# Physics 152:  Lecture Notes, Flashlets, etc.

Michael Fowler, U.Va.

### Gravity: Lecture Notes

Discovering Gravity: Galileo, Newton, Kepler   PDF

Galileo analyzes a cannonball’s trajectory, Newton imagines the cannon on a very high mountain shooting the cannonball into orbit, and sees the analogy to the Moon’s motion, which leads him to conjecture that the gravitational force extends to the moon and beyond, with strength proportional to the inverse-square of the distance. Analyzing Kepler's Laws of planetary motion indicates that a similar gravitational force keeps the planets in their orbit, suggesting a Universal Law of Gravitation. We give an (optional) calculus-based proof that the planets’ orbits are in fact ellipses.

Finding the gravitational attraction form a single mass, a pair of masses, a ring, a hollow sphere and finally a solid sphere, both inside and out.  How does gravity change on going down a mine?

Working with Gravity: Potential Energy  PDF

How potential energy relates to the gravitational field, near the earth’s surface and far away.  Potential energy and escape velocity.  Potential and kinetic energies in circular orbits.

Elliptic Orbits: Paths to the Planets  PDF

The interesting orbits are ellipses, or sequences of pieces of ellipses.  Some simple properties of the ellipse make it possible to understand these orbits well. We briefly discuss other (hyperbolic) orbits, and also the important role of the slingshot in actually reaching the outer planets.

Binary Stars and Tides  PDF

More gravitational phenomena: pairs of stars orbiting a common center; how a close gravitational source can distort a planet.

General Relativity

The Principle of Equivalence: a uniformly accelerating frame of reference is equivalent to a gravitational field.  How it necessarily follows that a gravitational field deflects light, and that a clock on the surface of a big planet runs slow.

### Gravity: Flashlets, etc.

Kepler’s Laws Flashlet

Create your own planetary orbit with the click of a mouse, and see Kepler’s Laws in action.

Shoot Mars Flashlet

How good is your aim at getting a spaceship to Mars?

This Excel spreadsheet will calculate planetary orbits over a wide range of initial conditions, and will work for gravitational forces that are not inverse square, producing some strange looking orbits.  Convenient numbers, such as GM = 8, correspond to mini solar systems with one-kilogram planets orbiting stars weighing only a hundred million tons or so, but the geometry of the orbits doesn’t depend on the scale, so we can gain intuition about real planetary systems.

### Fluids

Hydrostatics

A rather extensive historical lecture, originally for my Galileo and Einstein class, but you might find it interesting. I will not, however, expect you to know historical facts on the exams—just understand the physics!

Boyle’s Law   PDF

Some quotes from Boyle describing his discovery of the famous Law, followed by a derivation of the Law of Atmospheres—just how the air thins out at high altitudes.

The Bernoulli Effect  PDF

Viscosity   PDF

What is viscosity?  Exploring fluid friction at the molecular level, and explaining some surprising results—for example, why the viscosity of a gas is independent of its density.

Using Dimensions   PDF

M, L and T: all physics equations must have the same dimensions on both sides. How to make interesting physics predictions without doing much math.

Stokes’ Law   PDF

Dropping a ball through very viscous fluid. A dimensional prediction of the dependence of speed on radius. An experiment with glycerin.

Inertial Drag Force and the Reynolds Number   PDF

Another experiment, this time dropping coffee filters through air, with a very different result. The Reynolds Number:  the dimensionless ratio of inertial to viscous drag.

# Some Essential Math

The Exponential Function   PDF

Definition of the number e.  The series expansion of ex.   Solving the differential equation dy/dx = ay.   The Natural Logarithm.

Complex Numbers   PDF

Real numbers as vectors.  The square root of -1.   Polar Coordinates in the plane.  The  significance of the formula eiq = cosq + isinq

Some Exponential Integrals  PDF

An elementary discussion of various exponential integrals that arise in understanding Maxwell’s velocity distribution in the kinetic theory of gases.

### Oscillations

From Complex Numbers to the Simple Harmonic Oscillator  PDF

SHM as the Shadow of Circular Motion

A very simple spreadsheet animating the process. Press and hold the end of thescrollbar.

Crazy Way to Demonstrate Simple Harmonic Motion

From Vladimir Vasc, in the Czech Republic.

Damped and/or Driven Oscillators   PDF

You can safely skip the * sections: they treat the differential equations in more detail than we need at this point.

This spreadsheet plots the motion of a damped oscillator: you can specify the initial conditions and degree of damping, and find critical damping.

An external driving force is added to the previous spreadsheet: see how damping and resonance compete.

### Waves: Lecture Notes

Kinds of waves in one dimension: transverse, longitudinal, traveling, standing.

Analyzing Waves     PDF

How applying F = ma to a little piece of string leads to an equation that describes many different waves.

Sound Waves     PDF

Finding these forces and applying Newton’s Law to a small piece of  string is the way to the Wave Equation.  Flashlet by Patrick LeDuc.

Interference from Two Sources: Young’s Two Slit Experiment

Flashlet for The Doppler Effect

Great Website for Sound and Musical Instruments

Some Applets

Animated Spreadsheet for Two Traveling Waves

Building a Triangular Wave: Fourier Series

Spreadsheet for Beats, Group Velocity and Phase Velocity

### Heat

Heat and Temperature   PDF

Development of the concepts, some early applications, the Zeroth Law of Thermodynamics, calorimetry.

Thermal Expansion and the Gas Law  PDF

Linear and volume expansion coefficients, pressure versus temperature for a gas, the Kelvin scale, the gas law.

Early Attempts to Understand Heat: The Caloric Fluid Theory, and Rumford’s Experiment   PDF

Lavoisier suggests heat flow is a conserved fluid, like electricity.  Carnot analyzes the steam engine using this idea, and reaches many of the right conclusions.  Rumford manufactures cannons, and pours cold water on the caloric theory.

The Kinetic Theory of Gases    PDF

Ideal Gas Thermodynamics: Isotherms, Adiabats, Specific Heats   PDF

The Carnot Engine   PDF

Kinetic Theory Applets:  One Atom Gas,    Many Atom Gas,   Brownian MotionMaxwell Distribution.