Rotational Dynamics and Moment of Inertia

Fundamentals of Rotational Dynamics

• Revolution vs. Rotation:     * Revolution: During revolution, the object undergoes a circular path about some other object or a point outside the object.     * Rotation: During rotation, the motion is about an axis of rotation passing through the object itself.

• Definition and Characteristics of Circular Motion:     * Circular Motion: The motion of an object around a circular path is called circular motion.     * Accelerated Motion: As the direction of velocity changes at every instant, it is an accelerated motion.     * Periodic Motion: During the motion, the particle repeats its path along the same trajectory. Thus, the motion is periodic in space.

• Analogy Between Linear and Rotational Quantities:     * Displacement: Linear is $s$; Rotational/Angular is $\theta$.     * Velocity: Linear is $v = \frac{ds}{dt}$; Rotational/Angular is $\omega = \frac{d\theta}{dt}$.     * Acceleration: Linear is $a = \frac{dv}{dt}$; Rotational/Angular is $\alpha = \frac{d\omega}{dt}$.

Angular Kinematics and Relationships

• Tangential Velocity ($v$):     * It describes the motion of an object along the edge of the circle whose direction at any given point on the circle is always along the tangent to that point.     * Vector Form: $\vec{v} = \vec{\omega} \times \vec{r}$, where $\vec{r}$ is the position vector and $\vec{\omega}$ is the angular velocity.     * Magnitude: $v = r\omega$.

• Direction of Angular Velocity:     * The direction of angular velocity ($\omega$) can be determined using the Right Hand Thumb Rule.

• Relation Between Angular Velocity, Period, and Frequency:     * $n = \text{frequency}$     * $T = \text{Time period}$     * $\omega = \frac{2\pi}{T}$     * Since $n = \frac{1}{T}$, $\omega = 2\pi n$.

Uniform and Non-Uniform Circular Motion

• Uniform Circular Motion (UCM):     * During circular motion, if the speed of the particle remains constant, it is called Uniform Circular Motion (UCM).     * The direction of velocity changes at every instant, remaining always tangential to the path.     * The acceleration responsible for this is the centripetal or radial acceleration ($a_r$).     * In UCM, the magnitude of acceleration is constant: $a = \omega^2 r = \frac{v^2}{r} = v\omega$.     * This acceleration is always directed towards the center of the circular motion (along $-\vec{r}$), hence it is called centripetal.

• Non-Uniform Circular Motion (NUM):     * During circular motion, if the speed of the particle varies, it is called non-uniform circular motion.     * Example: The starting and stopping of a fan; the speed varies for some time, making it a non-uniform circular motion.     * The acceleration responsible for changing the magnitude of velocity is directed along or opposite to the velocity.     * This acceleration is always tangential and is called tangential acceleration ($a_t$).     * As the magnitude of tangential velocity changes, the corresponding angular velocity also changes at every instant due to angular acceleration: $\alpha = \frac{d\omega}{dt}$.

Dynamics of Circular Motion: Forces

• Centripetal Force vs. Centrifugal Force:     * Centripetal Force:         * Directed along the radius, towards the center of the circle.         * It is a real force.         * Considered in an inertial frame of reference.         * Vector form: $\mathbf{F} = -\frac{mv^2}{r} \mathbf{r}_0$.     * Centrifugal Force:         * Directed along the radius, away from the center of the circle.         * It is a pseudo force.         * Considered in a non-inertial frame of reference.         * Vector form: $\mathbf{F} = \frac{mv^2}{r} \mathbf{r}_0$.

• Direction of Angular Acceleration ($\alpha$):     * For increasing speed: $\alpha$ is in the same direction as $\omega$.     * For decreasing speed: $\alpha$ is in the opposite direction to $\omega$.

Applications of Circular Motion

• Vehicle on a Horizontal Unbanked Road:     * Forces acting on the car:         1. Weight ($mg$) acting downwards.         2. Normal reaction ($N$) that balances the weight ($N = mg$).         3. Force of static friction ($f_s$) between road and tires.     * Friction acts as the resultant centripetal force to prevent outward slipping or skidding: $f_s = \frac{mv^2}{r}$.     * Derivation:         * $\frac{f_s}{N} = \frac{\frac{mv^2}{r}}{mg} = \frac{v^2}{rg}$         * As speed increases, $f_s$ also increases. At maximum speed, $f_s = (f_s)_{max} = \mu_s N$.         * $\mu_s N = \frac{m v_{max}^2}{r}$         * $v_{max} = \sqrt{\mu_s rg}$, where $\mu_s$ is the coefficient of static friction.

