Let’s start with a definition: Work: a scalar quantity (it may be + or -)that is associated with a force acting on an object as it moves through some displacement. If the force tends to make the object go in the direction it traveled, then the force did + work. If the force opposed the motion, it did - work. If the force acts perpendicular to the motion, it does zero work. Work = (component of force in direction of motion)(displacement size) units of work are or Joules (J)

5 m 8 N 10 N F 1 =15 N F2F2

The previous definition is valid if the force does not change over the displacement. More generally, we can restrict our displacement to one that is very small (infinitesimal). Then the force will not change over the small displacement. Let us consider the case in one dimension with a force that depends on position: F x x1x1 x2x2 ∆x As ∆x gets smaller, the area of the rectangle has less “sticking out” above the curve. Thus we expect:

Fundamental Theorem of Calculus: Thus integration and differentiation are inverses of each other, sort of.

Exercise: A spring exerts a restoring force on an object attached to it directly proportional to the amount of “stretch” or compression. How much work is done by the spring as the object is stretched from x 1 to x 2 ? unstretched stretched x

In more than one dimension, we must take the direction into account:

Power If we calculate the work done over an interval, and divide by the time interval, we get the rate at which work is done, the power supplied (or consumed) by the force. In one dimension we have simply:

Why do we bother with the concept of work? Because when we look at the work done by the net force, i.e., the total work, a useful result is obtained. net F x vovo v

We then define the kinetic energy of an object as: Since our object started with KE =, the work done by the net F, i.e. the total work done, is equal to the change in the KE: Thus work is important because it is directly related to how the kinetic energy changes.

KE Energy of motion Total work done by all forces to get object from rest up to speed v

T=20 N fkfk v o = 4m/s T=20 N fkfk v = 6 m/s What was the change in the KE for our 4 kg mass? ∆KE = 1/2(4)(6) 2 - 1/2(4)(4) 2 ∆KE = = 40 J Find the value of f k : 3 m.

Conservation of Energy Suppose the system is under the influence of a single force F: Here the “∆” refers to the interval, i.e., the kinetic energy changed from K 0 to K as the force F did work over the interval. Let us suppose that the work done by F does not depend on the particular path taken between the ends of the interval, but depends only on the end points themselves. Such a force is said to be “ conservative ”. For such a force, we can always express the work done over an interval as the difference in the value of a new function at the ends of the interval: The function U is called the “ potential energy ”.

Then we may write: Define the total mechanical energy: Total mechanical energy is conserved (stays the same) if only conservative forces act.

Conservative Forces: Examples For a force to be conservative, the work it does must be independent of the path taken between the endpoints implying that the work depends on just the endpoints. Example 1: Any force that acts in 1-D and depends only on the position x. Denote the force as f(x). Clearly this depends only on the endpoints. Example 2: A mass changes position near the surface of the earth.

Depends only on endpoinjts

It is possible to do work on an object in such a way as to never give it any appreciable KE. Example: lift a mass slowing to some height. Example: slowly pull out a mass attached to a spring. In both examples some “agent” is applying a force just equal to the opposing force (gravity or spring force) and does work in positioning the object. Intuitively we might expect that the work done by the agent can be “gotten back” by allowing the system to return to where it started: let the mass fall release the mass/spring We say there is potential energy stored in the system. This PE is associated with the position of the system, not its motion.

U Energy due to position Formulae Work that some agent must do to slowly put the system into position starting from some (arbitrary) reference point.

Potential Energy Formulae Let’s stick to 1-D Only differences in energy can be measured, so we can pick the value of U(a) to cancel Then we have: Essentially an indefinite integral

For the spring we will always use x = 0 as the reference point, although in principle we could use other points.. For gravity we can choose the y = 0 point wherever convenient.

If a system moves only under the influence of gravity, elastic, or electric forces, then any change in KE always comes at the expense of PE in such a way that the total of the two remains the same. This is the principle of Conservation of Energy. Total mechanical energy

Example:A 4 kg ball is dropped from a height of 8 m. How fast is it moving just as it hits the ground?

Example: A spring with k = 50 N/m has a 4 kg mass attached such that it can move horizontally on a smooth surface. It is observed that when the spring is stretched 1.2 m, the mass is moving outward with a speed of 20 m/s. How much further out will the mass travel?

Force from Potential Energy We can find the various components of the force by taking the appropriate derivative.

Exercise: A 3-D potential energy is given by: Find F x at the point (1, 2, 4).

Graphs of Potential Energy Given the total energy and a graph of U vs x, one can qualitatively describe the types of motions available to the system E Turning points U K Negative slope of curve is force

Exercise: Describe possible motions for different total energies. Find force at x = 1 m