The flywheel of a steam engine runs with a constant angular speed of 321 rev/min (in the counterclockwise direction). When steam is shut off, the friction of the bearings stops the wheel in 1.9 h.
(a) What is the constant angular acceleration, in revolutions per minute-squared, of the wheel during the slowdown?
magnitude rev/min2
direction
(b) How many revolutions does the wheel make before stopping?
rotations
(c) At the instant the flywheel is turning at 75 rev/min, what is the tangential component of the linear acceleration of a flywheel particle that is 50 cm from the axis of rotation?
magnitude mm/s2
direction
(d) What is the magnitude of the net linear acceleration of the particle in (c)?
m/s2
Please don't post under multiple names. I will be happy to critique your thinking.
To solve this problem, we need to understand the relationship between angular acceleration, linear acceleration, and rotational motion.
(a) To find the constant angular acceleration, we first need to convert the time it takes for the flywheel to stop from hours to minutes.
1.9 hours = 1.9 * 60 minutes = 114 minutes
The angular acceleration can be found using the equation:
angular acceleration = (final angular velocity - initial angular velocity) / time
The initial angular velocity is given as 321 rev/min, and the final angular velocity is 0 rev/min since the flywheel stops.
angular acceleration = (0 rev/min - 321 rev/min) / 114 minutes
Simplifying this expression gives us the magnitude of the angular acceleration. The direction is counterclockwise, as mentioned in the question.
(b) To find the number of revolutions the wheel makes before stopping, we can use the relationship between angular acceleration, initial and final angular velocities, and time:
θ = (initial angular velocity * time) + (1/2 * angular acceleration * time^2)
Since the final angular velocity is 0 rev/min, we can solve for θ, which represents the number of revolutions:
θ = (321 rev/min * 114 minutes) + (1/2 * angular acceleration * (114 minutes)^2
This expression will give us the number of rotations.
(c) To find the tangential component of the linear acceleration, we can use the equation:
linear acceleration = radius * angular acceleration
Since the flywheel particle is located 50 cm from the axis of rotation, the radius is 0.5 m.
linear acceleration = 0.5 m * angular acceleration
(d) The magnitude of the net linear acceleration of the particle can be found by using the equation:
net linear acceleration = √[(tangential linear acceleration)^2 + (centripetal linear acceleration)^2]
We know the tangential linear acceleration from part (c), and the centripetal linear acceleration can be calculated using the equation:
centripetal linear acceleration = radius * (angular velocity)^2
The magnitude will give us the value of the net linear acceleration in m/s^2.