# Work done by the (changing) force of gravity

If you move vertically by a significant distance, the force of the Earth's gravity on you changes: it keeps decreasing as you move farther and farther away from the center of the Earth.

Therefore, in order to figure exactly how much work you do while, say, climbing up a ladder, you can't simply multiply "force times distance." Instead, you need to integrate. For example, if we want to compute the amount of work done by the force of gravity on Joe as he climbs the ladder, we would write down this integral. Note the signs of the two terms in the integral, which indicate the direction of the force, and the direction of the motion.

On the other hand, if we are interested in the work done by Joe as he climbs the ladder, it's a bit different. We know that the size of the force Joe must exert on the ladder to lift himself up is the same as the size of the force of gravity on him, but it points in the other direction -- AWAY from the Earth.

1. Joe finds a convenient ladder which goes vertically up from the base of Mount Everest (height 8000 meters) to its peak. Joe's mass is 60 kg. How much work must he do to climb from the bottom of this ladder to the top?
2. A single Big Mac contains about 540 calories. The ordinary "calorie" we use in conversation is equal to 4186 Joules of energy. How many Big Macs must Joe eat to replenish himself on his climb?

3. At the top of the mountain, Joe finds another ladder which climbs vertically into space, halfway to the Moon. "Cool!" thinks Joe, "I've always wanted to into space!" How much work must he do to climb from the bottom of this ladder to the top?
4. When Joe reaches the top of this ladder, he gets out his camera to take a picture. Unfortunately, he drops the lens cover, which falls back towards the Earth. How fast does it accelerate as it falls? (Yes, yes, there would be complicating effects due to the rotation of the Earth and the ladder. Let's ignore them for now ...)