A high-speed flywheel in a motor is spinning at 500 rpm when a power failure suddenly occurs. The flywheel has mass 40.0 kg and diameter 75.0 cm. The power is off for 30.0 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 200 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, and how many revolutions would the wheel have made during this time?
At $t =$ 0 a grinding wheel has an angular velocity of 24.0 rad/s. It has a constant angular acceleration of 30.0 rad/s$^2$ until a circuit breaker trips at $t =$ 2.00 s. From then on, it turns through 432 rad as it coasts to a stop at constant angular acceleration. (a) Through what total angle did the wheel turn between $t =$ 0 and the time it stopped? (b) At what time did it stop? (c) What was its acceleration as it slowed down?
A safety device brings the blade of a power mower from an initial angular speed of $\omega_1$ to rest in 1.00 revolution. At the same constant acceleration, how many revolutions would it take the blade to come to rest from an initial angular speed $\omega_3$ that was three times as great, $\omega_3 = 3\omega_1$?
So here it wants us to drop drive an equation for angular acceleration without solving for time So we can say our initial angular velocity will be twelve cleanser Radiance for seconds, our final angular velocity is going to be sixteen radiance per seconds. And then our change and angular displacement will be seven radiance. So we can first start off by saying omega half equals When they go, I plus out the tea and then we can say angular displacement Eagle singular initial in your basement, plus on mayor initial T plus half offa t squared and the fate of final minus data initial. It's something an equal Delta theta. And then here we can square we're going to square this term. So if we squared this term, this equation rather we would have Omega Final Squared equals Omega initial squared plus Alfa squared Times Square plus two off a national Omega initial Alfa team. So this would that would be this equation squared. And then what we're seeing here is that we can actually factor out a two Alfa so we can say Omega final squared equals omega initials there. Plus, we can say to Alfa, we could leave us with one over it. Two Alfa T squared plus omega initial T, and this is going to equal this So we can say that Omega initial T may serve no Omega initial squared, plus two Alfa times Delta A Fada will equal our final omega squared. And at this point, we have eliminated time so we can just solve for the angular acceleration and say that Alfa equals Omega Final squared minus omega initial squared, divided by two times Delta Theta. This is going to be sixteen squared, minus twelve squared call over two times seven radiance and the soon to be plus eight radiance per second squared. So this will be our angular acceleration without solving for time. And that's the end of the solution. Thank you for watching.