Work, Energy And Power

Work, Energy and Power Work is done when a force moves an object through a particular distance. The amount of work that is done is the amount of energy that has been transferred. Energy, then, is like the “currency” used in order to perform work. If you want to do 30J of work you must spend 30J of energy. Power is the rate at which work is done, or the rate at which energy is transferred.

Work Done There are two formulae for work done. W = ∆E W ∆E

= =

Work Done (in Joules, J) amount of energy transferred (in Joules, J) W = Fs

F s

= =

(1)

(2)

Force (in Newtons, N ) distance moved in direction of force (in metres, m)

Note that s is the distance moved in the direction of the force – you normally need to do some resolving in order to get both the distance and the force in the same direction. For this reason, this equation is often quoted as W = F s cos θ, where θ is the angle betwen F and s.

Energy Energy comes in many forms: gravitational potential energy (GP E), kinetic energy (KE), thermal energy, chemical energy, nuclear energy, electrical energy, elastic energy, . . . The SI unit for energy is the Joule, J. Kinetic Energy, KE Kinetic Energy is the energy due to movement: the faster an object is travelling the more kinetic energy it has. Imagine accelerating an object from rest (u = 0) to a velocity v. We can use two equations of motion, 1 1 s = (u + v)t = vt 2 2 And v−u v = a= t t Using N2L, mv F = ma = t Then, by equation 2, mv 1 1 W = Fs = × vt = mv 2 t 2 2 Now, because the gain in kinetic energy is due to the work done, 1 KE = mv 2 2

(3)

Gravitational Potential Energy, GPE GPE is energy due to the position or, more specifically, height. It is concerned with raising objects to a certain height, so here the relevant force is the object’s weight, and the relevant distance is the height through which that object has been raised. So, using equation 2, W = F s = mgh Again, the GPE gained is due to the work done, so GP E = mgh

(4)

KE and GPE In many situations (a falling object, a swinging pendulum, a rollercoaster, . . . ) KE increases as GPE decreases, and vice versa. The one implies the other. Due to the law of conservation of energy, we can equate these: 1 mv 2 = mgh (5) 2

Power Power is a measure of how quickly energy is transformed from one form to another. In other words, power is the amount of energy that is transformed per unit of time. Or, in mathematical terms: P =

∆E t

where P ∆E t

= = =

Power (in Watts, W) Amount of energy transferred (in Joules, J) Time (in seconds, s)

This means that 1W is exactly equivalent to 1 Js , or 1 Joule per second. Electrical and mechanical devices are designed to transform energy between different forms: cars are designed to turn chemical energy (petrol) into kinetic energy (movement); headphones are designed to convert electrical energy into sound energy; light bulbs are designed to convert electrical energy into light energy. As power is a measure of how quickly this energy transfer can take place, it is often used to rate different devices. Here are some typical values of power rating: A torch An electric light bulb An electric cooker A space rocket A large power station The Sun

1W 100W 10kW = 10,000W 100MW = 100,000,000W 10GW = 100,000,000,000W 100,000,000,000,000,000,000,000,000W