Find the moment of inertia of a uniform cube of mass M and edge l. Assume the axis of rotation passes through the center of the cube and is perpendicular to two of the faces.
What is the moment of inertia of a thin, flat plate in the shape of a semicircle rotating about the straight side (Fig. 12.28)? The mass of the plate is M and the radius is R.
The Correct Answer and Explanation is :
Moment of Inertia of a Uniform Cube
For a uniform cube of mass ( M ) and edge length ( l ), with an axis passing through its center and perpendicular to two of its faces, the moment of inertia is given by:
[
I = \frac{1}{6} M l^2
]
Derivation:
A cube can be considered as a solid of uniform mass distribution. The moment of inertia of a uniform solid cube about an axis through its center and perpendicular to two faces is derived using the integral definition of moment of inertia:
[
I = \int r^2 \, dm
]
Since the mass is uniformly distributed, we use standard results from integral calculus. By applying the parallel axis theorem and considering the cube as a sum of thin rods stacked along one axis, we arrive at:
[
I = \frac{1}{6} M l^2
]
This result is essential in physics and engineering, particularly in understanding the rotational dynamics of cubic objects.
Moment of Inertia of a Thin Semicircular Plate
For a thin, flat semicircular plate of mass ( M ) and radius ( R ), rotating about the straight edge, the moment of inertia is:
[
I = \frac{1}{4} M R^2
]
Derivation:
A semicircular plate can be thought of as half of a full circular disk. The moment of inertia of a full circular disk about its diameter is:
[
I_{\text{disk, center}} = \frac{1}{4} M R^2
]
Using the parallel axis theorem to shift the axis from the center of mass to the straight edge of the semicircle:
[
I = I_{\text{CM}} + M d^2
]
where ( d = \frac{4R}{3\pi} ) is the center of mass distance from the flat edge.
Applying this correction:
[
I = \frac{1}{4} M R^2 + M \left(\frac{4R}{3\pi}\right)^2
]
Solving the terms gives:
[
I = \frac{1}{4} M R^2
]
This result is widely used in structural and mechanical engineering, particularly in analyzing the rotational motion of semicircular plates.
These moments of inertia are essential in mechanics, particularly in calculating rotational kinetic energy, angular acceleration, and torque in various systems.