The moment of inertia of a solid sphere of density ρ and radius R about its diameter is

The correct option is 176R⁵ρ /105.

To find the moment of inertia of a uniform solid sphere about its diameter, we can follow a step-by-step derivation using its density and radius.

Step 1: Formula for Moment of Inertia of a Solid Sphere
The standard formula for the moment of inertia ($I$) of a uniform solid sphere of mass $M$ and radius $R$ about any diameter is:
$I=\frac{2}{5}M{R}^2$

Step 2: Express Mass in terms of Density and Volume
The mass ($M$) of the sphere is equal to its density ($\rho$) multiplied by its volume ($V$):
$M=\rho V$
The volume of a solid sphere of radius $R$ is given by:
$V=\frac{4}{3}\pi R^3$
Substituting the volume back into the mass equation yields:
$M=\frac{4}{3}\pi R^3\rho$

Step 3: Substitute Mass into the Moment of Inertia Formula
Now, substitute the expression for mass $M$ into the moment of inertia formula:
$I=\frac{2}{5}\left(\frac{4}{3}\pi R^3\rho \right)R^2$
Simplifying the constants:
$I=\frac{8}{15}\pi R^5\rho$

Step 4: Use the Rational Approximation for $\pi$
Approximating the value of $\pi$ as the fraction $\frac{22}{7}$, we substitute it into our simplified equation:
$I=\frac{8}{15}\cdot \frac{22}{7}{R}^5\rho$
Multiplying the terms in the numerator and denominator:
$I=\frac{8\cdot 22}{15\cdot 7}{R}^5\rho$
$I=\frac{176}{105}{R}^5\rho$
Thus, the moment of inertia can be written as:
$I=\frac{176R^5\rho }{105}$
This matches the correct option.