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]
