The moment of inertia of a solid sphere of mass M and radius R about a tangent to the sphere is

step 1 In this question we have to calculate the moment of inertia of a solid sphere about the tangent

The moment of inertia of a solid sphere of mass M and radius...

The moment of inertia of a solid sphere of mass $\mathrm{M}$ and radius $R$ about a tangent to the sphere is
(a) $\frac{2}{5} \mathrm{MR}^{2}$
(b) $\frac{6}{5} \mathrm{MR}^{2}$
(c) $\frac{4}{5} \mathrm{MR}^{2}$
(d) $\frac{7}{5} M R^{2}$

Key Concepts

Solid Sphere

A solid sphere is a three-dimensional object with constant density and a continuous, curved surface. In the context of rotational motion, the distribution of mass in a sphere influences its moment of inertia, and standard formulas have been derived assuming uniform mass distribution. The moment of inertia about its center for a solid sphere is a well-known result.

Parallel Axis Theorem

The parallel axis theorem is a useful concept that allows one to compute the moment of inertia of an object about any axis parallel to an axis through its center of mass. It states that the moment of inertia about the new axis is equal to the moment of inertia about the center of mass axis plus the product of the mass and the square of the distance between the two axes. This theorem is essential when dealing with rotations about axes that do not pass through the center of mass.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion about an axis. It depends on both the mass of the object and how that mass is distributed relative to the axis of rotation. Calculating the moment of inertia accurately is essential in analyzing rotational dynamics and angular acceleration.

Tangent to a Sphere

A tangent to a sphere is a line that touches the sphere at exactly one point and is perpendicular to the radius at the point of contact. In problems involving rotational motion, considering an axis that is tangent to the sphere requires the use of the parallel axis theorem to shift from the standard moment of inertia about the center of mass to that about the tangent axis.

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