Energy and Work

Work is done when force moves through a distance.

push box

If no movement, then no work!

lifting weights
These people are doing work, because they are lifting weights

weight hold
This guy did work to get the weights up,
but right now he is not doing any work as the weights aren't moving

weight lift statue
This guy is also not doing any work

tractor pull
This tractor is doing work as it pulls the hay bales along.

Same Direction!

The force and movement are measured in the same direction.

Work  =  Force × Distance × cos θ

Where θ is the angle between the force and the direction of motion.

So any force that is sideways to the movement is not included.

Example: John pushes a box 3 m straight forward using 200 N of force. But his push is a little upwards by 20°.

push box 20 degrees

Start with:

Work  = Force × Distance × cos θ

Put in the values we know:

Work  = 200 N × 3 m × cos 20°

Work  = 200 N × 3 m × 0.9397...

Work  = 564 N m (to nearest N m)

1 N m is 1 Joule (J) the preferred unit for work and energy (more on this later):

Work  = 564 J

(Without cos θ, the wrong value would be 600 J)

Here are some other angles:

cos(0°)=1   cos(60°)=0.5   cos(90°)=0
force along distance
  force at 60 degrees to distance
  force at 90 degrees to distance
W = Fd   W = Fd × 0.5   W = 0

So remember:


Energy is the capacity to do work.

It is like energy is stored up work.

Energy can be in many forms:

A coiled spring has elastic energy.

Batteries store chemical energy.
They convert that to electrical energy,
which we can use to do work.

Fuel also contains chemical energy. An engine can turn that fuel (with oxygen) into work, making your car go!


A hammer has mechanical energy:

Energy goes from one storage to another, or goes to heat:

energy work heat

Heat is a type of energy, too. In fact the total amount of energy stays the same:

Energy can't be created or destroyed.

This is called Conservation of Energy: energy just gets transformed and the total stays constant.

Energy is lost to a system when it leaves it. A battery loses energy when it powers a light. Our bodies lose heat all the time.

And when we say "energy is lost as heat" we really mean the energy is dissipated (spread out) into the environment, it is not gone from the Universe.

Sankey Diagrams

A "Sankey Diagram" shows where the energy goes. The width of the arrow shows how much goes where.

car moving

Here is a Sankey Diagram for a car's gasoline engine:

sankey diagram for car engine

Note: "MJ" is Megajoule (Million Joules). Add them up to make sure the totals match!

The Joule

The basic unit of energy and work is the Joule (J):

1 Joule is the work done by 1 Newton moving 1 meter

1 J = 1 N × 1 m

The Joule has the units N m, or kg m2/s2 (because a Newton is kg m/s2)

Only force and movement in the same direction count, so a more accurate description is:

1 J is the work done to an object when a force of 1 N acts on that object in the direction of its motion through a distance of 1 meter.

apple 1m

Example: How much energy is needed to lift an 0.1 kg apple up 1 meter?

gravity apple force

To hold a 0.1 kg apple against gravity needs 1 Newton of force:

F = mg

F = 0.1 kg × 9.8 m/s2

F1 N

But holding an apple is not work, the apple needs to move!

So, raising it using 1 N for 1 m (both in same direction!) gives:

Work  =  1 N × 1 m × cos 0°

  = 1 J

How Much?

One joule is about:



A kilojoule (kJ) is 1000 J:

A Megajoule (MJ) is 1 million J:

A Gigajoule (GJ) is 1 billion J:


Efficiency is how much of the energy is useful as a percent of the total energy.

Efficiency = Useful Energy Total Energy as a percentage

Example: For every 100 MJ (Megajoule) of energy a gasoline engine uses, only 25 MJ goes to driving it forward.

Efficiency = 25 MJ 100 MJ = 25%



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