## Mechanical working fills the kinetic store

Physics Narrative for 14-16

#### Power in a pathway fills a store

Filling or emptying kinetic stores necessarily involves speeding up or slowing down mass. It's an accumulation of the power in the pathway over time. The power in the pathway is just the rate at which the store is filling or emptying: how much energy you are shifting to or from the store per second. But which pathway?

The filling or emptying is most directly done by exerting a force on a mass. To fill the store up quickly, exert a large force. To fill the store up slowly, exert a smaller force. But if you remember the work about pathways, and in particular the power in a pathway, you'll remember that there are always two factors that determine the power in a pathway. (There is more on this in the SPT: Electricity and energy topic.)

The second factor is the velocity of the exerted force. If Alice is pushing, it's how fast she's moving with respect to Charlie as she pushes on the object. Charlie notices the object speeding up, and records a change in the kinetic store. Alice has a choice over how she provides the power in the mechanical working pathway (power = large force × small velocity, or power = small force × large velocity).

You can think about why it might be that the velocity is the second factor by thinking again about the units involved in the calculations, just as you did when we were considering momentum as a measure of motion.

You know that the power in the pathway must be measured in watts (that is, in joules per second). The force exerted is measured in newtons, and you may recall that the force times the distance gives you the energy in joules.

power = force × velocity

This implies a relationship amongst the units: watt = newton × metre second^{-1}

To see if that's correct, you need to connect force to more fundamental measures.

You may remember, from episode 01: force = mass × acceleration.

Again, you can reason with units: newton is equvalent to kilogram × metre second^{-2}. So force × velocity, can be expressed in units as kilogram × metre second^{-2} × metre second^{-1}. Earlier on in this episode you expressed joule as : kilogram × metre second^{-2} × metre, and you'll remember that a watt is just one joule every second. Putting these together you get: kilogram × metre second^{-2}second, which is kilogram × metre second^{-2} × metre second^{-1}.
This is consistent: the two sets of units agree. So velocity is the correct quantity to multiply by force to set the power in the pathway.

#### Choosing a force; choosing a velocity: setting the power

Power is a compensated quantity (a product of two independent contributing quantities – an increase in one contribution is compensated for by a decrease in the other to maintain a steady value of the compensated quantity). You saw that Alice had a choice:

power = large force × small velocity
,or
power = small force × large velocity.
But that's not all: the power in the pathway can fill or empty a store, so the power can effect an accumulation energy in the store, either positive or negative. As both force and velocity are vectors, you ought to write: Vector*P* = →F × →v

where ⋅

represents a special kind of multiplication (a dot product

) for vectors, giving a scalar as a result – remember that power is a scalar.

Even in thinking about simple, one-dimensional situations, there are four possibilities:

- Velocity is positive, force is negative.
- Velocity is negative, force is positive.
- Velocity is negative, force is negative.
- Velocity is positive, force is positive.

In the first pair of possibilities the power is changing the energy in the store in the opposite sense to that in the second pair (reversing the flow, from filling to emptying, or from emptying to filling), which, in either case, depends on the physical situation.