• Well of Death (Maut Ka Kua):     * This is a vertical cylindrical wall of radius $r$ inside which a vehicle is driven in horizontal circles (often seen in stunts).     * Forces acting on the vehicle:         1. Normal reaction ($N$) acting horizontally towards the center.         2. Weight ($mg$) acting vertically downwards.         3. Force of static friction ($f_s$) acting vertically upwards to prevent downward slipping.     * Derivation for Minimum Safest Velocity ($v_{min}$):         * $f_s = mg$         * $N = \frac{mv^2}{r}$         * To avoid slipping, $f_s \le \mu_s N$         * $mg \le \mu_s \frac{mv^2}{r}$         * $g \le \frac{\mu_s v^2}{r} \rightarrow v^2 \ge \frac{rg}{\mu_s}$         * $v_{min} = \sqrt{\frac{rg}{\mu_s}}$

Banking of Roads

• Purpose of Banking:     * On horizontal roads, centripetal force depends on friction. Frictional force has an upper limit and varies with road conditions (e.g., wet surfaces).     * To reduce dependency on friction, the surface of curved roads is tilted with the horizontal at an angle $\theta$. This is called banking of a road.

• Angle of Banking (Neglecting Friction):     * Weight ($mg$) acts downward; Normal reaction ($N$) is perpendicular to the road.     * Components of N:         * Vertical: $N \cos(\theta)$ balances $mg$.         * Horizontal: $N \sin(\theta)$ provides centripetal force $\frac{mv^2}{r}$.     * Equations:         * $N \sin(\theta) = \frac{mv^2}{r}$         * $N \cos(\theta) = mg$         * $\frac{N \sin(\theta)}{N \cos(\theta)} = \frac{\frac{mv^2}{r}}{mg} \rightarrow \tan(\theta) = \frac{v^2}{rg}$         * Angle of banking: $\theta = \tan^{-1}\left(\frac{v^2}{rg}\right)$.         * Most safe speed: $v_s = \sqrt{rg \tan(\theta)}$.

• Speed Limits on Banked Roads:     * Lower Speed Limit ($v_1$):         * For $v_1 < \sqrt{rg \tan(\theta)}$, the vehicle tends to slide down. Static friction ($f_s$) acts upwards along the incline.         * $\frac{v_1^2}{rg} = \frac{\tan(\theta) - \mu_s}{1 + \mu_s \tan(\theta)}$         * $v_{min} = \sqrt{rg \left( \frac{\tan(\theta) - \mu_s}{1 + \mu_s \tan(\theta)} \right)}$.     * Upper Speed Limit ($v_2$):         * For $v_2 > \sqrt{rg \tan(\theta)}$, the vehicle tends to skid outward. Static friction ($f_s$) acts downwards along the incline.         * $\frac{v_2^2}{rg} = \frac{\tan(\theta) + \mu_s}{1 - \mu_s \tan(\theta)}$         * $v_{max} = \sqrt{rg \left( \frac{\tan(\theta) + \mu_s}{1 - \mu_s \tan(\theta)} \right)}$.

Pendulums

• Definitions:     1. Pendulum: A tiny mass connected to a long, flexible, massless, inextensible string suspended from a rigid support.     2. Simple Pendulum: A pendulum where the string oscillates in a single vertical plane.     3. Conical Pendulum: A system where the string moves along the surface of a right circular cone and the point object performs uniform horizontal circular motion.

• Time Period of a Conical Pendulum:     * Let $L$ be the length of the string, $m$ the mass of the bob, and $\theta$ the angle with the vertical.     * Forces: Tension ($T_0$) and Weight ($mg$).     1. $T_0 \cos(\theta) = mg$     2. $T_0 \sin(\theta) = m r \omega^2$     * Combining these: $\tan(\theta) = \frac{r\omega^2}{g}$.     * From geometry: $r = L \sin(\theta)$.     * $\omega^2 = \frac{g \sin(\theta)}{r \cos(\theta)} = \frac{g \sin(\theta)}{L \sin(\theta) \cos(\theta)} = \frac{g}{L \cos(\theta)}$.     * $\omega = \sqrt{\frac{g}{L \cos(\theta)}}$.     * Time Period ($T$): $T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{L \cos(\theta)}{g}}$.     * Frequency ($n$): $n = \frac{1}{2\pi} \sqrt{\frac{g}{L \cos(\theta)}}$.

