(4)+Work,+Energy+&+Power

PAGE EDITORS: Michelle Cesarano, Matt Dughi, Will Meeks, Billy Frese

=__Energy__=

K = 1/2mv^2

 * Checkpoint: A particle moves along an //x// axis. Does the kinetic energy of the particle increase, decrease, or remain the same if the particle’s velocity changes (a) from −3 m/s to −2 m/s and (b) from −2 m/s to 2 m/s?
 * The equation for kinetic energy tells us that regardless of sign change, if the absolute value of velocity increases, kinetic energy increases, and vice versa.

__Spring Potential Energy__ (Usp) - The energy stored in a spring based on how much it has been either stretched or compressed from its equilibrium length
Usp = 1/2kx^2 __Mechanical Energy__ (ME) - Sum of the kinetic and potential energies of a system ME K + Ug + Usp. Mechanical energy cannot be lost without an outside force acting on the system, as represented in the equation Wnc= ΔME
 * Energy may transfer from one type to another, but energy is always conserved in our universe (principle of energy conservation)
 * Energy is measured in Joules, or kg*m^2/s^2

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=**__Work (W)__**=

__Conservative v. Nonconservative__
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__Conservative Forces__ - forces that are path independent --> relate to potential energy
ex: gravity, spring force (where the spring force is always toward the equilibrium position)

= =

__Nonconservative Forces__ - forces that are path dependent
ex: friction, applied forces, tension, air resistance, and just about everything else If a constant force is applied over a very small displacement dW = F dr r = x **i** + y **j** + z **k** If F is constant W = F **Δ r**

How does work done by a conservative force relate to a change in potential energy? Δ **U = -Wc** ex: What is the change in potential energy is I lift an apple from a height h1 to height h2? Δ Ug = - Wg Δ Ug = -∫ Fg dr (with limits h1 to h2) Δ Ug = - ∫ (-mg **y**)(dy **y**) Δ Ug = - [- mg(h2 - h1)] Δ Ug = mgh2 - mgh1 Wnet = Wnonconservative + Wconservative = Δ K - Wc = Δ Uc Wnc - <span style="font-family: sans-serif,helvetica,sans-serif; font-size: 11px; line-height: 17px;">Δ Uc = <span style="font-family: sans-serif,helvetica,sans-serif; font-size: 11px; line-height: 17px;">Δ K W = <span style="font-family: sans-serif,helvetica,sans-serif; font-size: 11px; line-height: 17px;">Δ K + <span style="font-family: sans-serif,helvetica,sans-serif; font-size: 11px; line-height: 17px;">Δ Uc

ME = K + U
Wnc = <span style="font-family: sans-serif,helvetica,sans-serif; font-size: 11px; line-height: 17px;">Δ<span style="font-family: arial,helvetica,sans-serif; font-size: 13px; line-height: 19px;">ME <span style="font-family: arial,helvetica,sans-serif; font-size: 13px; font-weight: normal; line-height: 19px;">**If no NC forces do any work on system:** <span style="font-family: arial,helvetica,sans-serif; font-size: 13px; line-height: 19px;">0 = Δ<span style="font-family: arial,helvetica,sans-serif; font-size: 13px; font-weight: normal; line-height: 19px;">ME = ME - MEo MEo = ME Ko + Uo = K + U <span style="font-family: arial,helvetica,sans-serif; font-size: 13px; font-weight: normal; line-height: 19px;">**If there are NC forces doing work on system:** Wnc = ME - MEo Wnc = (K + U) - (Ko + Uo) Wnc = <span style="font-family: sans-serif,helvetica,sans-serif; font-size: 11px; line-height: 17px;">Δ K + <span style="font-family: sans-serif,helvetica,sans-serif; font-size: 11px; line-height: 17px;">Δ U

The rate at which work is done or energy is used or transferred
energy/time = J/S = Watt (W) P = W / <span style="font-family: sans-serif,helvetica,sans-serif; font-size: 11px; line-height: 17px;">Δ t P = dW/dt = Fdxcos θ/dt = Fvcosθ (when F is a constant force) The relationship between force and potential energy in absence of nonconservative forces dW = F dr Wc = - <span style="font-family: sans-serif,helvetica,sans-serif; font-size: 11px; line-height: 17px;">Δ U dW = - dU F dx = - dU Fc = dUc/dx
 * Instantaneous Power:**

In graph of U v x, the slope is the force ME = K + U K = ME - U An object cannot move positions where U(x) > ME so it would have to stop (object is "bound")


 * <span style="font-family: Verdana,Helvetica,Arial,sans-serif; font-size: 12px; line-height: normal;">Checkpoint: A block moves with uniform circular motion because a cord tied to the block is anchored at the center of a circle. Is the power due to the force on the block from the cord positive, negative, or zero?
 * The power due to the force from the cord is zero, because the velocity of the block is perpendicular to the force.