Work, Energy, Power

k

"spring constant", unit: N/m, different for all springs

ij notation (5,6) to (7,8)

(2i + 2j)m (REMEMBER THE UNIT)

Power formula

1) If F is constant: P = W/∆t = F*∆x/∆t = FxVx --> P = FxVx. 2) In 3 dimensions, good for instantaneous / average): P = F(->)v(->)

∆E types

1) Transfer: one object to another. 2) Change of type. 3) Both (throw baseball- chemical energy in me --> Kinetic energy in baseball)

Solving spring problems hints

1) identify i and f positions. 2) Often Wfric = 0 and Wp = 0 --> W = 0 --> Emechf = Emechi "Conservation of E". 3) Look for v=0, y=0, x=0 --> can crossout/ignore in equation. 4) x is not on the number line. x means ∆L for spring.

The dot product

A dot B = |A| * (the length of part B pointing in the direction of A). A dot B = A * B(||)

Work when F and θ vary

APPROXIMATE: (1) break path into sections at points where F and θ change. (2) Wf ≈ ∆W1 + ∆W2. (3) So, Wf ≈ F1x∆x + F1y∆y + F2x∆x + F2y∆y EXACT: (1) ∞ number of sections, ∆x-->0 and ∆y-->0. (2) Wf = lim(∆x-->0, ∆y-->0) (F1x∆x + F1y∆y + F2x∆x + F2y∆y + ...) --> Wf = lim(∆x-->0) ∑(xi,xf) Fx∆x + lim(∆y-->0) ∑(yi,yf) Fy∆y. THINK OF EACH AXIS SEPARATELY. --> Wf = S(xi,xf) Fx(x)dx + S (yi,yf) Fy(y)dy. (USEFUL). --> Wf = S(ri,rf)F dot dr.

Spring force direction

Always towards rest position. Stretched pulls and compressed pushes.

Work done by spring

Can't multiply F*x for spring because F isn't constant. Direction of Fspring is opposite to direction of x. Wspring = S(xi,xf) Fsdx = S(xi,xf) (-kx)dx --> [W = S(ri->,rf->) F-> dot dr->] --> Ws = 1/2kxi^2 - 1/2kxf^2 (involves position, but path independent; like Wg; K is a constant like g).

Conservative force

EQUALLY helps you if you go in one direction and hinders you if you go in the other direction.

Delta (∆)

Especially when summing, delta means "a little bit of" a whole. So, a portion of a large amount.

Work and Kinetic Energy, derive equation

F = ma --> vf^2 = vi^2 + 2a∆x --> vf^2 = vi^2 + 2(F/m)∆x --> (m(vf^2-vi^2))/2 = F∆x --> F∆x = m/2(vf^2-vi^2) --> F∆x = 1/2mvf^2 - 1/2mvi^2 --> W = kf - ki --> W = ∆K

Spring force

F is applied by the spring to an object. Fs. Fs = 0 when spring is at "Normal length". Compressed spring pushes, stretched spring pulls (spring tries to return to normal length). 99% of springs have m = 0, so Fs same at both ends.

Spring force model

Fs = (-) kx; b is small, points in opposite direction of outside force

Work sign

Helps (+) or hinders (-). Doesn't include L/R/Up/Down.

Work

How much a force helps or hinders a thing to move

W, K, and Wg

If one F is Fg, Wg + ∑Wother = ∆K --> mgyi - mgyf + ∑Wother = 1/2mvf^2 - 1/2mvi^2 --> ∑Wother = 1/2mvf^2 - 1/2mvi^2 + [mgyf - mgyi] --> ∑Wother = ∆K + ∆Ug --> ∑Wother = (1/2mvf^2 + mgyf) - (1/2mvi^2 - mgyi) --> Mech E = K + Ug + Us --> ∑Wother = ∆Emech (how much useful amount of energy has changed)

Power

J/sec. Can be W/∆t or ∆E/∆t or dE/dt. Is a rate (A "rate of change with respect to time"). Symbol: P. Unit: J/sec aka watt (W) and nonmetric hp (1hp = 746W)

Power unit

Joules/sec

Kinetic energy

K = 1/2 * m * v^2

different types of energy

K, Ug, Us, chem, thermal

Oscillating spring givens

Left end: x=-A, v=0, k=0, Us=1/2kx^2. Middle: x=0, k=1/2kx^2, Us=0, Fs=-kx=0=ma-->a=0, v is max. Right: x=A, v=0, k=0, Us=1/2kx^2. L and R only Eelastic. M only Ekinetic.

