# The Work Equation

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## The Work Equation

**At the heart of energy is the equation:**

Work = force x distance moved in the direction of the force

**Or**

W = F s

The point that causes the most confusion here is that the distance moved is actually a vector because its direction is important.

*Look at these examples:*

**a) Just lifting - no friction.**

To lift this weight onto the shelf will require a force equal to the weight, **W**. The distance it will be lifted is **h**.

So work done = W x h

**b) Pushing up a slope - no friction.**

In this case once again you need to overcome the weight (due to gravity pulling down on the object). There is no friction so no other forces need to be overcome.

So Work done = W x h again!!

Even though the object has moved distance 's' in total, it has only moved distance 'h' in the direction of the force (i.e. up and down.)

**c) Pushing along the floor - no friction.**

Here is an object moving at constant velocity covering a distance s.

Work done = 0

Even though there are forces acting up and down, none are acting in the direction of the movement. Hence, no work is done overcoming a force.

**d) Pushing up a slope against friction.**

Here we're going to have to overcome friction (that'll take some energy) and gravity (that'll take some more energy) to lift this object to the top of the slope.

*In fact*

**Total Work done = work done overcoming friction + work done against gravity.**

But, let's deal with these one at a time.

First friction.

What's the friction force we've got to overcome? *Answer:* F

What's the distance moved in the direction of F? *Answer:* F acts down the slope and we've moved distance 's' up the slope. So we have had to overcome force F for a distance of s.

What's the work done against friction? *Answer:* Work = F x s

**Now gravity.**

What's the weight we've got to lift?* Answer:* W

What's the distance moved **in the direction of W**? Answer: W acts vertically down and we've moved distance **'h'** vertically up. So we have had to overcome W for a distance of h.

What's the work done against gravity? *Answer:* Work = W x h

Now add them together.

**Total work done = (F x s) + (W x h)**