Numerical Methods for Solving Equations

1.1 Solving an Algebraic System

This code defines a function f(V) and then solves the system f([x,y,z]) = [x,y,z] to find its fixed points using the algsys command.

faux(x,y,z) := [(x^2)/2 + 1/3, x+y+z/4, x^2-y^2+(z^2)/4];
f(V) := faux(V[1], V[2], V[3]);

ecus : f([x,y,z]) - [x,y,z];
algsys(ecus, [x,y,z]), numer;

1.2 Fixed-Point Iteration Method

This block implements the fixed-point iteration method. It starts with an initial guess (seed) and iteratively applies the function f until the norm of the difference between successive iterations is smaller than a given tolerance (10^-14).

nor(V) := sqrt(V.V);
fpprec:100;

block(
  semilla:[0,0,0],
  while nor(f(semilla)-semilla) > 10^(-14) do 
    semilla:bfloat(

Motion in a Plane

Magnitude of vector: |A| = √(a_x² + a_y² + a_z²)

Unit vector: â = vector a / |a|

Resultant: R = √(A² + B² + 2AB cos θ)

Time period of projectile motion (T): 2u sin θ / g

Maximum height (H): u² sin² θ / 2g

Range (R): u² sin 2θ / g

Uniform Circular Motion (UCM)

• Centripetal acceleration (a_c): v² / r
• Angular velocity (ω): θ / t

Relation between linear velocity and angular velocity: v = rω

Gravitation

The Universal Law of Gravitation: Masses attract each other with a force proportional to their masses and inversely proportional to the square of the distance between them.

Force (F): G m₁m₂ / r²

Gravity at height (g_h): g_h = g_s / (1 + h/r)² or g_h = g_s(1 - 2h/r) if h < 300 km

Gravity at depth (g_d): g_d = g_s(1 - d/R)

Escape

Thermodynamics Exam Cheat Sheet: Problem-Solving Recipes

🔹 Problem 1: Property Changes & T-s Sketch

(HW 9 Q1–Q3)
1a. Ideal-Gas Entropy Change

Statement: Air (ideal gas) in a piston-cylinder goes from State 1: T₁ = 350 K, p₁ = 100 kPa to State 2: T₂ = 550 K, p₂ = 700 kPa. Find Δs (kJ/kg·K) (a) reversible, (b) with irreversibilities.

1b. Isothermal Water Compression

Statement: 1 kg water in a piston-cylinder, initially saturated vapor at T = 160 °C, p = 150 kPa, isothermally compressed to saturated liquid. Work on water W = -471.5 kJ. Find (a) Q (kJ), (b) ΔS (kJ/K). Sketch T-s.

Steps for 1a (Ideal Gas)

1. Formula:

   

2. Compute

   • 

3. Evaluate

   

4. Irreversible:

   

Steps for 1b (Water Compression)

1. 1st Law:

   

2. Table Lookup at 160 °C (saturated):

   • 

3. Compute ΔU:

   

4. Compute

Mamíferos: Las hembras tienen mamas que producen leche para alimentar a las crías, el cuerpo está cubierto por pelos. La función principal es proteger del frio al animal.
Pelaje: raíz dentro del funículo piloso .Los pelos nuevos sacan los pelos viejos. La glándula sebácea lo hace más flexible el pelo. Lana (largo y fino), cerda (áspero y grueso), vello (suave y corto) y púa (largo y puntiagudo)
La temperatura: Es constante. Si sube o baja la temperatura siempre se mantiene constante.
Hibernación: desciende -10 Cº / Duerme no come y se esconde.
Invernación: baja la temperatura pero un duerme profundamente, come y hace algunas actividades.
La fecundación interna se produce en el ovulo con el esperma.
Sistema digestivo: Boca -faringe-

A/C/B/D/A/B/B/B/B/C/A/B/A/D/A/C/C/C/A/B/D/A/A/B/B/A/C/D/B/D

1. Stopwatch Accuracy

a) Stopwatch

b) To have more accuracy

c) i) 3rd

ii) (3.93 + 4.07 + 3.99) / 3 = 11.99 / 3 = 4

iii) 4 / 10 = 0.40

2. Water Speed Measurement

a) First, measure the distance between bridge X and bridge Y. Then, place the stick in the water at bridge X and time how long it takes to reach bridge Y. Calculate the speed by dividing the distance by the time. Repeat this process at least three times. Add all the results together and divide by three to get the average speed.

b) 2nd box

3. Acceleration and Force

a) i) 24 / 60 = 0.4 m/s²

ii) 7.5 × 10⁵ × 0.4 = 300,000 N

b) Speeding up / Slowing down / Steady speed / At rest

4. Resultant Force and Density

a) i) 280 - 250 = 30 N

ii) Up

b)... Continue reading "Physics Exam Answers and Explanations" »

Mean Free Path in Kinetic Theory

Mean free path is the average distance traveled by a gas molecule between two successive collisions with other molecules.

