Dynamics of Rotational Motion — Revision Notes

⚡ 30-Second Revision

Torque: — $\vec{\tau} = \vec{r} \times \vec{F}$, magnitude $\tau = rF\sin\theta$. Unit: N\cdot m.\n- Moment of Inertia (I): Rotational inertia. $I = \sum m_i r_i^2$ (discrete), $I = \int r^2 dm$ (continuous). Unit: kg\cdot m$^2$.\n- Newton's 2nd Law for Rotation: $\vec{\tau}_{net} = I\vec{\alpha}$.\n- Angular Momentum (L): $\vec{L} = I\vec{\omega}$ (rigid body), $\vec{L} = \vec{r} \times \vec{p}$ (particle). Unit: kg\cdot m$^2$/s or J\cdot s.\n- Conservation of Angular Momentum: If $\vec{\tau}_{net} = 0$, then $\vec{L} = \text{constant} \implies I_1\omega_1 = I_2\omega_2$.\n- Rotational Kinetic Energy: $K_{rot} = \frac{1}{2}I\omega^2$.\n- Total Kinetic Energy (Rolling): $K_{total} = \frac{1}{2}Mv_{CM}^2 + \frac{1}{2}I_{CM}\omega^2$.\n- Rolling without Slipping: $v_{CM} = R\omega$, $a_{CM} = R\alpha$.\n- Parallel Axis Theorem: $I = I_{CM} + Md^2$.\n- Perpendicular Axis Theorem (planar body): $I_z = I_x + I_y$.

2-Minute Revision

Rotational dynamics extends linear mechanics to spinning objects. The key players are **torque ($\tau$), the rotational equivalent of force, which causes angular acceleration ($\alpha$), and moment of inertia (I)**, the rotational equivalent of mass, which resists angular acceleration.

Newton's second law for rotation is $\tau = I\alpha$. Angular momentum ($L = I\omega$) is the rotational quantity of motion. Crucially, if no external torque acts on a system, its angular momentum is conserved ($I\omega = \text{constant}$).

This explains phenomena like a figure skater's spin. For objects rolling without slipping, total kinetic energy is the sum of translational and rotational components ($K_{total} = \frac{1}{2}Mv_{CM}^2 + \frac{1}{2}I_{CM}\omega^2$), with $v_{CM} = R\omega$.

Remember standard moments of inertia for common shapes and use the parallel axis theorem ($I = I_{CM} + Md^2$) when the axis is not through the center of mass. Pay attention to vector directions for torque and angular momentum.

5-Minute Revision

Dynamics of rotational motion governs how rigid bodies rotate under the influence of forces. The fundamental concept is **torque ($\vec{\tau}$)**, defined as $\vec{r} \times \vec{F}$, which is the rotational analogue of force.

It causes **angular acceleration ($\vec{\alpha}$). The resistance to this angular acceleration is quantified by the moment of inertia (I)**, which depends on both mass and its distribution relative to the axis of rotation ($I = \sum m_i r_i^2$).

Newton's second law for rotation states $\vec{\tau}_{net} = I\vec{\alpha}$. \n\n**Angular momentum ($\vec{L}$)**, the rotational analogue of linear momentum, is given by $\vec{L} = I\vec{\omega}$ for a rigid body.

A cornerstone principle is the conservation of angular momentum: if the net external torque on a system is zero, its total angular momentum remains constant ($I_1\omega_1 = I_2\omega_2$). This is vital for problems involving changing mass distribution.

\n\nRotational kinetic energy is $K_{rot} = \frac{1}{2}I\omega^2$. For rolling motion without slipping, an object has both translational and rotational kinetic energy, so $K_{total} = \frac{1}{2}Mv_{CM}^2 + \frac{1}{2}I_{CM}\omega^2$.

The condition for rolling without slipping is $v_{CM} = R\omega$ and $a_{CM} = R\alpha$. When objects roll down an incline, their acceleration depends on their moment of inertia, with objects having smaller $I/MR^2$ ratios (like a solid sphere) accelerating faster.

\n\nRemember the Parallel Axis Theorem ($I = I_{CM} + Md^2$) to find moment of inertia about an axis parallel to one through the center of mass. For planar bodies, the Perpendicular Axis Theorem ($I_z = I_x + I_y$) is useful.

Practice problems involving pulley systems with massive pulleys, where both linear and rotational dynamics are coupled, and energy conservation problems involving rotational motion.

Prelims Revision Notes

1
Rotational Analogues: — Understand the correspondence: Force $\leftrightarrow$ Torque ($\tau$), Mass $\leftrightarrow$ Moment of Inertia (I), Linear Velocity $\leftrightarrow$ Angular Velocity ($\omega$), Linear Acceleration $\leftrightarrow$ Angular Acceleration ($\alpha$), Linear Momentum $\leftrightarrow$ Angular Momentum (L), Kinetic Energy (Translational) $\leftrightarrow$ Kinetic Energy (Rotational). \n2. Torque: $\vec{\tau} = \vec{r} \times \vec{F}$. Magnitude $\tau = rF\sin\theta$. Direction by right-hand rule. \n3. Moment of Inertia (I): $I = \sum m_i r_i^2$. Standard formulas: \n * Ring/Hollow Cylinder (about central axis): $MR^2$ \n * Disc/Solid Cylinder (about central axis): $\frac{1}{2}MR^2$ \n * Solid Sphere (about diameter): $\frac{2}{5}MR^2$ \n * Hollow Sphere (about diameter): $\frac{2}{3}MR^2$ \n * Rod (about center, perpendicular): $\frac{1}{12}ML^2$ \n * Rod (about end, perpendicular): $\frac{1}{3}ML^2$ \n4. Theorems: \n * Parallel Axis Theorem: $I = I_{CM} + Md^2$. \n * Perpendicular Axis Theorem (planar body): $I_z = I_x + I_y$. \n5. Newton's Second Law for Rotation: $\vec{\tau}_{net} = I\vec{\alpha}$. \n6. Angular Momentum: $\vec{L} = I\vec{\omega}$. For a particle, $\vec{L} = \vec{r} \times m\vec{v}$. \n7. Conservation of Angular Momentum: If $\vec{\tau}_{net, ext} = 0$, then $L_{initial} = L_{final} \implies I_1\omega_1 = I_2\omega_2$. \n8. Rotational Kinetic Energy: $K_{rot} = \frac{1}{2}I\omega^2$. \n9. Rolling Motion (without slipping): \n * $v_{CM} = R\omega$, $a_{CM} = R\alpha$. \n * Total Kinetic Energy: $K_{total} = \frac{1}{2}Mv_{CM}^2 + \frac{1}{2}I_{CM}\omega^2$. \n * Acceleration down incline: $a = \frac{g\sin\theta}{1 + I_{CM}/MR^2}$. Object with smaller $I_{CM}/MR^2$ (e.g., solid sphere = 2/5) has greater acceleration. \n10. Work and Power: Rotational Work $W = \tau\theta$. Rotational Power $P = \tau\omega$.

Vyyuha Quick Recall

To remember the rotational analogues: 'For My Angular Teacher, I Always Learn Well.' \nForce $\rightarrow$ Torque ($\tau$) \nMass $\rightarrow$ Inertia (I) \nAcceleration (linear) $\rightarrow$ Acceleration (angular, $\alpha$) \nLinear momentum $\rightarrow$ L (Angular momentum) \nWork $\rightarrow$ Work (rotational)