The Moment of Inertia of a Flywheel

The moment of inertia of a flywheel is a measure of its resistance to changes in rotational motion. It is analogous to mass in linear motion and is denoted by the symbol I. The moment of inertia depends on the mass distribution and shape of the flywheel.

The formula for calculating the moment of inertia of a flywheel is given by:
I = m * r^2
where I is the moment of inertia, m is the mass of the flywheel, and r is the radius of gyration.

Fluctuation in Energy and Mean Speed

In the given problem, the flywheel is fluctuating at a speed of 40 rpm (revolutions per minute) and the fluctuation in energy is 2036 joules. To find the mean speed, we need to understand the relationship between energy and angular velocity.

The energy of a rotating object is given by:
E = (1/2) * I * ω^2
where E is the energy, I is the moment of inertia, and ω is the angular velocity.

To calculate the mean speed, we need to find the average energy over the given fluctuation. Since the energy is directly proportional to the square of the angular velocity, we can calculate the mean angular velocity and then convert it to mean speed.

Calculating Mean Angular Velocity

Given:
Moment of Inertia (I) = 19.6 kg NM^2
Fluctuation in Energy (E) = 2036 joules
Angular Velocity (ω) = 40 rpm

We can rearrange the energy formula to solve for ω:
ω = √(2 * E / I)

Substituting the given values:
ω = √(2 * 2036 / 19.6) = √206.73 = 14.37 rad/s

Calculating Mean Speed

To convert the mean angular velocity to mean speed, we need to know the radius of the flywheel. Let's assume the radius (r) is given as 0.5 meters.

The formula for calculating speed from angular velocity is:
v = ω * r

Substituting the values:
v = 14.37 * 0.5 = 7.18 m/s

Therefore, the mean speed of the flywheel is 7.18 m/s.

Conclusion

In this problem, we first calculated the mean angular velocity of the flywheel using the given moment of inertia, fluctuation in energy, and initial angular velocity. We then converted the mean angular velocity to mean speed by multiplying it with the radius of the flywheel. The calculated mean speed is 7.18 m/s.