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 analogous to linear motion but involve angular measures.

• 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: A wheel rotates from rest with a constant angular acceleration of 2 rad/s2. After 3 seconds, its angular velocity is \(\omega = 0 + (2)(3) = 6\) 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.

• Definition: \[ I = \sum m_i r_i^2 \] where \(m_i\) is the mass of the i-th particle and \(r_i\) is its distance from the axis.

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

Chapter 10: Dynamics of Rotational Motion

Torque

Torque is the rotational equivalent of force. It measures the tendency of a force to rotate an object about an axis.

• Definition: \[ \tau = r F \sin \theta \] where \(r\) is the lever arm, \(F\) is the force, and \(\theta\) is the angle between them.

• Rotational Dynamics: The rotational analog of Newton's second law: \[ \sum \tau = I \alpha \]

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 a measure of the quantity of rotation of an object and is conserved in the absence of external torques.

• Definition: \[ \vec{L} = I \vec{\omega} \]

• External Torque: The rate of change of angular momentum is equal to the net external torque: \[ \sum \vec{\tau}_{ext} = \frac{d\vec{L}}{dt} \]

• Conservation of Angular Momentum: If the net external torque is zero, angular momentum is conserved: \[ \vec{L}_{initial} = \vec{L}_{final} \]

Example: A figure skater pulling in her arms spins faster due to conservation of angular momentum.

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 the net torque on the object to be zero.

• \[ \sum \vec{F} = 0 \]

• \[ \sum \tau = 0 \]

• The choice of pivot point for calculating torque is arbitrary.

Center of Gravity

The center of gravity is the point at which the entire weight of an object can be considered to act for purposes of analyzing gravitational torque.

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 the stiffness of a material under tensile (stretching) stress. \[ Y = \frac{\text{Tensile Stress}}{\text{Tensile Strain}} = \frac{F/A}{\Delta L / L_0} \]

• Tensile Stress: Force per unit area applied perpendicular to the surface.

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

• Elastic Limit: Maximum stress that a material can withstand without permanent deformation.

• Bulk Modulus (B): Measures a material's 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 (\(\Delta V / V_0\)).

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 the shape of its container.

• Gas: No definite shape or volume; expands to fill its container.

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 defined as force per unit area. In fluids, pressure increases with depth due to the weight of the fluid above.

• Atmospheric Pressure: The pressure exerted by the weight of the atmosphere (\(P_{atm}\)).

• Absolute Pressure: The total pressure at a point, including atmospheric pressure. \[ P = P_0 + \rho g d \] where \(P_0\) is the surface 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: A practical application of Pascal's Principle, allowing a small force applied at one point to be transformed into a larger force at another point.

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: \[ F_b = \rho_{fluid} V_{displaced} g \]

• Finding Buoyant Force: Can be determined from the volume of displaced fluid or from the apparent weight loss of the object in fluid.

• Neutral Buoyancy vs Sink vs Float: An object floats if its average density is less than the fluid, sinks if greater, and is neutrally buoyant if equal.

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. \[ Q = A v \] where \(A\) is cross-sectional area and \(v\) is fluid speed.

• Equation of Continuity: For incompressible fluids, the product of cross-sectional area and velocity is constant along a streamline. \[ A_1 v_1 = A_2 v_2 \]

• Bernoulli's Principle: In steady flow, the sum of pressure energy, kinetic energy per unit volume, and potential energy per unit volume is constant. \[ P + \frac{1}{2} \rho v^2 + \rho g y = \text{constant} \]

• Torricelli's Equation: Gives the speed of fluid flowing out of an opening under gravity. \[ v = \sqrt{2 g h} \]