Mechanical Energy (With spring)

Mech E = (1/2*m*v^2) + (m*g*h) + (1/2*k*x^2)

Work unit

Newtons/meter aka Joule (J)

Work force diagram

Not a complete force diagram. Only shows the forces we're talking about.

Simple work cases

PARALLEL: A and B in same direction. θ = 0, cos(θ) = 1, A dot B = AB. ANTI PARALLEL: A and B in opposite direction, θ = 180, cos(θ) = -1, A dot B = -AB. PERPENDICULAR: A and B perpendicular, θ = 90, cos(θ) = 0, A dot B = 0.

S(xi->xf) f(x)dx means

SUM UP. Definite integral actually means: lim (∆x --> ∑(xi->xf) f(x)∆x

Hooke's law

Springs. Fs = (-) kx. The (-) reminds that the direction spring changes is opposite to the direction of its force.

Gravitational Potential Energy

Ug, UG = (1/2 * m * vf^2) - (1/2 * m * vi^2)

K

Unit: N/m. Bigger k --> stronger spring. Spring constant.

Elastic Potential Energy

Us = 1/2 * k * x^2

Total work and kinetic energy

W = (1/2*m*vf^2) - (1/2*m*vi^2) = kf - ki = ∆K

calculate work, 1 dimension with constant force

W = Fx*∆x, (Fx = Fcosθ)

Calculate work, most general definition

W = S(ri->, rf->) F->*dr->

Calculate work, 2 dimensions

W = S(xi,xf) Fxdx + S(yi,yf)Fydy

calculate work, 1 dimension

W = S(xi,xf) Fxdx. Fx = Fcosθ.

WORKother (with spring)

WORKother + WORKspring = [1/2*m*(vf^2) + m*g*(hf)] - [1/2*m*(vi^2) + m*g(hi)] --> WORKother = [1/2*m*(vf^2) + m*g*(hf) + 1/2*k*(xf^2)] - [1/2*m*(vi^2) + m*g(hi) + 1/2*k*(xi^2)]

WORKother (without spring)

WORKother = [1/2*m*(vf^2) + m*g*(hf)] - [1/2*m*(vi^2) + m*g(hi)]

Final WORKother equation

WORKother = [Mech Ef] - [Mech Ei]

WORKspring

WORKspring = -(1/2*k*xf^2 - 1/2*k*xi^2)

Work done by force

Wf = (Part of F in direction of ∆r) * (∆r) = (F)cos(θ)∆r. Fcosθ equals the displacement.

Solving spring problems, useful equation

Wfriction + Wpush = (1/2mvf^2 + mgyf + 1/2kxf^2) - (1/2mvi^2 + mgyi + 1/2kxi^2)

Work done by Gravity on a thing (Wg)

Wg = mg(yi-yf)

work vectors with ij notation

When given in ij form, A dot B = (Axi + Ayj)(Bxi + Byj) --> FOIL --> A dot B = AxBx + AyBy. (KEEP EACH AXIS SEPARATE)

Work with Force at an angle

Work = (F)cos(interior angle)(displacement)

Work always indicates

a change in energy (∆E) but not every ∆E involves work (ex. nuclear->electromagnetic)

x and y spring

don't mean xy-plane. X=0 is given to you. x is where spring can extend and compress. y=0 you can choose, where object on spring moves.

Spring A is stiffer than spring B. The "k" of spring A is _____ the "k" of spring B.

greater than

work with ij notation

i dot i =1; j dot j = 1; i dot j = 0 (REMEMBER COEFFICIENT) (KEEP EACH AXIS SEPARATE)

ij notation, sign of work determination factors

if i/j helps more than i/j hinders, (+). If i/j hinders more than i/j helps, (-). BECAUSE add works together, sign sensitive.

Spring x

not and never spring length. Is position of the end of the spring. Positive x --> negative Fs. Negative x --> positive Fs. Can graph Fs vs. x.

In Hooke's law, the "x" stands for

stretch

Energy

the amount of work that something could do

When dealing with springs, focus on

the position of the end of the spring

Fs vs x graph

x = stretch of compression. Nonlinear region, linear region (Fs = kx + b), plastic region, breaking point.

Work's sign depending on A and B angle

θ<90 (+), θ>90 (-), θ=90 (0)

∆r in three dimensions

∆r = ∆xi + ∆yj + ∆zk

∆r

∆xi + ∆yj

Mechanical energy without spring

∑W(other than Wg) = ∆Emech. Wfriction + Wpush = (1/2mvf^2 + mgyf) - (1/2mvi^2 + mgyi)

Final work equation

∑Wother = [1/2mvf^2 + mgyf + 1/2kxf^2] - [1/2mvi^2 + mgyi + 1/2kxi^2]