a flywheel has an angular speed of 1200rev/min when its motor is turned off.the wheel attains constant decelerations of 1.5rad/s2 due to friction in its bearing.determine the time required for the wheel to come to rest and the numer of revolutions the wheel makes before it comes to rest.
Wi= 1200 rev/min = 20 rev/sec
= 20*2*pi rad/sec
Wf = 0; alpha= -1.5 rad/s^2
Wf = Wi + alpha*t
0 = 40*pi - 1.5*t
t = 40*pi/1.5 = 84 sec
Now,
theta = Wi*t + (1/2)*alpha*t^2
Plug in the values in this eqn. to get theta - the rotation of the wheel (in radians)before it comes to halt. Divide the result by 2*pi to convert into no. of revolutions.
Please help me to elaborate the answer as i tried but couldn't
To determine the time required for the flywheel to come to rest and the number of revolutions it makes before coming to rest, we need to use the equations of motion for rotational motion.
The initial angular speed of the flywheel is given as 1200 rev/min. We need to convert this to radians per second by using the conversion factor:
1 revolution = 2π radians
So, the initial angular speed in radians per second is:
Initial angular speed = (1200 rev/min) x (2π radians/1 revolution) x (1 min/60 sec)
= (1200 x 2π) / 60 radians/sec
= 40π radians/sec
The flywheel has a constant deceleration of 1.5 rad/s^2 due to friction. We can use the equation of motion for rotational motion:
Final angular speed^2 = Initial angular speed^2 + 2αθ
Here, Final angular speed is 0 (since the flywheel comes to rest), Initial angular speed is 40π radians/sec, α (deceleration) is -1.5 rad/s^2 (negative because it is a deceleration), and θ is the angle covered in radians.
Simplifying the equation:
0 = (40π)^2 + 2(-1.5)θ
0 = 1600π^2 - 3θ
3θ = 1600π^2
θ = (1600π^2) / 3 radians
The number of revolutions the wheel makes can be calculated by dividing the angle θ by 2π:
Number of revolutions = θ / 2π
= [(1600π^2) / 3] / 2π
= (1600π^2) / (6π)
= 800π / 3 revolutions
To find the time required for the wheel to come to rest, we can use the equation:
Final angular speed = Initial angular speed + αt
Since the final angular speed is 0, the equation becomes:
0 = 40π + (-1.5)t
1.5t = 40π
t = (40π) / 1.5 seconds
Now we can calculate the time required for the wheel to come to rest and the number of revolutions it makes before coming to rest:
Time required for the wheel to come to rest = (40π) / 1.5 seconds
Number of revolutions the wheel makes = 800π / 3 revolutions