Moment of Inertia — Revision Notes

⚡ 30-Second Revision

Definition: — Rotational analogue of mass. $I = \sum m_i r_i^2$ (discrete), $I = \int r^2 dm$ (continuous).
Unit: — kg m$^2$.
Parallel Axis Theorem: — $I = I_{CM} + Md^2$.
Perpendicular Axis Theorem: — $I_z = I_x + I_y$ (for planar bodies).
Radius of Gyration: — $k = \sqrt{I/M}$.
Standard Formulas (Central Axis):

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

2-Minute Revision

Moment of Inertia ($I$) is a crucial concept in rotational dynamics, representing an object's resistance to angular acceleration, similar to how mass resists linear acceleration. Unlike mass, $I$ depends on both the total mass and its distribution relative to the axis of rotation. The fundamental definition for discrete particles is $I = \sum m_i r_i^2$, where $r_i$ is the perpendicular distance from the axis. For continuous bodies, it's an integral. Remember the unit: kg m$^2$.

Two powerful theorems simplify calculations: the Parallel Axis Theorem ($I = I_{CM} + Md^2$) allows you to find $I$ about any axis parallel to a known CM axis. The Perpendicular Axis Theorem ($I_z = I_x + I_y$) is for planar bodies, relating $I$ about an axis perpendicular to the plane to two axes within the plane.

Memorize standard formulas for common shapes like rings ($MR^2$), discs ($1/2 MR^2$), rods ($1/12 ML^2$ about CM), and spheres ($2/5 MR^2$ for solid, $2/3 MR^2$ for hollow, about diameter). The radius of gyration ($k = \sqrt{I/M}$) is a useful concept for comparing mass distribution.

Always identify the correct axis of rotation and apply the appropriate formula or theorem.

5-Minute Revision

Moment of Inertia ($I$) is the cornerstone of rotational dynamics, acting as the rotational counterpart to mass. It quantifies an object's opposition to changes in its angular velocity. Crucially, $I$ is not a fixed property; it varies with the chosen axis of rotation and the distribution of mass. The further mass is from the axis, the greater its contribution to $I$ (due to the $r^2$ dependence).

Key Formulas & Theorems:

1
Definition: — For discrete particles, $I = \sum m_i r_i^2$. For continuous bodies, $I = \int r^2 dm$.
2
Parallel Axis Theorem: — $I = I_{CM} + Md^2$. Use this when the axis of interest is parallel to an axis passing through the center of mass ($I_{CM}$), and $d$ is the perpendicular distance between them. This is vital for problems like a rod rotating about its end.
3
Perpendicular Axis Theorem: — $I_z = I_x + I_y$. Applicable *only* for planar bodies. $I_x$ and $I_y$ are Moments of Inertia about two perpendicular axes lying in the plane, and $I_z$ is about an axis perpendicular to the plane, passing through their intersection.
4
Radius of Gyration ($k$): — $I = Mk^2 \implies k = \sqrt{I/M}$. It's the effective distance at which the entire mass could be concentrated to yield the same $I$.

Standard Shapes (Memorize these!):

Thin Ring/Hoop: — $I_{CM} = MR^2$ (axis perp. to plane, through center)
Uniform Disc/Solid Cylinder: — $I_{CM} = \frac{1}{2}MR^2$ (axis perp. to plane, through center)
Uniform Thin Rod: — $I_{CM} = \frac{1}{12}ML^2$ (axis perp. to length, through CM); $I_{\text{end}} = \frac{1}{3}ML^2$ (axis perp. to length, through end)
Solid Sphere: — $I_{CM} = \frac{2}{5}MR^2$ (axis through diameter)
Hollow Sphere/Spherical Shell: — $I_{CM} = \frac{2}{3}MR^2$ (axis through diameter)

Problem-Solving Tips:

Always identify the axis of rotation first.
Determine if the object is a point mass system or a continuous body.
If the axis is not through the CM, consider using the Parallel Axis Theorem.
If it's a planar body and you need $I$ perpendicular to the plane, consider the Perpendicular Axis Theorem.
For rolling motion, the instantaneous axis of rotation is at the point of contact, requiring the Parallel Axis Theorem.
Pay attention to units and perform calculations carefully. Moment of Inertia is fundamental to understanding rotational kinetic energy, angular momentum, and torque, so a strong grasp here will benefit other related topics.

Prelims Revision Notes

Moment of Inertia ($I$) is the measure of an object's resistance to angular acceleration. It's the rotational analogue of mass. The SI unit is kg m$^2$.

Key Definitions & Formulas:

For discrete particles: — $I = \sum m_i r_i^2$, where $m_i$ is the mass of the $i$-th particle and $r_i$ is its perpendicular distance from the axis of rotation.
For continuous bodies: — $I = \int r^2 dm$.
Radius of Gyration ($k$): — $I = Mk^2 \implies k = \sqrt{I/M}$. It represents the effective distance of mass distribution from the axis.

Important Theorems:

1
Parallel Axis Theorem: — $I = I_{CM} + Md^2$. Used to find Moment of Inertia about an axis parallel to one passing through the center of mass ($I_{CM}$), where $d$ is the perpendicular distance between the axes. This is extremely useful for shifting the axis of rotation.
2
Perpendicular Axis Theorem: — $I_z = I_x + I_y$. Applicable *only* for planar bodies. $I_x$ and $I_y$ are Moments of Inertia about two mutually perpendicular axes lying in the plane of the body, and $I_z$ is about an axis perpendicular to the plane and passing through their intersection.

Standard Moments of Inertia (about CM or principal axis):

**Thin Ring/Hoop (mass $M$, radius $R$):**

* Axis perpendicular to plane, through center: $I = MR^2$ * Axis along diameter: $I = \frac{1}{2}MR^2$

**Uniform Disc/Solid Cylinder (mass $M$, radius $R$):**

* Axis perpendicular to plane, through center: $I = \frac{1}{2}MR^2$ * Axis along diameter: $I = \frac{1}{4}MR^2$

**Uniform Thin Rod (mass $M$, length $L$):**

* Axis perpendicular to length, through CM: $I = \frac{1}{12}ML^2$ * Axis perpendicular to length, through one end: $I = \frac{1}{3}ML^2$

**Solid Sphere (mass $M$, radius $R$):**

* Axis through diameter: $I = \frac{2}{5}MR^2$

**Hollow Sphere/Spherical Shell (mass $M$, radius $R$):**

* Axis through diameter: $I = \frac{2}{3}MR^2$

Key Points for NEET:

Moment of Inertia depends on mass distribution and the axis of rotation.
Mass further from the axis contributes more due to $r^2$.
Be proficient in applying both theorems.
Understand the relationship between Moment of Inertia, rotational kinetic energy ($E_k = \frac{1}{2}I\omega^2$), and angular momentum ($L = I\omega$).
For rolling motion, the instantaneous axis of rotation is at the point of contact, requiring the Parallel Axis Theorem.

Vyyuha Quick Recall

To remember the order of Moments of Inertia for common shapes (smallest to largest coefficient for $MR^2$ about central axis):

Solid Sphere (0.4) Does Have Radius (1.0)

Solid Sphere: $I = \frac{2}{5}MR^2 = 0.4 MR^2$
Disc: $I = \frac{1}{2}MR^2 = 0.5 MR^2$
Hollow Sphere: $I = \frac{2}{3}MR^2 \approx 0.67 MR^2$
Ring: $I = MR^2 = 1.0 MR^2$