• It is denoted by the Greek letter λ (lambda).
• It depends on the size of the molecules, pressure, and temperature.
• A higher pressure decreases the mean free path, while a higher temperature generally increases it.
• Formula:
  [ λ = \frac{1}{\sqrt{2}\pi d^2 n} ]
  where d represents the diameter of a molecule and n represents the number of molecules per unit volume.

Gibbs Free Energy and Thermodynamic Stability

Gibbs free energy (G) is the thermodynamic function that indicates the maximum useful work obtainable from a system at constant temperature and pressure.

• It is defined by the equation:
  [ G = H - TS

Energy: Units and Types

• Unit of energy: Joule (J)
• Conservation of Energy: Energy cannot be created or destroyed, only transferred or transformed.
• Kinetic Energy: Energy of motion.
• Potential Energy: Stored due to position (e.g., at the top of a pendulum).

Conduction

• Occurs mainly in solids.
• Particles transfer energy by vibrating and passing it to neighbors.
• Metals are good conductors.
• Poor conductors (insulators), e.g., air and layers of clothing.

Convection

• Happens only in fluids (liquids and gases).
• Heated fluid becomes less dense → rises.
• Cooler fluid becomes more dense → sinks.
• Creates convection currents.
• Example: Hot water in a pan rises to the top.

Radiation

• Transfer by infra-red waves.
• Only method that works in a vacuum (space).
• Best absorber/emitter:

Core Thermodynamic Principles

• Calorimetry: Measures heat transfer in a process.
• Absolute Zero: Lowest temperature; no molecular motion (0 K).
• Clausius Statement: Heat does not flow from cold to hot by itself.
• Enthalpy: Total heat content (H = U + PV).
• Entropy: Measure of disorder; increases in natural processes.
• Diffusion: Particles move from high to low concentration.
• Efficiency: Useful output divided by energy input.
• Gibbs Free Energy: Energy available to do work (G = H - TS).
• Internal Energy: Energy stored inside a system.
• Heat Capacity: Heat needed to raise an object's temperature by 1°C.
• Latent Heat: Heat for phase change without temperature change.
• Quasistatic Process: Very slow, system stays in equilibrium.
• Reversible Process: Can go backward with

Chapter 29: The Magnetic Field

Magnetic Poles and Materials: Like poles repel (+ +), opposite poles attract (+ -). Paramagnetic materials are magnetized only in the presence of a strong external magnetic field. Magnetic flux flows from north to south.

Gauss's Law for Magnetism

The magnetic flux through any closed surface is always zero. There is no way to isolate a north or south magnetic pole (no magnetic monopoles).

The most elementary electric field is from a point charge (a static charge), while the most elementary magnetic flux density is from a magnetic dipole (a small magnet or magnetic bar).

The induced magnetic dipole always has an opposite pole facing the solenoid.

Solenoid:

An electric charge experiences a magnetic force in a magnetic field,... Continue reading "Physics Formulas and Concepts: Magnetism, Induction, and Optics" »

1. Ohm’s Law (V–I Relationship)

Aim

To verify Ohm’s Law and determine the resistance of a metallic conductor.

Apparatus

Battery, ammeter, voltmeter, rheostat, key, and resistance wire.

Theory

Ohm’s Law states that at a constant temperature, the current is directly proportional to the potential difference: V = IR.

Procedure

• Connect the circuit (ammeter in series, voltmeter in parallel).
• Adjust the current using the rheostat.
• Record the V and I readings.
• Plot a graph of V versus I.

Observation

A straight-line graph passing through the origin.

Result

Ohm’s Law is verified. Resistance R = V/I (slope of the graph).

2. Parallel Combination of Resistances

Aim

To verify the law of parallel combination using a meter bridge.

Theory

1/Rp = 1/R1 + 1/R2

Result

The... Continue reading "Physics Lab Experiments and Activities Manual" »