Parallel and Perpendicular Axis Theorem — Revision Notes

⚡ 30-Second Revision

Moment of Inertia (I): — Rotational equivalent of mass. $I = \sum m_i r_i^2$ or $I = \int r^2 dm$. Units: $\text{kg} cdot \text{m}^2$.
Parallel Axis Theorem: — $I = I_{CM} + Md^2$

* Applies to any rigid body (2D or 3D). * Axes must be parallel. * One axis must pass through the Center of Mass (CM). * $d$ = perpendicular distance between axes.

Perpendicular Axis Theorem: — $I_z = I_x + I_y$

* Applies ONLY to planar bodies (laminae). * Axes $x, y$ lie in the plane, $z$ is perpendicular to the plane. * All three axes must intersect at a common point.

**Standard $I_{CM}$ values:**

* Rod (perp. to length): $I_{CM} = \frac{1}{12}ML^2$ * Disc (perp. to plane): $I_{CM} = \frac{1}{2}MR^2$ * Ring (perp. to plane): $I_{CM} = MR^2$ * Solid Sphere (about diameter): $I_{CM} = \frac{2}{5}MR^2$ * Hollow Sphere (about diameter): $I_{CM} = \frac{2}{3}MR^2$

2-Minute Revision

The Parallel and Perpendicular Axis Theorems are vital shortcuts for calculating moment of inertia. The Parallel Axis Theorem ($I = I_{CM} + Md^2$) is universally applicable to any rigid body, whether 2D or 3D.

It states that the moment of inertia about any axis ($I$) equals the moment of inertia about a parallel axis through the center of mass ($I_{CM}$) plus the product of the body's total mass ($M$) and the square of the perpendicular distance ($d$) between the two axes.

Remember, one of the axes *must* be the CM axis. This theorem is frequently used to find the moment of inertia of a rod about its end or a disc about a tangent.

The Perpendicular Axis Theorem ($I_z = I_x + I_y$) is more specific, applying *only* to planar bodies (thin laminae). It states that the moment of inertia about an axis perpendicular to the plane ($I_z$) is the sum of the moments of inertia about two mutually perpendicular axes ($I_x, I_y$) lying within the plane, provided all three axes intersect at a common point.

This is crucial for finding the moment of inertia of a disc or ring about its diameter. Always check the applicability conditions carefully before using either theorem. Mastering these theorems, along with standard $I_{CM}$ values, is key to solving rotational dynamics problems efficiently in NEET.

5-Minute Revision

A solid grasp of the Parallel and Perpendicular Axis Theorems is non-negotiable for NEET Physics. Let's consolidate the key aspects.

1. Parallel Axis Theorem ($I = I_{CM} + Md^2$):

This theorem is a workhorse for any rigid body. It allows you to 'shift' the axis of rotation. You *must* know the moment of inertia ($I_{CM}$) about an axis passing through the body's center of mass. The new axis must be *parallel* to this CM axis. The term $Md^2$ accounts for the additional inertia due to the mass being distributed further from the new axis, where $M$ is the total mass and $d$ is the perpendicular distance between the two parallel axes.

Example: — For a uniform rod of mass $M$ and length $L$, $I_{CM} = \frac{1}{12}ML^2$ (axis perpendicular to rod, through center). To find $I_{end}$ (axis perpendicular to rod, through one end), $d = L/2$. So, $I_{end} = \frac{1}{12}ML^2 + M(L/2)^2 = \frac{1}{12}ML^2 + \frac{1}{4}ML^2 = \frac{1}{3}ML^2$.

2. Perpendicular Axis Theorem ($I_z = I_x + I_y$):

This theorem is more specialized, applying *only* to planar bodies (laminae), which are essentially 2D objects with negligible thickness (e.g., thin discs, rings, square plates). If the body lies in the $xy$-plane, and $I_x$ and $I_y$ are moments of inertia about the x and y axes (which are mutually perpendicular and lie in the plane), then $I_z$ (moment of inertia about the z-axis, perpendicular to the plane) is their sum. Crucially, all three axes must intersect at a common point.

