Rotational Motion, Equilibrium, and Fluid Mechanics: Study Notes

Chapter 9: Rotation of Rigid Bodies

Rotational Kinematics

Rotational kinematics describes the motion of objects as they rotate about a fixed axis. The key quantities are angular displacement, angular velocity, and angular acceleration.

• Angular Displacement (\(\theta\)): The angle through which a point or line has been rotated in a specified sense about a specified axis. Measured in radians.

• Angular Velocity (\(\omega\)): The rate of change of angular displacement with respect to time. $\omega = \frac{d\theta}{dt}$

• Angular Acceleration (\(\alpha\)): The rate of change of angular velocity with respect to time. $\alpha = \frac{d\omega}{dt}$

Example: If a wheel rotates from 0 to 2 radians in 4 seconds, its average angular velocity is $\omega_{avg} = \frac{2\ \text{rad}}{4\ \text{s}} = 0.5\ \text{rad/s}$.

Moment of Inertia

The moment of inertia quantifies an object's resistance to changes in its rotational motion, analogous to mass in linear motion. It depends on the mass distribution relative to the axis of rotation.

• Formula: $I = \sum m_i r_i^2$ (for discrete masses), $I = \int r^2 dm$ (for continuous mass distributions)

Rotational Kinetic Energy: The energy due to rotation is given by $K_{rot} = \frac{1}{2} I \omega^2$

Example: A solid disk of mass $M$ and radius $R$ has $I = \frac{1}{2}MR^2$ about its center.

Chapter 10: Dynamics of Rotational Motion

Torque

Torque is the rotational equivalent of force; it causes changes in rotational motion.

• Definition: $\tau = r F \sin\theta$

• Rotational Dynamics: Newton's second law for rotation: $\tau_{net} = I \alpha$

Example: Applying a force of 10 N at a distance of 0.5 m from the axis, perpendicular to the lever arm, gives $\tau = 0.5 \times 10 = 5\ \text{N} \cdot \text{m}$.

Torque as a Vector and the Cross Product

Torque is a vector quantity, calculated using the cross product:

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

Angular Momentum

Angular momentum is the rotational analog of linear momentum and is conserved in isolated systems.

• Definition: $\vec{L} = I \vec{\omega}$

• External Torque: The rate of change of angular momentum equals the net external torque: $\vec{\tau}_{ext} = \frac{d\vec{L}}{dt}$

• Nonzero External Torque: Changes angular momentum.

• Zero External Torque: Angular momentum is conserved.

Example: A spinning figure skater pulls in her arms, reducing $I$ and increasing $\omega$ to conserve $L$.

Chapter 11: Equilibrium & Elasticity

Static Equilibrium

An object is in static equilibrium if it is at rest and remains at rest. This requires both the net force and net torque to be zero.

• Net Force: $\sum \vec{F} = 0$

• Net Torque: $\sum \tau = 0$ (about any pivot point)

Example: A balanced seesaw with equal weights at equal distances from the pivot is in static equilibrium.

Center of Gravity

The center of gravity is the point where the total weight of a body is considered to act for purposes of analysis.

Elasticity: Young's Modulus and Bulk Modulus

Elasticity describes how materials deform under stress and return to their original shape when the stress is removed.

• Young's Modulus (Y): Measures stiffness under tensile (stretching) or compressive (squeezing) stress. $Y = \frac{\text{Tensile Stress}}{\text{Tensile Strain}} = \frac{F/A}{\Delta L/L_0}$

• Tensile Stress: Force per unit area ($F/A$)

• Tensile Strain: Relative change in length ($\Delta L/L_0$)

• Elastic Limit: Maximum stress that can be applied without permanent deformation.

• Bulk Modulus (B): Measures resistance to uniform compression. $B = -\frac{\Delta P}{\Delta V/V_0}$

• Volume Stress: Change in pressure ($\Delta P$)

• Volume Strain: Relative change in volume ($\Delta V/V_0$)

Example: Steel has a high Young's modulus, meaning it is very stiff and resists stretching.

Chapter 12: Fluid Mechanics

States of Matter

Matter exists in three primary states: solid, liquid, and gas. Each state has distinct physical properties.

• Solid: Definite shape and volume

• Liquid: Definite volume, takes shape of container

• Gas: No definite shape or volume

Fluids

Fluids are substances that flow and take the shape of their container. Both liquids and gases are considered fluids.

Pressure in Fluids

Pressure is the force exerted per unit area. In fluids, pressure increases with depth due to the weight of the fluid above.

• Atmospheric Pressure: Pressure exerted by the weight of the atmosphere (at sea level, $1.01 \times 10^5$ Pa)

• Absolute Pressure: Total pressure at a point, including atmospheric pressure and pressure due to fluid depth. $P = P_0 + \rho g d$

• At Depth d: $P = P_0 + \rho g d$, where $P_0$ is atmospheric pressure, $\rho$ is fluid density, $g$ is acceleration due to gravity, and $d$ is depth.

Pascal's Principle

Pascal's Principle states that a change in pressure applied to an enclosed fluid is transmitted undiminished to all portions of the fluid and to the walls of its container.

• Hydraulic Lift: Uses Pascal's Principle to multiply force. If $F_1$ is applied to area $A_1$, the output force $F_2$ on area $A_2$ is $F_2 = F_1 \frac{A_2}{A_1}$.

Buoyant Force and Archimedes' Principle

Archimedes' Principle states that a body immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced.

• Buoyant Force: $F_b = \rho_{fluid} V_{disp} g$

• Finding Buoyant Force: From displaced fluid or from the apparent weight loss of the object.

• Neutral Buoyancy: Occurs when the buoyant force equals the object's weight (object neither sinks nor floats).

• Sink vs Float: If $F_b > W$, object floats; if $F_b < W$, object sinks.

Example: A block of wood floats because the weight of water displaced equals the block's weight.

Fluid Dynamics

Fluid dynamics studies the motion of fluids and the forces that affect them.

• Laminar Flow: Smooth, orderly flow of fluid in parallel layers.

• Flow Rate (Q): Volume of fluid passing a point per unit time. $Q = A v$

• Equation of Continuity: For incompressible fluids, $A_1 v_1 = A_2 v_2$

• Bernoulli's Principle: For an ideal fluid, the sum of pressure energy, kinetic energy per unit volume, and potential energy per unit volume is constant along a streamline. $P + \frac{1}{2} \rho v^2 + \rho g y = \text{constant}$

• Torricelli's Equation: The speed of efflux of a fluid under gravity from a hole at depth $h$ is $v = \sqrt{2gh}$

Example: Water flows faster through a narrow pipe than a wide one, as per the equation of continuity.