Rotational Dynamics, Fluid Dynamics, and Thermodynamics: Structured Study Notes

Rotational Dynamics and Angular Motion

Angular Coordinates and Conversion

Rotational motion describes the movement of objects around a fixed axis. The angular position θ is measured in radians, where one full revolution equals $2\pi$ radians or $360^\circ$.

• Angular position (θ): $\theta = \frac{s}{r}$, where s is arc length and r is radius.

• Conversion: $1\text{ revolution} = 2\pi \text{ radians} = 360^\circ$

Angular Kinematics

Angular displacement, velocity, and acceleration are analogous to their linear counterparts.

• Angular displacement: $\Delta \theta = \theta_2 - \theta_1$

• Average angular velocity: $\omega_{avg} = \frac{\Delta \theta}{\Delta t}$

• Instantaneous angular velocity: $\omega = \frac{d\theta}{dt}$

• Average angular acceleration: $\alpha_{avg} = \frac{\Delta \omega}{\Delta t}$

• Instantaneous angular acceleration: $\alpha = \frac{d\omega}{dt}$

Linear vs. Angular Motion

There is a direct correspondence between linear and angular motion variables.

Linear	Angular
x (displacement)	θ (angular displacement)
v (velocity)	ω (angular velocity)
a (acceleration)	α (angular acceleration)
F (force)	τ (torque)
m (mass)	I (moment of inertia)

Rotational Kinematics Equations

For constant angular acceleration:

• $\omega = \omega_0 + \alpha t$

• $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$

• $\omega^2 = \omega_0^2 + 2\alpha (\theta - \theta_0)$

Rotational Kinetic Energy

The rotational kinetic energy of a rigid body is given by:

• $K = \frac{1}{2} I \omega^2$

• Moment of inertia (I): $I = \sum m_i r_i^2$ or $I = \int r^2 dm$ for continuous bodies

Torque and Rotational Dynamics

Torque (τ) is the rotational equivalent of force and causes angular acceleration.

• $\vec{\tau} = \vec{r} \times \vec{F}$

• Magnitude: $|\tau| = r F \sin \theta$

• Direction: Determined by the right-hand rule

• Newton's Second Law for Rotation: $\tau = I \alpha$

Work and Power in Rotational Motion

• Work by torque: $W = \tau \Delta \theta$

• Power: $P = \tau \omega$

Angular Momentum

Angular momentum is a measure of the rotational motion of an object.

• $\vec{L} = I \vec{\omega}$

• For a system of particles: $\vec{L} = \sum \vec{r}_i \times m_i \vec{v}_i$

• Conservation of Angular Momentum: If net external torque is zero, $\frac{d\vec{L}}{dt} = 0$

Fluid Dynamics

Phases of Matter and Density

Fluids include liquids and gases. Density is a key property:

• Density (ρ): $\rho = \frac{m}{V}$

• Water: $\rho = 1.0 \times 10^3 \text{ kg/m}^3$

• Specific gravity: Ratio of density to water

Pressure in Fluids

Pressure is the force per unit area exerted by a fluid.

• $P = \frac{F}{A}$

• Pressure at depth: $P = P_0 + \rho g h$

• Pressure is exerted equally in all directions at a point in a fluid at rest.

Atmospheric Pressure

Atmospheric pressure decreases with altitude.

• $P(y) = P_0 e^{-\frac{\rho g y}{P_0}}$ (derived from integrating $\frac{dP}{dy} = -\rho g$)

Pascal's Principle

If an external pressure is applied to a confined fluid, it is transmitted undiminished throughout the fluid.

• $P = P_0 + \rho g h$

Buoyancy and Archimedes' Principle

Buoyant force is the upward force exerted by a fluid on a submerged object.

• $F_b = \rho_{fluid} V g$

• Archimedes' Principle: The buoyant force equals the weight of the fluid displaced.

Fluid Dynamics: Continuity and Bernoulli's Equation

Describes the flow of fluids.

• Continuity equation: $A_1 v_1 = A_2 v_2$ (for incompressible fluids)

• Bernoulli's equation: $P + \frac{1}{2} \rho v^2 + \rho g y = \text{constant}$

Thermal Expansion and Kinetic Theory of Gases

Temperature and Thermometers

Temperature measures the average kinetic energy of particles.

• Kinetic energy: $E_k = k_B T$

• Temperature scales: Celsius, Fahrenheit, Kelvin

• Conversion: $T(K) = T(^\circ C) + 273.15$

Thermal Expansion

Materials expand when heated.

• Linear expansion: $\Delta L = \alpha L_0 \Delta T$

• Volume expansion: $\Delta V = \beta V_0 \Delta T$

Kinetic Molecular Theory for Gases

Describes the behavior of ideal gases.

• Ideal gas assumptions: Point particles, no interactions, random motion, elastic collisions

• State variables: Pressure (P), Volume (V), Temperature (T)

• Boyle's Law: $P \propto \frac{1}{V}$ (at constant T)

• Charles' Law: $V \propto T$ (at constant P)

• Ideal Gas Law: $PV = nRT$

• Alternate form: $PV = Nk_B T$

Calorimetry and Thermal Equilibrium

Heat and Internal Energy

Heat is energy transferred due to temperature difference.

• Change in internal energy: $\Delta E_{int} = Q - W$

• Specific heat: $Q = mc\Delta T$

• Latent heat: $Q = mL$ (for phase changes)

Thermal Equilibrium and Zeroth Law

Objects in thermal contact reach the same temperature.

• Zeroth Law: If A is in thermal equilibrium with B, and B with C, then A is with C.

First Law of Thermodynamics

Energy Conservation in Thermodynamic Processes

The first law relates changes in internal energy to heat and work.

• $\Delta E_{int} = Q - W$

• Q: Heat added to the system

• W: Work done by the system

Examples and Applications

• Calculating heat required for temperature change and phase change

• Using $PV = nRT$ to find mass or moles of gas

• Applying Bernoulli's equation to fluid flow problems

Additional info: These notes cover topics from Ch.8 (Rotational Dynamics and Angular Momentum), Ch.9 (Fluid Dynamics), Ch.13 (Thermal Expansion and Kinetic Molecular Theory for Gases), Ch.14 (Calorimetry and Thermal Equilibrium), and Ch.15 (First Law of Thermodynamics) as outlined in the Physics college course.