Question

How do you derive the moment of inertia about a point on a shape?

a) Divide the shape into infinitesimally small elements, calculate the moment of inertia for each element, and integrate over the entire shape.

b) Multiply the mass of the shape by the square of its distance from the point.

c) Take the derivative of the shape's mass distribution with respect to distance.

d) Sum the masses of all points on the shape.

Answer 1

To derive the moment of inertia, divide the shape into small elements, use the formula I = mr² to calculate each element's inertia, and integrate over the entire shape.

Step-by-step explanation:

To derive the moment of inertia about a point on a shape, you typically use the following approach:

1. Divide the shape into infinitesimally small elements.
2. Calculate the moment of inertia for each element by using the formula I = mr², where 'm' is the mass of the element and 'r' is the perpendicular distance from the element to the axis of rotation.
3. Integrate this expression over the entire shape to sum up the contributions of all infinitesimal elements.

This process requires knowledge of calculus, as it involves an integral over the mass distribution of the object. For regular-shaped rigid bodies, moments of inertia can also be found in physics reference materials where they have been already derived from such integrations.