The moment of inertia about a transverse axis through the...

The moment of inertia about a transverse axis through the center of a disc radius is $20 \mathrm{~cm}$, density $9 \mathrm{~g} \mathrm{~cm}^{-3}$ and thickness $7 \mathrm{~cm}$ is (a) $1584 \times 10^{7} \mathrm{~g} \mathrm{~cm}^{2}$ (b) $1.584 \times 10^{6} \mathrm{~g} \mathrm{~m}^{2}$ (c) $1.584 \times 10^{7} \mathrm{~g} \mathrm{~cm}^{2}$ (d) $1.584 \times 10^{4} \mathrm{~g} \mathrm{~cm}^{2}$

The moment of inertia about a transverse axis through the center of a disc radius is $20 \mathrm{~cm}$, density $9 \mathrm{~g} \mathrm{~cm}^{-3}$ and thickness $7 \mathrm{~cm}$ is
(a) $1584 \times 10^{7} \mathrm{~g} \mathrm{~cm}^{2}$
(b) $1.584 \times 10^{6} \mathrm{~g} \mathrm{~m}^{2}$
(c) $1.584 \times 10^{7} \mathrm{~g} \mathrm{~cm}^{2}$
(d) $1.584 \times 10^{4} \mathrm{~g} \mathrm{~cm}^{2}$

Key Concepts

Moment of Inertia

The moment of inertia is a physical quantity that measures an object's resistance to rotational acceleration about an axis. It is calculated by integrating the mass elements of the object multiplied by the square of their distances from the rotation axis. This concept is central when analyzing the rotational dynamics of any rigid body.

Mass Calculation from Density

Mass density relates mass to volume, and for objects with uniform density, the mass can be determined by multiplying the density by the volume. This is particularly useful when dealing with three-dimensional objects of simple geometries, where the volume calculation simplifies finding the total mass.

Standard Moment of Inertia Formulas for Regular Shapes

For common geometries like discs, standard formulas exist that provide the moment of inertia about various axes. These formulas are derived by integrating across the object's geometry and take into account its mass distribution. They serve as essential shortcuts to avoid recalculating the integral for every similar object.

Unit Analysis and Conversions

In problems involving physical quantities such as density, mass, and moment of inertia, it is vital to ensure that all units are consistent. Converting between units, when necessary, guarantees that the final computed values are dimensionally accurate and physically meaningful.

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