Buoyancy and Volume Displacement

Objects that float have something known as buoyancy which is an upward force that exists when the specific object has less density than the surrounding medium.

The mass of the object is important, but the displaced volume is what affects upward force.

Thus a cement canoe can float and a wooden boat can sink- if filled with water– since the effective volume decreases.

Also a ball of clay could be formed into an object with enough volume to float, but Prof. Kesten didn’t quite pull it off in class today while the cap did float- just the other guy and I predicted.

Pressure, equilibrium, and energy conservation

When pressure is exerted it will act and then resolve to equilibrium quickly.

If you squeeze toothpaste it will come out the other end and then stop moving.

Pressure coming from a tire jack can have a very large force, but energy is conserved, so it will be across a small distance.

Fluids and Fluid Properties

Fluids behave and interact with forces differently than solids might.

A fluid cannot sustain a shear force (a force parallel to the surface).

When you’re talking about a fluid it is also more useful to observe bulk properties, not the actions of a single molecule.

When water is going through a garden hose, a single molecule might travel at 8000 m/s, but it doesn’t really matter since the overall flow of the water might be at 4 m/s.

Earth Orbit, Comets and Seasons

The Earth travels around the sun in an elliptical pattern, but it is still fairly close to a circular orbit.

The seasons on Earth are determined by the angle of sunlight that hits the planet according to the axis of rotation of the planet, not the proximity of the planet to the Sun.

A comet might heat up when closer to the Sun on an ellipse with a high eccentricity (deviation from perfect circle), but the Earth’s orbit does not have a high eccentricity.

Also, if you know part of an elliptical path you can figure out the period and axes of the ellipse.

It can thus be calculated that Halley’s comet actually does not go out as far from the Sun as Pluto.

Gravity Potential Energy in Space

In space there is potential energy that is gravitational, similar in some ways to the potential energy associated with height on earth, but also different.

Two objects must be present and the potential energy increases with less distance between them, instead of more height like on earth.

The potential energy will be at a maximum absolute value when the radius between is smallest, but it is considered negative since the objects are moving toward each other.

They will also have the most kinetic energy right before they collide rather than zero kinetic energy when an object on earth hits the ground.

Gravity Drilling to the Center of the Earth

If you were to drill a hole through the center of the earth the result might seem a little counter intuitive.

Gravity decreases with distance, away from the earth, ie in outer space, and it also decreases as you travel toward the center of the earth.

The reason is that the force of gravity on an object will only come from the inner layers (relative to position of the object).

The volume of the sphere with radius of the distance from the center will affect you, not any outside layers, since the outside layers will cancel forces.

The part of the outside layer near to you will be closer with less mass and the part farther away will have more mass (proportional inversely to square of distance).

Newton Derives Universal Gravitation Constant and Mass of Earth

Skeptics did not believe Newton’s theories of gravitation to be true, but his figures proved valid.

The radius of the Earth has been known since ancient times and the distance to the moon has also been calculated.

Therefore Newton could use these numbers in a clever way to calculate both the universal gravitation constant and the mass of the earth.

When he plugged in numbers it showed that they resulted in the period of the moon being 1/13 of a year, which is accurate.

In a similar way you can often times manipulate multiple variables using known data (like using the calculus technique of summing shells to obtain a volume) to obtain solutions.

Natural Frequency Introduction

All objects that oscillate have something called a natural frequency.

At this frequency they will experience resonance and start oscillating with increasing magnitude towards infinity, or in the case of something like a bridge until it breaks.

In another instance, a snare drum can be sitting motionless, but if a saxophone player near it plays a certain tone that happens to match its natural frequency (or a multiple of it) then the snare drum will start vibrating.

Physics Mathematics Notation Euler Sine Cosine

When writing mathematical expressions, numbers and formulas that look quite different can mean the same thing.

For instance, power series (an infinite series of terms added together) can be used to represent things like the natural log (ln).

That is actually how devices such as calculators compute numbers.

Likewise e with imaginary exponents is used as part of an expressions by Euler for sine and cosine.

Pendulum Motion and Simple Harmonic Approximation

Technically the movement of a pendulum doesn’t quite match simple harmonic motion.

However, if the angle theta is small enough,  θ~sin(θ)

In fact for angles less than 15 degrees the error will be less than one percent.

Thus a simple pendulum essentially displays SHM (simple harmonic motion).

Such approximations are very useful in certain applications, especially in engineering, where such a small amount of error can be tolerable.