Vectors, components, sine and cosine in power series

Back in algebra II or so you probably started going over vectors.

To start off, you could describe a vector using an angle (like that on a compass) and a magnitude or as two components (i and j).

Later on, in three dimensions you could use i, j, and k components to describe a vector.

In earlier grades, teachers taught cosines and sines only in their relations to triangles, but it turns out that the sine and cosine functions can describe any function (this is related to power series).

Furthermore, a combination of sines and cosines can act as the basis of n-dimensional space.

Thus they prove to be useful far beyond describing triangles.

Tranverse and Longitudinal Waves

From qwerty_gauri on flickr

From qwerty_gauri on flickr

Waves exist in two forms– transverse (like an ocean wave) and longitudinal (like sound waves).

sound wave

sound wave

A transverse wave has a displacement perpendicular to the medium while the displacement of the medium in a longitudinal wave is parallel to the medium.

Thus an ocean wave goes up and down while sound waves compress the air in a parallel way towards the target.

Waves can be periodic, similar to simple harmonic motion, or a single pulse, similar to a pulse moving along a slinky.

And if someone asks you what’s nu, you can say v/lambda.  (physics joke)

Physics of White Water Rafting Bernoulli Pressure

If you go ever go white water rafting you’ll start out in a nice calm area, kind of like the happy singing place on Splash Mountain in Disneyland.

The river is wide at this point and the velocity of the water is fairly low.

Further down the river though the river will narrow and rocks will be there.

According to Bernoulli, p1 + (1/2)ρv12 = p2 + (1/2)ρv22

Notation:

p is pressure
ρ (rho) is density

and the narrowness will therefore increase the velocity to allow the same amount of water to flow in the same time it takes at the calmer points in the river.

More water has to go through a smaller area so it will need to go faster.

Buoyancy Continued Ice, Salt Water, Pressure

When an object is immersed within a fluid, for example air or water, the volume of the fluid displaced will contribute to an upward force.

The amount of fluid displaced will cause a force of equal magnitude to the weight of that fluid to push upwards.

Ice is about 91.7% the weight of water so therefore 8.3% of a piece of ice would be above water in the case of pure water.

Salt water is a bit more dense and therefore more like 10% of the ice would be above water.

The difference in pressure between the top and bottom of an object makes this movement possible.

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).