1. A fly wheel with a mass of 50 kg is accelerated by an electric motor from rest to an angular velocity of 120rads/s in 14 seconds, if the value of k is 0.3m and there is a frictional torque of 0.07Nm calculate the torque applied, total work and the power from the motor.
Assume the acceleration is constant
Angular acceleration of flywheel:
alpha = (120 rad/s - 0 rad/s)/14 s
alpha ~ 8.571 rad/s^2
Moment of inertia of flywheel:
I = mk^2
I = (50kg)(0.3 m)^2
I = 4.5 kg*m^2
Torque applied by motor:
Ta - Tf = I*alpha
Ta - 0.07 N*m = (4.5 kg*m^2)(8.571 rad/s^2)
Ta = 38.64 N*m
Total work done by motor on flywheel:
Work = applied torque*angle
angle = 120 rad/s * 14 s = 168 rad
Work = (38.64 N*m) (168 rad)
Work = ______ J
Power output of motor:
P = work/time
P= ______ J / 14 s
P = _______ W
To solve this problem, we can use the equation for torque, work, and power in rotational motion.
1. Torque:
The equation for torque is given by:
Torque = Moment of Inertia x Angular Acceleration
In this case, the flywheel is accelerated from rest, so the initial angular velocity is zero. The angular acceleration can be calculated using the formula:
Angular Acceleration = (Final Angular Velocity - Initial Angular Velocity) / Time
Given:
Mass of the flywheel (m) = 50 kg
Radius of the flywheel (k) = 0.3 m
Final Angular Velocity (ω) = 120 rad/s
Time (t) = 14 s
First, we need to calculate the moment of inertia of the flywheel. The moment of inertia for a solid cylinder or disk is given by:
Moment of Inertia (I) = (1/2) x mass x radius^2
So,
I = (1/2) x m x k^2
Plugging in the values:
I = (1/2) x 50 kg x (0.3 m)^2
Now that we have the moment of inertia, we can calculate the angular acceleration:
Angular Acceleration = (ω - 0) / t
Plugging in the values:
Angular Acceleration = (120 rad/s - 0) / 14 s
Once we have the angular acceleration, we can calculate the torque:
Torque = I x Angular Acceleration
2. Total Work:
The work done by a torque is given by the formula:
Work = Torque x Angle Rotated
As the flywheel is accelerated from rest to a final angular velocity, the angle rotated can be calculated using the formula:
θ = (1/2) x Angular Acceleration x Time^2
Plugging in the values:
θ = (1/2) x Angular Acceleration x Time^2
Now that we have the angle rotated, we can calculate the total work:
Total Work = Torque x Angle Rotated
3. Power:
The power can be calculated using the formula:
Power = Work / Time
Now, let's calculate each value step by step:
Step 1: Calculate the moment of inertia:
I = (1/2) x 50 kg x (0.3 m)^2
Step 2: Calculate the angular acceleration:
Angular Acceleration = (120 rad/s - 0) / 14 s
Step 3: Calculate the torque:
Torque = I x Angular Acceleration
Step 4: Calculate the angle rotated:
θ = (1/2) x Angular Acceleration x Time^2
Step 5: Calculate the total work:
Total Work = Torque x Angle Rotated
Step 6: Calculate the power:
Power = Total Work / Time
By following these steps and plugging in the given values, you can find the torque applied, total work, and the power from the motor.