the power had not come back on, we can use the equation for angular deceleration:

alpha = (w_final - w_initial) / t

where alpha is the angular deceleration, w_final is the final angular velocity, w_initial is the initial angular velocity, and t is the time.

(c) To find how many revolutions the wheel would have made during this time, we can use the formula:

theta = w_initial * t + 0.5 * alpha * t^2

where theta is the angular displacement, w_initial is the initial angular velocity, t is the time, and alpha is the angular acceleration.