Rotational Motion, Torque, and Dynamics

Torque ( $\tau$ ): Defined as the tendency of an applied force to rotate an object about an axis of rotation.
Influencing Factors:
- Magnitude of the applied force ( $F$ ).
- Position/distance from the axis of rotation ( $r$ ).
- Angle ( $\theta$ ) between the position vector and the line of action of the force.
Torque Formula: $\tau = r F \sin(\theta) = F d = r_\perp F$
- The lever arm ( $d$ or $r_\perp$ ) is the perpendicular distance from the axis of rotation to the line along which the force acts: $d = r \sin(\theta)$ .
Angular Limits:
- $\theta = 0^\circ$ or $180^\circ$ : Torque is zero ( $\tau = 0$ ) because the force is parallel to the position vector.
- $\theta = 90^\circ$ or $270^\circ$ : Torque achieves maximum magnitude.
Units & Sign Convention:
- SI Unit: $\text{N}\cdot\text{m}$
- Counterclockwise (anticlockwise) torque: Positive ( $+\tau$ ).
- Clockwise torque: Negative ( $-\tau$ ).

Definition: The tendency of a body to resist angular acceleration; represents rotational inertia.
Governing Factors:
- Mass of the body ( $M$ or $m$ ).
- Distribution of mass relative to the axis of rotation.
- Selection/location of the axis of rotation.

Mathematical Definitions:
- Single point mass: $I = m r^2$
- Discrete collection of particles: $I = \sum_i m_i r_i^2 = m_1 r_1^2 + m_2 r_2^2 + m_3 r_3^2 + \dots$
- Continuous mass distribution: $I = \int r^2\,dm$
- SI Unit: $\text{kg}\cdot\text{m}^2$
Formulas for Common Geometries:
- Hoop / thin-walled cylinder (about cylinder axis): $I = M R^2$
- Annular cylinder / ring (about cylinder axis): $I = \frac{1}{2} M (R_1^2 + R_2^2)$
- Solid cylinder or disk (about cylinder axis): $I = \frac{1}{2} M R^2$
- Solid cylinder or disk (about central diameter): $I = \frac{1}{4} M R^2 + \frac{1}{12} M l^2$
- Thin rod (axis through center, perpendicular to length): $I = \frac{1}{12} M l^2$
- Thin rod (axis through one end, perpendicular to length): $I = \frac{1}{3} M l^2$
- Solid sphere (about any diameter): $I = \frac{2}{5} M R^2$
- Thin spherical shell (about any diameter): $I = \frac{2}{3} M R^2$
- Hoop (about any diameter): $I = \frac{1}{2} M R^2$
- Flat slab (perpendicular axis through center): $I = \frac{1}{12} M (a^2 + b^2)$

Rotational Formulation: $\tau_{\text{net}} = I \alpha$ $\alpha = \frac{\tau_{\text{net}}}{I}$
- Angular acceleration ( $\alpha$ ) is directly proportional to net torque ( $\tau_{\text{net}}$ ) and inversely proportional to rotational inertia ( $I$ ).
Derivation from Linear Dynamics:
- Linear tangential form: $F_T = m a_T = m(r \alpha)$
- Multiplying both sides by $r$ : $F_T r = m r^2 \alpha \implies \tau = I \alpha$
Equilibrium State Independence:
- A system can have zero net translational force ( $\mathbf{F}_{\text{net}} = 0$ ) while experiencing a non-zero net rotational torque ( $\boldsymbol{\tau}_{\text{net}} \neq 0$ ).
Problem-Solving Framework:
- Draw free-body diagrams showing points of application for each force.
- Apply translational equations: $\mathbf{F}_{\text{net}} = \sum \mathbf{F}_i = m \mathbf{a}$
- Apply rotational equations about a common axis: $\boldsymbol{\tau}_{\text{net}} = \sum \boldsymbol{\tau}_i = I \boldsymbol{\alpha}$
- Use constraint conditions (such as $a_T = r \alpha$ ) to link translational and rotational variables.

Angular Momentum ( $L$ ):
- Defined as the rotational equivalent of linear momentum: $L = I \omega$
- For a point particle moving at radius $r$ : $L = m r^2 \omega$
- SI Unit: $\text{kg}\cdot\text{m}^2/\text{s}$
Torque and Rate of Change: $\tau_{\text{net}} = \frac{\Delta L}{\Delta t} = \frac{dL}{dt}$
Law of Conservation of Angular Momentum:
- If the net external torque acting on a rotating system is zero ( $\tau_{\text{net}} = 0$ ), the total angular momentum remains constant: $\frac{dL}{dt} = 0 \implies L_i = L_f$ $I_0 \omega_0 = I \omega$
- Decreasing the moment of inertia ( $I$ ) results in a proportional increase in angular velocity ( $\omega$ ).

Kinetic Energy Formulations:
- Translational Kinetic Energy: $K_{\text{trans}} = \frac{1}{2} m v^2$
- Rotational Kinetic Energy: $K_{\text{rot}} = \frac{1}{2} I \omega^2$
- Total Mechanical Kinetic Energy for an object rolling without slipping ( $v = r \omega$ ): $K_{\text{total}} = \frac{1}{2} m v^2 + \frac{1}{2} I \omega^2$
Conservation of Mechanical Energy: $E_i = E_f$ $U_i + K_{\text{linear,initial}} + K_{\text{rotational,initial}} = U_f + K_{\text{linear,final}} + K_{\text{rotational,final}}$
Objects Rolling Down an Incline of Height $h$ : $m g h = \frac{1}{2} m v^2 + \frac{1}{2} I \left(\frac{v}{r}\right)^2$

Velocities at Bottom of Incline:
- Frictionless sliding box: $v = \sqrt{2 g h}$
- Solid sphere ( $I = \frac{2}{5} m r^2$ ): $v = \sqrt{\frac{10}{7} g h} \approx 1.20 \sqrt{g h}$
- Solid cylinder ( $I = \frac{1}{2} m r^2$ ): $v = \sqrt{\frac{4}{3} g h} \approx 1.15 \sqrt{g h}$
- Hoop / hollow cylinder ( $I = m r^2$ ): $v = \sqrt{g h} = 1.00 \sqrt{g h}$
Speed Ranking: Sliding Box > Solid Sphere > Solid Cylinder > Hollow Cylinder / Hoop. Objects with a smaller fraction of inertia concentrate less potential energy into rotational motion and reach the bottom faster.

Displacement: $s$ (Linear) $\longleftrightarrow$ $\theta$ (Rotational)
Velocity: $v$ (Linear) $\longleftrightarrow$ $\omega$ (Rotational)
Acceleration: $a$ (Linear) $\longleftrightarrow$ $\alpha$ (Rotational)
Inertia / Mass: $m$ (Linear) $\longleftrightarrow$ $I$ (Rotational)
Dynamic Cause: Force $F = m a$ (Linear) $\longleftrightarrow$ Torque $\tau = I \alpha$ (Rotational)
Momentum: $p = m v$ (Linear) $\longleftrightarrow$ Angular Momentum $L = I \omega$ (Rotational)
Kinetic Energy: $K = \frac{1}{2} m v^2$ (Linear) $\longleftrightarrow$ $K_R = \frac{1}{2} I \omega^2$ (Rotational)
Work: $W = F \cdot s$ (Linear) $\longleftrightarrow$ $W = \tau \cdot \theta$ (Rotational)