Rotational Motion – Moment of Inertia, Torque & Angular Momentum | Entrance Exam Physics Notes

ROTATIONAL MOTION — SHORT NOTES (Entrance Exam Focus)

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1. Basic Definitions

Rigid Body

A body that does not deform during motion.

Rotational Motion

Motion of a rigid body about a fixed axis.

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2. Angular Quantities

Same as circular motion but applied to rigid bodies:

$$\omega = \frac{d\theta}{dt} \quad (\text{angular velocity})$$
$$\alpha = \frac{d\omega}{dt} \quad (\text{angular acceleration})$$

Relation with linear quantities:

$$v = r\omega$$
$$a_t=r\alpha$$

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3. Moment of Inertia (I)

Rotational analog of mass. Measures resistance to angular acceleration.

$$I=\sum m_i{r}_i^{\mathrm{2}}$$

Unit: kg m²

Standard Moments of Inertia (Must Memorize)

• Thin rod (axis through center):

$$I=\frac{1}{12}M{L}^2$$

• Thin rod (axis through end):

$$I=\frac{1}{3}M{L}^2$$

• Ring (axis through center & perpendicular):

$$I=M{R}^2$$

• Disc (axis through center):

$$I=\frac{1}{2}M{R}^2$$

• Solid sphere:

$$I=\frac{2}{5}M{R}^2$$

• Hollow sphere:

$$I=\frac{2}{3}M{R}^2$$

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4. Parallel Axis Theorem

$$I=I_{cm}+M{d}^2$$

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5. Perpendicular Axis Theorem

(For plane lamina only)

$$I_z=I_x+I_{\mathrm{y}}$$

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6. Torque (τ)

Rotational analog of force.

$$\tau =rF\sin \theta =I\alpha$$

Unit: N·m

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7. Angular Momentum (L)

Rotational analog of linear momentum.

$$L = I\omega$$
$$\text{Torque}=\frac{dL}{d\mathrm{t}}$$

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8. Rolling Motion

Rolling = rotation + translation without slipping.

$$v=r\omega$$

Kinetic Energy of Rolling Body

$$K=\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2$$

For rolling without slipping:

$$\omega =\frac{v}{\mathrm{r}}$$

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9. Pure Rolling vs. Sliding

• Pure rolling → no slipping → 

  $v = r\omega$
  

• Sliding → friction acts to oppose motion

• Rolling → static friction does no work

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10. Work–Energy in Rotation

$$W = \tau\theta$$
$$K_{\text{rot}}=\frac{1}{2}I{\omega }^2$$

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11. Rotational Dynamics (Newton’s Second Law for Rotation)

$${\tau }_{\text{net}}=I\alpha$$

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12. Conservation of Angular Momentum

If net external torque = 0:

$$I_1{\omega }_1=I_2{\omega }_{\mathrm{2}}$$

Applications:

• Diving gymnast

• Figure skater

• Neutron stars spinning faster

• Ice skater pulling in arms

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13. Disc & Cylinder Rolling Down Incline

Acceleration:

$$a=\frac{g\sin \theta }{1+\frac{I}{m{R}^{\mathrm{2}}}}$$

Common results:

• Solid sphere → fastest

• Hollow sphere → slowest

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14. Most Expected Exam Questions

1. Moment of inertia using parallel/perpendicular axis

2. Rolling without slipping problems

3. Torque–angular acceleration numericals

4. Angular momentum conservation

5. Race of rolling bodies on slope (sphere, ring, disc)

6. Rod rotation about ends or center

7. Flywheel energy storage