Energy Methods

 



You
need at your fingertips the following formulae

kinetic energy
Ekinetic = mv
m = mass, v = velocity
(gravitational) potential energy (change)
Egravitational = mgΔh m = mass, h = height (change),
g
= gravitational field strength (= 9.8 N/kg)
elastic (potential) energy

λ = modulus of elasticity,
l
= unstretched length,
x
= extension (= current length - l )
work done against a force (often resistive or frictional, but not always)
Ework = Fs
Ework(= F.s = Fs cosθ)
F = force, s = distance moved, using the component along the line of the force. The the vector formulation is not often needed.
power

E can stand for any form of energy
power P = F v
F = force, P = power, v = velocity

In all of these formulae, you must use a consistent system of units (metres, kilograms, seconds, joules, etc.)

Sum of energy changes  =  0

Worked example

A skier sets off down a slope with a starting velocity of 5 m/s. The slope is 500 m long and involves a drop in height of 200 m. During the run, there is a total resistive force of one tenth of the skier's weight. Calculate the velocity at the bottom of the run.

Notice that we are not given the mass of the skier, and yet it appears in most of the energy formulae. With any luck it will cancel out, so we put it in as
m whenever we need to and keep our fingers crossed.

KE at start =
m 5 = 12.5m
KE at finish = mv
GPE change = m 9.8 200 = 1960 m
Work done against resistive forces = (mg/10) 500 = 490 m

We have to be a little careful about the sign of the various energy changes:

ΔKE + ΔGPE + Δwork = 0 The signs are all positive at this stage because this is just the standard equation
(mv - 2.5m) - 1960 m + 490 m = 0   
Signs have become negative where our logic has told us that the value we are substituting is negative
v = 1472.5
Cancelling by m and re-arranging
v = 54 m/s
This is quite fast - about 100 mph.