Example: — For a uniform circular disc of mass $M$ and radius $R$, $I_z = \frac{1}{2}MR^2$ (axis through center, perpendicular to plane). Due to symmetry, $I_x = I_y$ (about any diameter). Applying the theorem: $I_z = I_x + I_y \implies \frac{1}{2}MR^2 = 2I_x \implies I_x = \frac{1}{4}MR^2$. So, the moment of inertia about a diameter is $\frac{1}{4}MR^2$.

Key Takeaways for NEET:

Memorize standard $I_{CM}$ values: — This is non-negotiable for speed.
Understand conditions: — Know when each theorem can be applied (planar body for perpendicular axis, parallel axes for parallel axis).
Practice combined problems: — Many NEET questions require using both theorems in sequence.
Be careful with 'd': — Always use the *perpendicular* distance between the parallel axes.
Symmetry: — Utilize symmetry (e.g., $I_x = I_y$ for discs/rings about diameters) to simplify calculations.

Prelims Revision Notes

Parallel Axis Theorem ($I = I_{CM} + Md^2$)

Definition: — Moment of inertia about any axis ($I$) equals moment of inertia about a parallel axis through the center of mass ($I_{CM}$) plus $Md^2$.
Applicability: — Universal for any rigid body (2D or 3D).
Conditions:

1. The two axes must be parallel. 2. One of the axes *must* pass through the center of mass (CM). 3. $d$ is the perpendicular distance between the two parallel axes.

Common Use Cases:

* Rod: $I_{CM} = \frac{1}{12}ML^2$ (perp. to length). $I_{end} = \frac{1}{3}ML^2$ (using $d=L/2$). * Disc: $I_{CM} = \frac{1}{2}MR^2$ (perp. to plane). $I_{tangent, perp} = \frac{3}{2}MR^2$ (using $d=R$). * Ring: $I_{CM} = MR^2$ (perp. to plane). $I_{tangent, perp} = 2MR^2$ (using $d=R$).

Perpendicular Axis Theorem ($I_z = I_x + I_y$)

Definition: — For a planar body, moment of inertia about an axis perpendicular to its plane ($I_z$) equals sum of moments of inertia about two mutually perpendicular axes ($I_x, I_y$) lying in its plane.
Applicability: — STRICTLY for planar bodies (laminae, effectively 2D, e.g., thin disc, ring, square plate).
Conditions:

1. Body must be planar. 2. Axes $x, y$ must lie in the plane of the body. 3. Axes $x, y, z$ must be mutually perpendicular. 4. All three axes must intersect at a common point on the body.

Common Use Cases:

* Disc: $I_z = \frac{1}{2}MR^2$. By symmetry, $I_x = I_y$ (about diameter). So, $I_{diameter} = \frac{1}{4}MR^2$. * Ring: $I_z = MR^2$. By symmetry, $I_x = I_y$ (about diameter). So, $I_{diameter} = \frac{1}{2}MR^2$. * Square Plate: $I_x = \frac{1}{12}Ma^2$ (through CM, parallel to side). $I_z = I_x + I_y = \frac{1}{6}Ma^2$.

Key Points for NEET:

Memorize Standard $I_{CM}$ values: — Essential for quick problem-solving.
Identify Body Type: — Crucial for deciding which theorem to use (planar vs. 3D).
Axis Orientation: — Carefully note if the axis is parallel, perpendicular, through CM, or at an edge/corner.
Combined Problems: — Be prepared for problems requiring both theorems sequentially (e.g., square plate about a corner).

Vyyuha Quick Recall

For Parallel Axis Theorem, think: Parallel Axes Together, Inertia CM Mass Distance Squared. ($I = I_{CM} + Md^2$). For Perpendicular Axis Theorem, think: Planar Always Three Axes, In Zero X Y (meaning $I_z = I_x + I_y$ for planar bodies, with $x,y$ in plane and $z$ perpendicular, all intersecting at origin).