Physics 7A Final Exam Study Guide: Classical Mechanics, Fluids, and Oscillations

Classical Mechanics: Kinematics and Dynamics

One- and Two-Dimensional Kinematics

Kinematics describes the motion of objects without reference to the forces causing the motion. It includes both one-dimensional and two-dimensional motion.

• Position, Velocity, and Acceleration: The position of an object as a function of time is given by $x(t)$ or $y(t)$. Velocity is the rate of change of position, and acceleration is the rate of change of velocity.

• Equations of Motion (Constant Acceleration):

  • $x(t) = x_0 + v_0 t + \frac{1}{2} a t^2$

  • $v(t) = v_0 + a t$

  • $v^2 = v_0^2 + 2a(x - x_0)$

• Projectile Motion: Involves two-dimensional motion under gravity, with horizontal and vertical components analyzed separately.

Forces and Newton's Laws

Newton's laws of motion form the foundation of classical mechanics, describing the relationship between forces and motion.

• Newton's First Law: An object remains at rest or in uniform motion unless acted upon by a net external force.

• Newton's Second Law: $\vec{F}_{\text{net}} = m \vec{a}$

• Newton's Third Law: For every action, there is an equal and opposite reaction.

• Types of Forces: Gravitational, normal, tension, friction, and spring forces.

Circular Motion and Gravitation

Uniform Circular Motion

Objects moving in a circle experience a centripetal acceleration directed toward the center of the circle.

• Centripetal Acceleration: $a_c = \frac{v^2}{r}$

• Centripetal Force: $F_c = m \frac{v^2}{r}$

Gravitational Force

Newton's law of universal gravitation describes the attractive force between two masses.

• Gravitational Force: $F_g = G \frac{m_1 m_2}{r^2}$

• Applications: Satellite motion, orbital mechanics, and planetary motion.

Work, Energy, and Power

Work and Kinetic Energy

Work is done when a force causes displacement. Energy is the capacity to do work.

• Work: $W = F d \cos \theta$

• Kinetic Energy: $K = \frac{1}{2} m v^2$

• Work-Energy Theorem: $W_{\text{net}} = \Delta K$

Potential Energy

• Gravitational Potential Energy: $U_g = mgh$

• Elastic Potential Energy (Spring): $U_s = \frac{1}{2} k x^2$

Conservation of Energy

• Mechanical Energy Conservation: $E = K + U$ (in the absence of non-conservative forces)

Momentum and Collisions

Linear Momentum

Momentum is the product of mass and velocity. It is conserved in isolated systems.

• Momentum: $\vec{p} = m \vec{v}$

• Impulse: $\vec{J} = \Delta \vec{p} = F_{\text{avg}} \Delta t$

• Conservation of Momentum: $\sum \vec{p}_{\text{initial}} = \sum \vec{p}_{\text{final}}$

Collisions

• Elastic Collisions: Both momentum and kinetic energy are conserved.

• Inelastic Collisions: Only momentum is conserved; kinetic energy is not.

Rotational Dynamics and Angular Momentum

Rotational Kinematics

Describes the motion of rotating bodies using angular displacement, velocity, and acceleration.

• Angular Displacement: $\theta$ (radians)

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

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

Rotational Inertia (Moment of Inertia)

The moment of inertia quantifies an object's resistance to changes in rotational motion.

Shape	Moment of Inertia ($I$)
Hoop (radius $R$)	$I = M R^2$
Solid cylinder/disk	$I = \frac{1}{2} M R^2$
Solid sphere	$I = \frac{2}{5} M R^2$
Rectangular plate (about center)	$I = \frac{1}{12} M (a^2 + b^2)$
Long thin rod (center)	$I = \frac{1}{12} M L^2$
Long thin rod (end)	$I = \frac{1}{3} M L^2$

Rotational Dynamics

• Torque: $\tau = r F \sin \theta$

• Rotational Analog of Newton's Second Law: $\sum \tau = I \alpha$

• Angular Momentum: $L = I \omega$

• Conservation of Angular Momentum: $L_{\text{initial}} = L_{\text{final}}$ (if net external torque is zero)

Fluid Dynamics

Fluid Flow and Bernoulli's Principle

Fluid dynamics studies the motion of liquids and gases. Bernoulli's equation relates pressure, velocity, and height in a flowing fluid.

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

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

• Applications: Manometers, flow velocity, pressure differences.

Simple Harmonic Motion

Oscillatory Motion

Simple harmonic motion (SHM) describes systems where the restoring force is proportional to displacement.

• Equation of Motion: $x(t) = A \cos(\omega t + \phi)$

• Angular Frequency: $\omega = \sqrt{\frac{k}{m}}$

• Period: $T = \frac{2\pi}{\omega}$

• Mechanical Energy: $E = \frac{1}{2} k A^2$

Waves and Sound

Standing Waves in Tubes

Standing waves are formed by the interference of two waves traveling in opposite directions. The boundary conditions depend on whether the tube is open or closed at the ends.

• Open-Open Tube: Antinodes at both ends; fundamental frequency $f_1 = \frac{v}{2L}$

• Open-Closed Tube: Node at closed end, antinode at open end; fundamental frequency $f_1 = \frac{v}{4L}$

• Frequency of Harmonics: $f_n = n f_1$ (open-open), $f_n = (2n-1) f_1$ (open-closed)

Additional Mathematical Tools

Useful Integrals and Trigonometry

• Integrals: $\int \frac{dx}{x} = \ln|x| + c$

• Trigonometric Identities: $\sin^2 \theta + \cos^2 \theta = 1$

Summary Table: Key Equations and Concepts

Topic	Key Equation	Application
Kinematics	$x(t) = x_0 + v_0 t + \frac{1}{2} a t^2$	Projectile motion, free fall
Newton's Laws	$F = m a$	Force analysis, equilibrium
Energy	$E = K + U$	Conservation of energy
Momentum	$p = m v$	Collisions, impulse
Rotation	$\tau = I \alpha$	Rotational motion
Fluids	$P + \frac{1}{2} \rho v^2 + \rho g h = \text{constant}$	Bernoulli's principle
SHM	$x(t) = A \cos(\omega t + \phi)$	Oscillators, springs
Waves	$f_n = n \frac{v}{2L}$	Standing waves in tubes

Example Applications

• Satellite in Orbit: Use conservation of angular momentum and energy to analyze elliptical orbits.

• Torsion Balance: Apply rotational equilibrium and oscillatory motion to measure gravitational constant $G$.

• Fluid Flow in Tubes: Use Bernoulli's equation and continuity to determine velocities and pressure differences.

• Spring-Mass Oscillator: Analyze SHM to find frequency, position, and energy.

• Rolling Disk on Track: Apply conservation of energy and rotational dynamics to determine speed and trajectory.

• Water Cannon Motion: Use differential equations and conservation of momentum to analyze variable mass systems.

Additional info: These study notes are based on a final exam covering topics from classical mechanics, rotational dynamics, fluid mechanics, oscillations, and waves, as indicated by the problems and provided equation sheet.