Work and Energy II

Previous: Work and Energy I

***Work and Energy II*** is the seventh lecture in the **Mechanics** section of PH1011. It covers work done in the context of non-constant forces such as the spring force, the differences between conservative and non-conservative forces, and conservation of energy.

Next: Impulse and Momentum

## Content

### Spring Force and Elastic Potential Energy

The spring force is the non-constant force exerted by springs in resistance to stretching. This is opposite and proportional to the displacement of the spring by Hooke's law - **F = -kx**, *k* being the spring constant (Nm-1). The spring constant can be found as the gradient of a graph of force vs extension, and varies per spring.

The energy input from the work done by the spring force is elastic potential energy - it can be converted to kinetic energy so long as the spring remains intact as the object attached to the spring is accelerated. The work done in this case cannot be given by Fd as F is not constant - it must be an integration; **Ws = ∫x0xf Fs(x)dx**. Solving the integral finds W = -k/2Δx2. Again the potential energy is the negative of work done - **Us = k/2Δx2**. As before with gravitational potential, F = -d/dxU.

### Conservative and Non-conservative Forces

Integrating *d/dtK = F.v* with ΔK = W gives W = ∫Fdr (as dr = vdt). This shows that the total work done across a path with unequal forces must be calculated for each small displacement rather than as an overall value - giving rise to the difference between forces which do and do not depend on path taken; non-conservative and conservative forces respectively. For conservative forces, WA->B = -WB->A; Fconservative ∮F.dr = 0. Forces are conservative if its associated work done is 0 across any path (eg it does not lose any energy in travel). Gravitational and spring forces are conservative.

A field of potential U(r) can be aligned with all spatial points. Setting a staring point to U(r0) = 0 can allow the U(r) of another point to be calculated via the work done to get to that point; as **W = ∫r0rF.dr** then U(r) = U(r0 - ∫r0rF.dr. This again gives *ΔU = -W*.

Non-conservative forces cause energy to be lost as work done - eg friction.

In systems where more than one conservative force act, Ek + Ug + Us = constant. This is excellent for simplifications, however in almost all real systems a combination of conservative and non-conservative forces take effect. As work done is lost here **ΔEk + ΔU = Wk**. This can be simplified by inputting the change in mechanical energy (EK + U) and the conversion of work done to heat (W=-ΔEth) to give **ΔEmec + ΔEth = 0**. In event of an external force acting on the system, ΔEmec + ΔEth = Wext.

### Examples

* Without friction: A roller coaster travelling from point A to point B need only take into account its heights (to find potential energy) and velocities (kinetic energy) at these points to calculate unknown values (such as mass, height, final/initial velocities).

* With friction: Mechanical energy must be considered. The difference between the final and initial energies must be what is lost in friction - given as W = -Fd. This assumes a constant force of friction and given distance of travel.

## Summary

Non-constant forces such as the spring force require integrals to find their work done. Elastic potential energy occurs in spring systems, and is again equal to kinetic and gravitational energies. Conservative forces do not lose energy due to work done and therefore can regain the sum of their energy regardless of path taken; non-conservative forces such as friction do lose energy due to work done and must be calculated using mechanical energy (the sum of kinetic and potential, equal to the negative energy lost as heat.