What is the theorem of parallel axes? A uniform disc has a mass of 4kg and radius 2cm. Calculate the moment of inertia about an axis perpendicular to it's plane 1. Through its centre. 2. Through a point of its circumference
The theorem of parallel axes states that the moment of inertia of an object about an axis parallel to and a distance "d" away from an axis through its center of mass is equal to the moment of inertia about the axis through its center of mass plus the product of its mass and the square of the distance "d".
1. Moment of inertia through its center: The moment of inertia of a uniform disc about an axis through its center is given by:
I = (1/2) * mass * radius^2
Substituting the given values:
I = (1/2) * 4kg * (0.02m)^2
= 0.0024 kg * m^2
2. Moment of inertia through a point on its circumference: To calculate the moment of inertia through a point on its circumference, we need to apply the parallel axis theorem.
Let's assume the distance "d" from the center axis to the point on the circumference is equal to the disc's radius (0.02m).
Using the theorem of parallel axes:
I' = I + mass * d^2
I' = (1/2) * 4kg * (0.02m)^2 + 4kg * (0.02m)^2
= 0.0024 kg * m^2 + 0.0024 kg * m^2
= 0.0048 kg * m^2
Therefore, the moment of inertia about an axis perpendicular to the plane of the uniform disc is:
1. Through its center: 0.0024 kg * m^2
2. Through a point on its circumference: 0.0048 kg * m^2