An experimental solid disc flywheel has a 300m diameter and a mass of 70kg. Experiments show that, at all speeds of rotation, the frictional resistance to motion is equivalent to a torque of 0.15Nm. if when the flywheel is rotating at 50 rev/min, the driving power is removed, calculate;
a- the angular retardation
b - time taken for the flywheel to come to rest
c - the total number of revolutions made by the flywheel during retardation.
To calculate the angular retardation of the flywheel, you need to use the equation for torque.
a) The equation for torque is given by: torque = moment of inertia * angular acceleration.
The moment of inertia (I) of a solid disc can be calculated using the formula: I = (1/2) * m * r^2, where m is the mass of the disc and r is the radius.
Given:
- Diameter of the disc = 300m, so the radius (r) = 300m / 2 = 150m
- Mass of the disc (m) = 70kg
The moment of inertia (I) can be calculated as follows:
I = (1/2) * m * r^2
I = (1/2) * 70kg * (150m)^2
I = 1/2 * 70kg * 22500m^2
I = 787,500 kg m^2
The torque due to the frictional resistance (τ) is given as 0.15Nm.
Then, torque = I * angular acceleration
0.15Nm = 787,500 kg m^2 * angular acceleration
Solving for angular acceleration, we get:
angular acceleration = 0.15Nm / 787,500 kg m^2
b) To calculate the time taken for the flywheel to come to rest, we need to find the time it takes for the angular velocity (ω) to decrease to zero.
The formula relating angular acceleration (α), time (t), and initial angular velocity (ω0), and final angular velocity (ω) is:
ω = ω0 + α * t
At rest, the final angular velocity (ω) is zero, so the equation becomes:
0 = ω0 + α * t
Solving for time (t), we get:
t = -ω0 / α
In this case, the initial angular velocity, ω0, is given as 50 rev/min. Convert this to radians per second:
ω0 = (50 rev/min) * (2π rad/rev) * (1 min/60s)
ω0 = 5.24 rad/s
Substituting the values into the equation, we have:
t = -(5.24 rad/s) / (angular acceleration)
c) The total number of revolutions made by the flywheel during retardation can be found using the formula:
Total number of revolutions = (initial angular velocity - final angular velocity) / (2π)
In this case, the initial angular velocity is 50 rev/min, and the final angular velocity is zero.
Substituting these values into the formula, we get:
Total number of revolutions = (50 rev/min - 0) / (2π rev)
Solving this equation will give you the total number of revolutions made by the flywheel during retardation.
Please substitute the values into the respective equations and evaluate them to get the final answers.