6.3 Rotational Motion, Gravitation & Fluid Statics/Dynamics

6.3 Rotational Motion, Gravitation & Fluid Statics/Dynamics

Rotational kinematics, gravitational mechanics, orbital dynamics, and fluid physics represent essential core concepts tested in FSc Physics and the AMC initial examination. This section presents comprehensive explanations, key physical equations, and worked quantitative examples.

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1. Angular Motion Variables & Kinematics

Angular Kinematic Quantities

When a body rotates about a fixed axis:

• Angular displacement ($\theta$): Angle swept by radius vector (measured in radians, where $1\text{ rev} = 2\pi\text{ rad} = 360^\circ$). Arc length relation: $s = r \theta$.
• Angular velocity ($\omega$): Time rate of change of angular displacement: $\omega = \frac{d\theta}{dt}$ (SI unit: $\text{rad/s}$).
• Angular acceleration ($\alpha$): Time rate of change of angular velocity: $\alpha = \frac{d\omega}{dt}$ (SI unit: $\text{rad/s}^2$).

Linear-Angular Equivalences & Acceleration

• Tangential velocity: $v = r \omega$
• Tangential acceleration: $a_t = r \alpha$
• Centripetal (radial) acceleration: $a_c = \frac{v^2}{r} = r \omega^2$ (directed toward rotational center)
• Total linear acceleration: $a = \sqrt{a_t^2 + a_c^2}$

Rotational Kinematic Equations (Constant $\alpha$)

1. $\omega_f = \omega_i + \alpha t$
2. \theta = \omega_i t + \frac{1}{2} \alpha t^2
3. $2 \alpha \theta = \omega_f^2 - \omega_i^2$

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2. Dynamics of Rotation & Angular Momentum

Centripetal Force & Moment of Inertia

• Centripetal Force ($F_c$): Inward net force compelling circular motion: $F_c = m a_c = \frac{m v^2}{r} = m r \omega^2$
• Moment of Inertia ($I$): Measure of rotational inertia, dependent on mass distribution relative to axis: $I = \sum m_i r_i^2$ SI Unit: $\text{kg}\cdot\text{m}^2$.

Rotational Kinetic Energy & Torque

• Rotational Kinetic Energy: $K_{rot} = \frac{1}{2} I \omega^2$
• Rolling without slipping: $K_{total} = K_{trans} + K_{rot} = \frac{1}{2} m v^2 + \frac{1}{2} I \omega^2$
• Newton's 2nd Law for Rotation: $\tau_{net} = I \alpha$

Angular Momentum Conservation

Angular momentum $\mathbf{L}$ is defined as: $\mathbf{L} = \mathbf{r} \times \mathbf{p} \implies L = I \omega = m v r \sin\theta$ Law of Conservation of Angular Momentum: If net external torque is zero ($\boldsymbol{\tau}_{ext} = 0$), total angular momentum remains strictly constant: $I_1 \omega_1 = I_2 \omega_2$

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3. Universal Gravitation & Orbital Mechanics

Newton's Law of Universal Gravitation

$F = G \frac{m_1 m_2}{r^2}$ where $G = 6.674 \times 10^{-11}\text{ N}\cdot\text{m}^2/\text{kg}^2$. Acceleration due to gravity at surface: $g = \frac{G M}{R^2}$.

Satellite Orbital Speed & Escape Velocity

• Orbital Speed ($v_{orb}$): Equating gravitational force to centripetal force ($G \frac{M m}{r^2} = \frac{m v^2}{r}$): $v_{orb} = \sqrt{\frac{G M}{r}}$ For near-Earth orbit ($r \approx R$): $v_{orb} = \sqrt{g R} \approx 7.9\text{ km/s}$.
• Escape Velocity ($v_{esc}$): Minimum launch speed to permanently break free from gravitational pull: $v_{esc} = \sqrt{\frac{2 G M}{R}} = \sqrt{2 g R} \approx 11.2\text{ km/s}$
• Ratio: $v_{esc} = \sqrt{2} \times v_{orb} \approx 1.414 \times v_{orb}$.

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4. Fluid Statics & Fluid Dynamics

Fluid Statics: Pressure, Pascal & Archimedes

• Hydrostatic Pressure: $P = \rho g h$
• Pascal's Principle: Applied pressure in an enclosed fluid is transmitted undiminished in all directions: $\frac{F_1}{A_1} = \frac{F_2}{A_2}$.
• Archimedes' Principle: Buoyant force equals weight of displaced fluid: $F_B = \rho_{fluid} V_{displaced} g$.

Viscosity & Stokes' Law

• Stokes' Law: Retarding drag force on sphere of radius $r$ moving at speed $v$ through fluid of viscosity $\eta$: $F_d = 6 \pi \eta r v$
• Terminal Velocity ($v_t$): Steady speed reached when effective weight equals drag force: $v_t = \frac{2 r^2 g (\rho_{sphere} - \rho_{fluid})}{9 \eta}$

Fluid Dynamics: Continuity & Bernoulli's Principle

• Equation of Continuity (Mass Conservation): For incompressible fluid flow: $A_1 v_1 = A_2 v_2 = \text{constant flow rate } Q$
• Bernoulli's Principle (Energy Conservation): For steady, non-viscous, incompressible fluid flow: $P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}$
• Torricelli's Law: Speed of efflux from a tank orifice at depth $h$: $v = \sqrt{2 g h}$

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5. Worked AMC Numerical Examples

Example 1: Escape Velocity vs Orbital Speed Ratio

Question: Calculate circular orbital speed $v_{orb}$ and escape velocity $v_{esc}$ near Earth's surface taking $g = 9.8\text{ m/s}^2$ and Earth radius $R = 6.4 \times 10^6\text{ m}$. Verify their ratio.

Solution:

1. Orbital speed: $v_{orb} = \sqrt{g R} = \sqrt{9.8 \times 6.4 \times 10^6} = \sqrt{62.72 \times 10^6} \approx 7920\text{ m/s} = 7.92\text{ km/s}$.
2. Escape velocity: $v_{esc} = \sqrt{2 g R} = \sqrt{2 \times 62.72 \times 10^6} \approx 11200\text{ m/s} = 11.2\text{ km/s}$.
3. Ratio verification: $\frac{v_{esc}}{v_{orb}} = \frac{11.2}{7.92} = \sqrt{2} \approx 1.414$.

Example 2: Equation of Continuity

Question: Water flows through a horizontal pipe of area $A_1 = 10\text{ cm}^2$ at speed $v_1 = 2\text{ m/s}$. If the pipe narrows to area $A_2 = 2.5\text{ cm}^2$, find flow speed $v_2$.

Solution:

1. Apply Continuity equation: $A_1 v_1 = A_2 v_2$.
2. Solve for $v_2$: $v_2 = \frac{A_1 v_1}{A_2} = \frac{10 \times 2}{2.5} = 8\text{ m/s}$.

Example 3: Moment of Inertia Comparison

Question: Compare moments of inertia of a solid sphere and a thin hoop, both of mass $M = 5\text{ kg}$ and radius $R = 0.2\text{ m}$.

Solution:

1. Hoop: $I_{hoop} = M R^2 = 5 \times (0.2)^2 = 0.20\text{ kg}\cdot\text{m}^2$.
2. Solid Sphere: $I_{sphere} = \frac{2}{5} M R^2 = 0.4 \times 0.20 = 0.08\text{ kg}\cdot\text{m}^2$.
3. The solid sphere has a significantly smaller moment of inertia due to central mass concentration.