NEET Physics: parallel and perpendicular axes theorems Practice Questions

Correct answer: B — Iₓ² + I_y² = I_z²

The perpendicular axis theorem, when applied to the relationship between the principal moments of inertia (or components related to them), results in the Pythagorean relationship: Iₓ² + I_y² = I_z². This is the defining relationship derived from the concept of perpendicular axes.

Correct answer: D — I = I₍CM) + Md²

The Parallel Axis Theorem states that I = I₍CM) + Md², where M is the total mass and d is the perpendicular distance between the axes.

Correct answer: C — Mass times the square of the distance between the axes

In the Parallel Axis Theorem, Md² represents the contribution to the moment of inertia due to the displacement of the axis from the center of mass.

Correct answer: D — I_z = Iₓ + I_y

For perpendicular axes, the theorem states that the square of the moment of inertia about the third axis is the sum of the squares of the moments of inertia about the other two axes (I_z² = Iₓ² + I_y²), which is a more complex form of the relationship often used. However, the simplest application for the axis theorem itself is I_z = Iₓ + I_y when considering the moment of inertia tensor components, ensuring the components are related. For the simple theorem application, if we consider the diagonal relation, the relationship is additive.

Correct answer: A — Parallel

The Parallel Axis Theorem is specifically used to relate the moment of inertia about two parallel axes (one being the centroidal axis). Therefore, it directly applies to axes that are parallel to each other.

Correct answer: B — the axis is displaced from the center of mass

The theorem is specifically used when the axis of interest is displaced from the center of mass (I = I₍CM) + Md²).

Correct answer: C — Iₓ + I_y

According to the Perpendicular Axis Theorem, the sum of the squares of the moments of inertia about the three mutually perpendicular axes is equal to the moment of inertia of the body about any plane perpendicular to the third axis. More simply, for a planar body, I_z = Iₓ + I_y is not the direct theorem. The correct Perpendicular Axis Theorem states that Iₓ + I_y = I_z if the axes are chosen correctly based on the geometry, but the standard application for three orthogonal axes is Iₓ² + I_y² + I_z² = constant. However, the most direct application related to perpendicular axes is Iₓ + I_y = Iₚₗₐₙₑ where Iₚₗₐₙₑ is the moment of inertia about the axis perpendicular to the plane formed by X and Y. In the context of introductory problems involving perpendicular axes, the sum of the moments of inertia is often implied or tested conceptually, but the core theorem relates the moments of inertia about the perpendicular axes directly. Let's rephrase to test the direct theorem: If Iₓ and I_y are moments about perpendicular axes, the theorem relates Iₓ, I_y, and I_z. The most direct application often tested is that the sum of the products of inertia is zero, but for the perpendicular axes theorem, for a planar object, I_z = Iₓ + I_y is often used conceptually in simple contexts or a related theorem application. The intended conceptual recall here relies on the property that for a planar body, Iₓ + I_y = Iₚₗₐₙₑ. In the context of the specific options provided, the relationship tested is often related to the perpendicularity itself. For a planar body, the perpendicular axis theorem implies Iₓ + I_y = I_z where I_z is the moment of inertia about the axis perpendicular to the plane defined by X and Y.

Correct answer: C — Iₓ + I_y = I_z

The Perpendicular Axis Theorem for three axes states that I_z = Iₓ + I_y. This establishes the relationship between the moments of inertia about perpendicular axes.

Correct answer: D — Parallel Axis Theorem

The Parallel Axis Theorem is specifically designed to handle the shift of the axis by relating the moment of inertia about the new axis to the moment of inertia about the parallel axis passing through the center of mass.

Correct answer: A — I' = I + Md²

Using the Parallel Axis Theorem: I' = I₍cm) + Md². Since I₍cm) = I, the new moment of inertia is I' = I + Md².

Correct answer: D — Parallel Axis Theorem

The Parallel Axis Theorem is specifically used to relate the moment of inertia about a parallel axis to the moment of inertia about the center of mass, which is the foundation for calculating inertia when the axis is shifted but parallel.

Correct answer: A — 50 kg·m²

According to the Perpendicular Axis Theorem, I_z = Iₓ + I_y. Therefore, I_z = 20 + 30 = 50 kg·m²

Correct answer: A — The center of mass

The Parallel Axis Theorem is used to find the moment of inertia about a parallel axis. The distance (d) must be measured from the center of mass of the body to the new parallel axis.