physics midterm

quantitative science

quantity (measurable)

qualitative science

descriptive (not measurable)

examples of quantities

height

weight

age

examples of qualities

complexion

nature

physical quantity

(precise measurements are possible)

eg. height/age

abstract quantity

(estimate is possible)

eg. rate your pain 1-10

quantification

using numbers and units to describe a vlue

unit

standard measurement

number

amount of times a unit is contained

(56 y/o. = 1 yr 56 times)

extensive quantity

size-dependent

eg. length, mass, time

intensive quantity

not size-dependent

eg. volume, density

intrinsic quantity

not dependent on amount of material present

eg. melting/boiling point

extrinsic quantity

dependent on amount of material present

eg. mass, volume

scalar

no direction

vector

includes built in direction

geometrical dimension

size

eg. 1D, 2D, 3D

physical dimension

composition

principle of homgenity

in any valid physical equation, the dimensions of both sides must be the same

flaw of dimensional analysis

cant detect constants

cant detect scalar/vector quatities

angular displacement

s

angular velocity

w

angular acceleration

a

force

experience of the region

built in sense of direction

vector

field

region

vector mechanics

study of the effect of forces

distance

total units travelled

displacement

how much changed from the starting point

vector algebra (x) consists of what

vector addition

vector multiplication

vector calculus (dx) consists of what

vector differentiation

vector integration

vector multiplication has what qualities

uses scalar/number value

multiplied by 2 or more vectors

types of products used

dot product

cross product

methods of representing a vector

symbolic (analytical) and picture (graphical)

vector symbol

A with an arrow above it

magnitude symbol

|A| or just A

5M is a what quantity

scalar

5M N is a what quantity

vector

free vectors

can be moved parallel to itself with no direction or change

co-initial vector

same initial point

concurrent vector

same ending (terminal) point

parallel vectors

\----→

\-→

equal vector

\----→

\----→

negative vector

\-----→

anti-parallel vector

\---→

coplanar vector

on the same plane

resultant

two or more vectors added together

law of triangle

if you have two vectors, you can find their sum by creating a triangle with the two vectors as two sides of the triangle. The third side of the triangle represents the sum of the two vectors.

law of the parallelogram

a graphical method used to add two vectors. To use this method, you draw the two vectors with their tails at the same point and form a _______ by drawing lines from their heads. The vector sum of the two vectors is represented by the diagonal of the _____ passing through the common tail of the vectors.

vector A multiplied by a positive number

n \* A, same direction as A

vector A multiplied by a negative number

n \* A, opposite direction to A

vector A multiplied by 0

null vector, any direction

vector A multiplied by 1

A, same direction (equal vector)

vector A multiplied by -1

\-A, opposite direction (negative vector)

unit vector

assigns direction to a number

dimensionless entity

n̂

formula for unit vector

vector/magnitude

x, y, z axes are represented by?

i, j, k

rectangular components

like breaking a vector into parts that go up-and-down and side-to-side. instead of thinking about the whole arrow, we can just look at how much of it goes up-and-down and how much goes side-to-side.

vector formula

Ax i + Ay j + Az k

vector magnitude formula

sqrt(Ax^2 + Ay^2 + Az^2)

Ax = ? (in terms of trig and θ)

in a 2D vector

Acos(θ)

Ay = ? (in terms of trig and θ)

in a 2D vector

Asin(θ)

how to calculate θ?

θ = tan^-1(Ay/Ax)

in a 3D vector, vector A is composed of what 3 angles?

α, β, γ

in a 3D vector, vector A is composed of what 3 rectangular components?

Ax i, Ay j, Az k

cos(α) = ?

Ax/A

cos(β) = ?

Ay/A

cos(γ) = ?

Az/A

calculate the angles with this info

Ax = 5, Ay = -3, Az = 1

cos(α) = 32.3

cos(β) = 120.5

cos(γ) = 80.3

multiplication is basically what?

a special case of addition

**combined effect**

dot product concept

vector A • Vector B → cos

cross product concept

vector A x vector B → sin

dot product definition

vector A • Vector B = AB(cos(θ))

dot product physical idea

push/flux

dot product nature

scalar quantity

dot product algebraic calculation

AxBx + AyBy + AzBz

dot product geometrical idea

vector A • Vector B = A \* (shadow of b on vector A)

vector A • Vector B = ?

AB cos(θ)

vector B • Vector A = ?

BA cos(θ)

solve for cos(θ)

AB cos(θ) = AxBx + AyBy + AzBz

cos(θ) = (AxBx + AyBy + AzBz)/(AB)

cross product nature

vector quantity

cross product analytical idea

vector A x Vector B = AB sin(θ)\*n̂

cross product physical idea

rotational

geometrical idea of cross product

△ = 1/2(AB(sin(θ)))

vector area

regarded as an area perpendicular to a given area

angle between vector A x vector B

180 degrees

magnitude of vector A x vector B

AB sin(θ)

3 ways to generate a null vector?

zero \* vector A

vector A + (-) vector A

vector A x vector A

review how to calculate cross product with 2nd and 3rd order matrices

ok, i’ve reviewed how to calculate cross product with 2nd and 3rd order matrices

calculate the cross product

vector A = 2i - j

vector B = 3i + 4k

(5sqrt(5)/2) k

vector calculus

division by number/scalar

not division by another vector

small changes

why cant we divide a vector by a vector?

cant divide by directions

independent variable examples in vector calculus

time

mass

dependent variable examples in vector calculus

displacement

velocity

acceleration

displacement formula with derivative

dr

velocity formula with derivative

dr/dt

acceleration formula with derivative

dv/dt

position vector

the position of a point in space relative to an origin

xi + yj + zk = resultant vector

parametric equations

x = x(t)

y = y(t)

z = z(t)

given

r = (3t^2)i + (5t-1)j + (4t^3-2t^2+1)k

solve for position, velocity, and acceleration and the position angles at t=0

ok ive solved for position, velocity, and acceleration and the position angles at t=0

a field is ?

a region in which a physical quantity has different calues at different points

if a fields quantity is scalar?

the field is scalar