Work and Energy I

Previous: Friction

Work and Energy I is the sixth lecture in the Mechanics section of PH1011. It covers the relationship between work done and kinetic energy, the work-energy theorem, instantaneous power and definitions of energy.

Next: Work and Energy II

Motivation
The relationship of kinetic energy and gravitational energy can be found thus: from the derivative of v2 - dv2/dt = catch you later derivation

unfinished page

∫ The conservation of energy therefore states: Ekg = constant.

Points proven by the above derivation:
 * Forces acting in directions perpendicular to velocity have no effect
 * Gravitational potential is affected only by the final and initial states; intermediate values have no bearing on the result
 * The constant of motion is a scalar quantity, and the sum of the kinetic and potential energies.

Work
Substituting in a generic constant force instead of the gravitational force gives (m/2)v2 = ∫Fr dt = F(rf - ri) = Fd. As the force is only relevant where acting parallel to velocity, W = Fdcosθ. Work is equal to the change in kinetic energy (due to the above derivation); this is stated in the work-kinetic energy theorem: W = ΔEk.

An object with associated kinetic energy can do work on another object by losing some of its own kinetic energy; the work done on the pushing object is negative. Energy input into an object is positive work done; output is negative. If more than one force acts upon a body, the net work done is the sum of all fractional works done.

Energy
Energy can be defined as a system's ability to do work; however this is not all encompassing, as it is dependent on distance and energies such as chemical energy have no dealings with position changes at all.

Instantaneous power is dW/dt - given here as P = F.v (Watts).

Gravitational potential energy is equal to the integral of F by r - eg Fgz = -d/dz Ug.

Examples

 * 1: Mass on a pendulum
 * Setting Ug and Ek equal allows calculation of a maximum velocity so long as a maximum height is known (and maximum height can be calculated if a triangle is provided.


 * 2: Mass sliding down a plane
 * Finding the stopping distance of an object having slid down a slope requires first calculation of the final kinetic energy; from here W = Ek = Fd.

Summary
The energy conservation law can be derived from derivation of velocity squared, using substituted forces, Newton's second law, the scalar rule and the product rule of derivation. This derivation can also be used to find the work done, which is dependent on displacement in the direction of velocity and the relative force. Instantaneous power is related here by Fv, and W = Fdcosθ.