• Factors Governing Frequency of Conical Pendulum:     * Length ($L$): $n \propto \frac{1}{\sqrt{L}}$. As length decreases, frequency increases.     * Acceleration due to gravity ($g$): $n \propto \sqrt{g}$. As $g$ increases, frequency increases.     * Angle ($\theta$): $n \propto \frac{1}{\sqrt{\cos(\theta)}}$. As $\theta$ increases, $\cos(\theta)$ decreases and $n$ increases.     * Mass: Frequency is independent of the mass of the bob.

Vertical Circular Motion (VCM)

• Types of VCM:     1. Controlled: Speed is kept constant or not totally controlled by gravity (e.g., giant wheel).     2. Gravity-Controlled: Energy is supplied at the lowest point, and kinematics are governed by gravitation (interconversion of kinetic and gravitational potential energy).

• Minimum Speeds in Gravity-Controlled VCM:     * Topmost point (A): $T_A = 0$. Centripetal force provided by weight: $\frac{mv_A^2}{r} = mg \rightarrow v_A = \sqrt{rg}$.     * Lowermost point (B): By conservation of energy ($KE_B = KE_A + PE_{loss}$): $v_B = \sqrt{5rg}$.     * Middle point (C or D): By conservation of energy: $v_C = \sqrt{3rg}$.

• Tension Differences:     * $T_B - T_A = 6mg$     * $T_C - T_A = 3mg$

• Convex Over-bridge:     * A vehicle at the top of a convex bridge experience weight ($mg$) downwards and Normal reaction ($N$) upwards.     * Resultant force: $mg - N = \frac{mv^2}{r}$.     * For the vehicle to maintain contact, $N \ge 0$. This imposes an upper speed limit:     * $N = mg - \frac{mv^2}{r}$. Setting $N = 0$ gives $v_{max} = \sqrt{rg}$.

Moment of Inertia (M.I.)

• Definition: The moment of inertia ($I$) is the sum of the product of the mass of each particle and the square of its perpendicular distance from the axis of rotation: $I = \sum m_i r_i^2$.

• Kinetic Energy of a Rotating Body:     * A rigid object consists of particles $m_1, m_2, \dots, m_n$ at distances $r_1, r_2, \dots, r_n$.     * All particles have the same $\omega$ but different linear speeds $v_i = r_i \omega$.     * Individual $\text{K.E.}_i = \frac{1}{2} m_i v_i^2 = \frac{1}{2} m_i r_i^2 \omega^2$.     * $\text{Total Rotational K.E.} = \sum \frac{1}{2} m_i r_i^2 \omega^2 = \frac{1}{2} \left( \sum m_i r_i^2 \right) \omega^2$.     * $\text{Rotational K.E.} = \frac{1}{2} I \omega^2$.

• Physical Significance of M.I.:     * M.I. is the rotational analogue of mass ($m$). It represents rotational inertia.     * It depends on individual masses and the distribution of these masses about the axis of rotation.

• Moment of Inertia of Specific Objects:     * Uniform Ring: All mass $M$ is at distance $R$ from the axis. $I = MR^2$.     * Uniform Disc: A two-dimensional circular object. Surface density $\sigma = \frac{M}{A} = \frac{M}{\pi R^2}$.         * Considered as concentric rings of radius $r$ and width $dr$.         * Mass of elemental ring $dm = \sigma(2\pi r dr)$.         * $dI = dm \times r^2 = 2\pi \sigma r^3 dr$.         * Integration from $0$ to $R$: $I = \int_0^R 2\pi \sigma r^3 dr = 2\pi \sigma [ \frac{r^4}{4} ]_0^R = \frac{\pi \sigma R^4}{2}$.         * Substituting $\sigma = \frac{M}{\pi R^2}$: $I = \frac{\pi R^4}{2} \frac{M}{\pi R^2} = \frac{1}{2} MR^2$.

• Radius of Gyration ($K$):     * Defined as the distance from the axis where the entire mass of the object could be concentrated to have the same moment of inertia: $I = MK^2$.     * A larger $K$ value indicates that the mass is distributed farther from the axis.