exam two

seven steps of problem presentation method (PPM)

given, find, representation, model, assumptions, solution, validation

3 common examples of assumptions for PPM

1. frictionless / no drag / no air resistance

2. massless strings and wheels

3. modeling objects as particles

brainstorming

dynamic problem-solving process that builds upon spontaneous ideas generated by a group with the purpose of ideation (idea generation)

what are 4 sources of brainstorming

nature, prior art, random input, and the problem itself

what are 4 methods of brainstorming

plain brainstorming, random entry sheet, brainwriting, and ‘the worst method'‘

what are the 4 rules for quality brainstorming

focus on quantity, withhold criticism, welcome unusual ideas, combine and improve ideas

decision matrix (QFD: Quality Function Deployment)

tool to provide evaluation of completing needs, generative and unbiased, weighing importance factor based on performance data and normalizing functions (nice to have=5, good to have=10, must have=15)

normalizing function equation

(1-) (performance data value)/(upper bound on performance)

classic mechanics

describes how engineering systems behave at the human scale

particle model

set of assumptions that simplifies analysis of engineering systems in motion (point masses, forces, linear momentum)

kinematics (of a particle)

geometry/shape of motion in space under influence of forces

Newton’s first law of motion

law of inertia: if ΣF=0, then there is no change in motion of object (static equilibrium)

Newton’s second law of motion

law of force: if ΣF≠0, then ΣF=ma

Newton’s third law of motion

law of action+reaction: F_{A on B}=-F_{A on B} (equal but opposite)

what four components do free body diagrams (FBDs) contain

object chosen to analyze, all external forces acting on object (normal, friction, applied, etc.), any geometric info, set of coordinate axes

6 steps of solving force problems

1. Draw a picture

2. Construct a free body diagram for all bodies

3. Define axes on each FBD

4. Define knowns and unknowns

5. Set up sum of forces in x and y directions

6. Solve for unknowns; use kinematic equations to solve for position and velocity

spring/elastic forces

stretching/compressing of certain types of objects (usually metal)

direction and forces properties of spring forces

direction is always parallel to spring and opposite of change in length, forces tensile or compressive (depending on deform)

two equations of magnitude properties of spring forces (Δl and |F_spring|)

• Δl = |current length - L₀|

• aka change in length = |current length - unstretched length|

• |F_spring| = k * Δl (Hooke’s law)

• aka magnitude of spring force = spring constant (elastic stiffness) * change in length

friction forces

forces that arise due to contact between rough surfaces (static vs. kinetic)

static friction equation

|f|=|F|=μ_s|N| because the block is in static equilibrium (f and F have opposite directions)

kinetic/sliding friction equation

|f|=μ_k|F| (direction of f is opposite that of motion, |f| is constant but <|F|)

static/kinetic friction coefficient

constant value in which μ_s>μ_k

ê_r

time-dependent unit vector that points in radially outward direction from origin (in polar coordinates, points at particle)

ê_θ

time-dependent unit vector that points perpendicular to ê_r in direction of increasing angle

r(t) equation in polar coordinates

r(t) ê_r(t) + z(t) k̂

v(t) equation in polar coordinates

ṙ(t) ê_r(t) + r(t)ɵ̇(t) ê_θ(t) + ż(t) k̂

a(t) equation in polar coordinates

{r ̈(t)-r(t)ɵ̇²} ê_r(t) + {r(t)Ɵ̈(t)+2ṙ(t)ɵ̇(t)} ê_θ(t) + z ̈(t) k̂

centripetal acceleration

-r(t)ɵ̇²

Euler acceleration

r(t)Ɵ̈(t)

Coriolis acceleration

2ṙ(t)ɵ̇(t)

extensive properties (4 examples)

scalar physical properties for which individual property values can be added together to get a total property value (ex: mass, volume, energy, electric charge)

3 examples of physical quantities that are not extensive

temperature, density, color

universal accounting equation

final amount - initial amount = input - output + generation - consumption (tracks changes in extensive quantities within a system with a defined boundary)

final amount

amount of quantity in system at the end of the time period

initial amount

amount of quantity in system at the end of the time period

input

amount of quantity passing through the boundary into the system during time period

output

amount of quantity passing through the boundary out of the system during time period

generation

amount of quantity produced during the time period within the system boundary

consumption

amount of quantity lost during the time period within the system boundary

3 major considerations when using the universal accounting equation

known boundary, start and end time, accounting on extensive properties

state quantity

physical property with change in start to end (displacement)

path quantity

physical property with change in start to end AND process details (distance)

universal accounting equation system representation diagram

initial and final inside box, generated in from top, consumed out from bottom, input in from left, output out from left

work

applying a force to a particle in a way that causes the particle to move through some distance, only the portion of the force parallel to the motion can do work

energy

what a system has if it has the ability to do work on a particle

what is work done by a net force equal to

change in kinetic energy

work equation with constant force F causing displacement d (work energy theorem)

W = F ⋅ d = |F| * |d| * cosθ, d = Δr = r_final - r_initial

work equation with nonconstant force F causing particle to move

W = (from r_final to r_initial) ∫(F ⋅ d*r)

conservative force characteristics

path-independent state quantity, only the initial and final states matter (ex: gravity force, elastic/spring force, electric force)

conservative force equation

W_C=PE_initial-PE_final

non-conservative force characteristics

path quantity, distance travelled relates to the amount of work (ex: friction force, tension force, push/pull)

non-conservative force (general) equation

W_NC=∑E_final-∑E_initial

potential energy equation for gravity work

W_grav=mgH

potential energy equation for spring work

W_spring=(1/2)kx²

kinetic energy equation for work

W = F ⋅ Δx = (1/2)m|v|²

what is the simplified equation for work done by friction force

−μmg(*1 or *cosθ or *sinθ)*Δx

if friction work is done on the system, is the force an input or output

input

if friction work is done by the system, is the force an input or output

output