More practice computing work done by forces which vary ...

But first, just a bit more review of integration.

• Some special functions

Let's look a very simple situation in which the wind pushes an iceboat across a frozen lake. The wind is constant in size and pushes exactly in the direction of the boat's motion.

• Example: a constant force in the direction of motion

But real life is rarely so simple. You need to know how to deal with more complicated situations. Let's take a look at several ...

The force is constant, but not exactly in the direction of motion. This isn't so bad; you just need to use the dot product to find the component of the force in the direction of motion.

• Example: Constant force NOT in direction of motion
• Example: Constant force not always in direction of motion
• Example: Constant force not always in direction of motion (again)

The magnitude of the force changes as the object moves (though the direction is always in the direction of motion)

There's no way to avoid integration when this happens.

• Work done by a force which falls as 1/r
• Work done by a force which falls as 1/r²

But what if the direction of the force, or the object, CHANGES as the object moves? Then there's no way around it. You need to carry out an integration, using the proper force, and the proper component, at every point.

• Example: Constant force, changing direction of motion
  • solution in PDF format
  • Solution in GIF format: Part 1 Part 2
• Your turn: multiple forces and changing direction of motion

If we have time, we might try some additional examples ...

• A constant force, just use the dot product

• A constant force, but over a curved path

Copyright © Michael Richmond. This work is licensed under a Creative Commons License.