3- An electric motor can accelerate a Ferris wheel of moment of inertia (I =20 000 kg.m2) from rest to (10.0 rev/min) in (12.0 s). When the motor is turned off, friction causes the wheel to slow down from (10.0 to 8.00 rev/min) in (10.0 s). Determine (a) the torque generated by the motor to bring the wheel to 10.0 rev/min and (b) the power that would be needed to maintain this rotational speed
To solve this problem, we need to use the formulas for torque and power in rotational motion.
(a) To determine the torque generated by the motor to bring the wheel to 10.0 rev/min, we can use the formula for torque:
Torque = Moment of inertia * Angular acceleration
First, let's find the angular acceleration. We know that the wheel is initially at rest and then accelerates to 10.0 rev/min in 12.0 s. We can calculate the angular acceleration using the formula:
Angular acceleration = Change in angular velocity / Time taken
Change in angular velocity = Final angular velocity - Initial angular velocity
= 10.0 rev/min - 0 rev/min
= 10.0 rev/min
Time taken = 12.0 s
Angular acceleration = 10.0 rev/min / 12.0 s
Next, we plug in the given moment of inertia: I = 20,000 kg.m^2.
Torque = 20,000 kg.m^2 * (10.0 rev/min / 12.0 s)
Note that we need to convert the angular velocity from rev/min to rad/s. There are 2π radians in one revolution and 60 s in one minute. So, the conversion factor is (2π rad / 1 rev) * (1 rev / 60 s) = π/30 rad/s.
Converting the angular velocity to rad/s:
Angular velocity = 10.0 rev/min * (2π rad / 1 rev) * (1 min / 60 s)
= 10.0 * π/30 rad/s
= π/3 rad/s
Now, we can calculate the torque:
Torque = 20,000 kg.m^2 * (π/3 rad/s / 12.0 s)
(b) To find the power needed to maintain this rotational speed, we can use the formula for power:
Power = Torque * Angular velocity
We've already calculated the torque in part (a). Now, we need to find the angular velocity at a rotational speed of 10.0 rev/min.
Angular velocity = 10.0 rev/min * (2π rad / 1 rev) * (1 min / 60 s)
Finally, we can substitute these values into the power formula to calculate the power needed.
Power = Torque * Angular velocity
Remember to convert the angular velocity to rad/s.
I hope this explanation helps you understand how to solve this problem!