10.10:
Moment of Inertia of Compound Objects
Moment of inertia of an object about an axis is defined as the sum of the products of the mass of each particle in the object with the square of its distance from the axis of rotation.
In case of compound objects, the total moment of inertia of the object is the sum of moment of inertia of individual components along the axis of rotation.
Consider a wheel of radius 0.5 m having a rim of mass 10 kg and twelve spokes each having mass 0.3 kg and length 0.5 m.
Now, to determine the total moment of inertia of the wheel, first, individually determine the moment of inertia of the rim rotating along its cylindrical axis and the spokes rotating about their ends. Then, add the moment of inertia of all the components of the wheel.
By substituting the known values in the equation, the total moment of inertia for the wheel is obtained to be 2.8 kg⋅m2.
10.10:
Moment of Inertia of Compound Objects
The moment of inertia is a quantitative measure of the rotational inertia of an object. It is defined as the sum of the products obtained by multiplying the mass of each particle of matter in a given body by the square of its distance from the axis. The total moment of inertia for compound objects can be found by determining and adding the moment of inertia of individual components together.
Consider a child of mass (mc) 25 kg standing at a distance (rc) of 1 m from the axis of a rotating merry-go-round. The merry-go-round is approximated as a uniform solid disk with a mass (mm) of 500 kg and a radius (rm) of 2 m. Find the moment of inertia of the compound system.
The total moment of inertia of the system can be determined by adding up the individual moments of inertia of the merry-go-round and the child rotating on the axis. Since the mass and size of the child are much smaller than the merry-go-round, the child can be considered as a point mass.
- The moment of inertia (I) for the child is calculated as
- The moment of inertia (I) for the merry-go-round is calculated as
- By substituting and adding both values, the total moment of inertia of the system is determined to be 1025 kg⋅m2.
This text is adapted from Openstax, University Physics Volume 1, Section 10.5: Calculating Moments of Inertia.