Rotational Motion, Fluids, and Thermodynamics: Structured Study Notes

Rotational Motion

Angular Coordinates and Conversion

Rotational motion describes the movement of objects around a fixed axis. The angular position is measured in radians, which relate to degrees as follows:

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

• Radians: $2\pi$ radians = $360^\circ$

• Conversion: $x = r\theta$ (arc length), $v = r\omega$ (linear velocity), $a = r\alpha$ (linear acceleration)

Angular Kinematics

Angular motion parallels linear motion, with analogous quantities:

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

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

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

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

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

Linear vs. Angular Motion Comparison

Linear	Angular
x	$\theta$
v	$\omega$
a	$\alpha$
$v = \frac{dx}{dt}$	$\omega = \frac{d\theta}{dt}$
$a = \frac{dv}{dt}$	$\alpha = \frac{d\omega}{dt}$

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

• Kinetic energy of rotation: $K = \frac{1}{2} I \omega^2$

• Moment of inertia (I): $I = \int r^2 dm$

Torque and Angular Acceleration

Torque is the rotational analog of force and causes angular acceleration.

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

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

• Right-hand rule: Determines direction of torque vector.

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

Work and Power in Rotational Motion

• Work by torque: $W = \int \tau d\theta$

• Power: $P = \tau \omega$

Angular Momentum

• Angular momentum: $\vec{L} = I \vec{\omega}$

• Conservation: If $\tau_{net} = 0$, then $\frac{d\vec{L}}{dt} = 0$

Fluids

Phases and Density

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

• Density: $\rho = \frac{m}{V}$

• Specific gravity: Ratio of density to water's density.

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

Pressure in Fluids

• Pressure: $P = \frac{F}{A}$

• Pressure acts equally in all directions at a point in a fluid.

Hydrostatic Pressure

• Hydrostatic pressure: $P = P_0 + \rho g h$

• Atmospheric pressure decreases with altitude: $P(y) = P_0 e^{-\frac{\rho g y}{P_0}}$

Pascal's Principle

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

Buoyancy and Archimedes' Principle

• Buoyant force: $F_b = \rho_{fluid} V g$

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

• Example: Determining if an object is made of gold by measuring its apparent weight in water.

Fluid Dynamics

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

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

Temperature and Thermal Expansion

Temperature Scales and Measurement

• Temperature: Measure of average kinetic energy.

• Scales: Celsius, Fahrenheit, Kelvin

• Conversions:

  • $T(K) = T(^\circ C) + 273.15$

  • $T(^\circ F) = \frac{9}{5} T(^\circ C) + 32$

Zeroth Law of Thermodynamics

• If two systems are each in thermal equilibrium with a third, they are in equilibrium with each other.

Thermal Expansion

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

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

• $\alpha$ and $\beta$ are coefficients of linear and volume expansion, respectively.

Kinetic Theory of Gases and Ideal Gas Law

State Variables and Ideal Gas

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

• Ideal gas assumptions:

  1. Point particles

  2. No interactions

  3. Random motion

  4. Elastic collisions

Gas Laws

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

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

• Ideal Gas Law: $PV = nRT$

• Microscopic form: $PV = Nk_B T$

• n: number of moles, $N$: number of molecules

Example Calculation

• Given $P$, $V$, $T$, solve for $n$ using $n = \frac{PV}{RT}$

Thermodynamics

Internal Energy and Heat

• Internal energy: $E_{int} = \frac{3}{2} nRT$ (for monatomic ideal gas)

• Heat capacity: $Q = mc \Delta T$

• Latent heat: $Q = mL$ (fusion or vaporization)

First Law of Thermodynamics

• $\Delta E_{int} = Q - W$

• Differential form: $dE_{int} = dQ - dW$

Thermodynamic Processes

Process	Condition	Heat (Q)	Work (W)
Isothermal	$\Delta T = 0$	$Q = W$	$W = nRT \ln \frac{V_f}{V_i}$
Adiabatic	$Q = 0$	$Q = 0$	$\Delta E_{int} = -W$
Isovolumetric	$\Delta V = 0$	$Q = \Delta E_{int}$	$W = 0$
Isobaric	$\Delta P = 0$	$Q = \Delta E_{int} + W$	$W = P \Delta V$

Work Done by a Gas

• $W = \int_{V_i}^{V_f} P dV$

• For isobaric: $W = P(V_f - V_i)$

Molar Specific Heats

• At constant volume: $Q_v = nC_v \Delta T$

• At constant pressure: $Q_p = nC_p \Delta T$

• For monatomic gas: $C_v = \frac{3}{2}R$, $C_p = \frac{5}{2}R$

• For diatomic gas: $C_v = \frac{5}{2}R$, $C_p = \frac{7}{2}R$

• Ratio: $\gamma = \frac{C_p}{C_v}$

Examples and Applications

• Calculating heat required for temperature change and phase change.

• Using $PV = nRT$ to solve for unknowns in gas problems.

• Applying the first law to different thermodynamic processes.

Additional info: Some equations and examples have been expanded for clarity and completeness. All major topics from the provided notes have been included and organized for exam preparation.