# Self-Assessment Solutions

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### Self-Assessment Solutions

#### Rotational Dynamics

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## Chapter A. List of Formulas

Symbols

Chapter 1, "Points and Lines"

Chapter 2, "Geometry Snippets"

Chapter 3, "Trigonometry Snippets"

Chapter 4, "Vector Operations"

Chapter 5, "Matrix Operations"

Chapter 6, "Transformations"

Chapter 7, "Unit Conversions"

Chapter 8, "Motion in One Dimension"

Chapter 9, "Derivative Approach to Motion in One Dimension"

Chapter 10, "Motion in Two and Three Dimensions"

Chapter 11, "Newton's Laws"

Chapter 12, "Energy"

Chapter 13, "Momentum and Collisions"

Chapter 14, "Rotational Motion"

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### Symbols

Symbol

Name

Description

a

Alpha

Often used to represent an angle

D

Delta

Means "change in"

e

Epsilon

Coefficient of restitution

f '( t )

The derivative of function f with respect to t

m

Mu

Coefficient of friction

w

Omega

Used here to represent angular velocity

p

Pi

Constant 3.14

q

Theta

Often used to represent an angle

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### Chapter 1, "Points and Lines"

#### Equation of a Line

The graph of an equation of the form A x + B y = C, where A and B are not both 0, is a straight line.

#### Slope

For any line in standard form, A x + B y = C, the slope m = “A/B.

#### Parallel Lines

If two lines are parallel, their slopes must be equal.

#### Perpendicular Lines

If two lines are perpendicular,

#### Equation of a Line

Slope-intercept form: y = mx + b

Point-slope form: ( y y 1 ) = m( x x 1 )

where ( x 1 , y 1 ) is a point on the line.

#### System of Linear Equations

A system of two linear equations in the same plane has

• Exactly one solution if the two graphs have different slopes.

• An infinite set of solutions if both graphs have the same slope and y-intercept.

• No solution if the graphs have the same slope but different y-intercepts.

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### Chapter 2, "Geometry Snippets"

#### Pythagorean Theorem

In a right triangle, the square of the length c of the hypotenuse is equal to the sum of the squares of the lengths a and b of the other two sides: c 2 = a 2 + b 2 .

#### Distance Formula in 2D

where P 1 ( x 1 , y 1 ) and P 2 ( x 2 , y 2 ) are points on the line.

#### Distance Formula in 3D

where P 1 ( x 1 , y 1 , z 1 ) and P 2 ( x 2 , y 2 , z 2 ) are points on the line.

#### Midpoint Formula in 2D

is the midpoint between P 1 ( x 1 , y 1 ) and P 2 ( x 2 , y 2 ).

#### Midpoint Formula in 3D

is the midpoint between P 1 ( x 1 , y 1 , z 1 ) and P 2 ( x 2 , y 2 , z 2 ).

#### Parabola with a Vertical Axis

y = a ( x h ) 2 + k , with vertex ( h , k ) and axis of symmetry x = h .

#### Parabola with a Horizontal Axis

x = a ( y k ) 2 + h , with vertex ( h , k ) and axis of symmetry y = k .

#### Equation of a Circle

( x h ) 2 + ( y k ) 2 = r 2

where the center is ( h , k ) and the radius is r .

#### Equation of a Circle Centered at the Origin

x 2 + y 2 = r 2

where the center is (0,0) and the radius is r .

#### Equation of a Sphere

( x h ) 2 + ( y k ) 2 + ( z l ) 2 = r 2

where the center is ( h , k , l ) and the radius is r .

#### Equation of a Sphere Centered at the Origin

x 2 + y 2 + z 2 = r 2

where the center is (0,0,0) and the radius is r .

#### Circle-Circle Collision Detection

Given two circles and , if , a collision occurs.

#### Optimized Circle-Circle Collision Detection

Given two circles and , if ( h 2 h 1 ) 2 + ( k 2 k 1 ) 2 ( r 1 + r 2 ) 2 , a collision occurs.

#### Optimized Sphere-Sphere Collision Detection

Given two spheres and , if ( h 2 h 1 ) 2 + ( k 2 k 1 ) 2 + ( l 2 i 1 ) 2 ( r 1 + r 2 ) 2 , a collision occurs.

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