Physics: study of the physical world

Apple falls under gravity: a simple physics problem

Physics: study of the physical world

Rock climbing: a physics problem

gravity

friction

energy

Force

Physics: study of the physical world

Predicting Global Warming:

A complicated physics problem

Physics: study of the physical world

Nano-technology: physicists do it

Physics: study of the physical world

Origin of the universe: a physics problem

Main Branches of Physics

Mechanics

Electromagnetics

Thermodynamics

Quantum Mechanics

Relativity

Nuclear Physics

Grand Unified Theory?

Rigid body mechanicsFluid mechanics

Light & opticsElectrical engineering

Atomic & molecular physicsNanotechnology

Astrophysics

Statistical mechanics

Problem Solving

Outline of a useful problem-solving strategy

•can be used for most types of physics problems

•Can also be used for many types of problems in life

Important!

Math BasicsScalars & Vectors

Dimensions & UnitsGeometry & Trigonometry

Aϑ

f(x)

Scalars and Vectors

Scalar: magnitude only

e.g.: 30 cookies

Vector: magnitude and direction

e.g.: 45 Newtons of force, upwards

In physics, various quantities are either scalars or vectors.

Distance: Scalar Quantity

Distance is the path length traveled from one location to another. It will vary depending on the path.

Distance is a scalar quantity – it is described only by a magnitude.

Speed: Scalar Quantity

Speed is distance ÷ time

Since distance is a scalar, speed is also a scalar (and so is time)

Instantaneous speed is the speed measured over a very short time span. This is what a speedometer in a car reads.

Average speed is distance ÷ some larger time interval

Distance and Speed: Scalar Quantities

Average speed is the distance traveled divided by the elapsed time:

• A bar over something usually denotes the average, also sometimes used: < S >

• A Δ (Greek: delta) is usually used to indicate a difference or interval (Δt = t2-t1)

Displacement: a Vector

Displacement is a vector that points from the initial position to the final position of an object.

Vectors & dimensions

A vector has both magnitude and direction

•In one dimension, a vector has one component.

•In two dimensions, a vector has two components.

•In three dimensions, vectors have three components…

A vector is usually drawn as an arrow.It is often symbolized with a small arrow over (or sometimes under) a symbol:

A

Addition of vectors

Graphical addition

Vector Quantities: Velocity

Note that an object’s position coordinate may be negative, while its velocity may be positive – the two are independent.

Velocity is a vector that points in the direction that an object is moving in

2-D Geometry review

Curves of functions: y = f(x)

f(x)

xSlope of curve Δy/Δx

2-D Geometry review

Linear functions: slope is always the same, constant

General functions: slope depends on position

Vector Quantities

Different ways of visualizing uniform velocity:

Vector Quantities

This object’s velocity is not uniform.

Does it ever change direction, or is it just slowing down and speeding up?

Visualizing non-uniform velocity:

More 2-D Geometry

Pythagorean Theorem:

a2 + b2 = h2

More 2-D Geometry

Angles are in units of degrees or radians(on calculator, be sure to know which is used!) How to convert?

Ψ

Angles

More 2-D Geometry

Ψ

Trigonometry:

More 2-D Geometry

Sin and Cos are always < 1

More 2-D Geometry

Properties of sine and cosine

Use:•All periodic things•Angles and directions

One last mathematical tidbit

Quadratic Formula: memorize it

If faced with an equation where the unknown variable is squared, re-arrange things to look like this:

Then x is given by:

(There are two possible solutions)

Vectors in 2-DVectors have components

The magnitude of a vector and the direction of a vector are related to the components

Use trigonometry and Pythagoras

Ax= A cos()

Ay= A sin()

Manipulating Vectors in 2-D

Adding things in one dimension is easy:

3 Apples + 2 Apples = 5 Apples

But in two (or more) dimensions: we add the components: if we have a vector {x Apples, y Oranges}

{2 Apples, 3 Oranges} + {5 Apples, 2 Oranges}

= (2+5) Apples, (3+2) Oranges

= 7 Apples, 5 Oranges

Vector Components Review

If you know A and B, here is how to find C:

The components of C are given by:

And

Vector components Review

Vector Addition and Subtraction

Vectors are resolved into components and the components added separately; then recombined to find the resultant vector.

Example

Addition of vectors: adding components

So what’s length of R, and direction of R?

Example

Addition of vectors: adding components