# Mechanics

calculus-based physics for scientists and engineers

## Conservation of Energy

### Conservation of Mechanical Energy

The mechanical energy of a system of objects is the kinetic energies plus all the potential energy. (When we move beyond gravity, there can be more potential energies.) If no net work is done by non-conservative forces, the mechanical energy is conserved. That means the sum of kinetic plus potential never changes. The fraction of energy that is kinetic or potential changes, but the total energy does not.

Working with this conservation law utilizes that insight. After you have a picture, schematic, coordinate system, and a functional form of the potential energy function, choose two points in time. List the potential energies and kinetic energies for the initial and final times. Their respective sums have to be the same.

### Conservation of Mechanical Energy: Example

Here is an example using conservation of mechanical energy. It goes through the following steps:

1. Determine objects of system.
2. Find forces on objects.
3. Confirm no net work done by non-conservative forces.
4. Establish a coordinate system.
5. Derive potential energy function from conservative force.
6. Choose two points in time.
7. List energies at each point in time.
8. The sums of the energies at each point in time are equal.

### Conservation of Mechanical Energy: Coordinate Systems

When doing these problems, choices have to be made. You have to choose your coordinate system: both the location of the zero and the direction of the positive x axis. You also have to choose the location where the potential energy is equal to zero. These choices change how the problem is done. They change the mathematical representations of both the forces and the potential energy functions. They do not, however, change the physics. Judicious choices can make solving problems easier, and skill in making these choices only comes with practice.

### Conservation of Energy with Nonconservative Forces

Energy is always conserved, but it may not be conserved for your system. A system interacts with its environment. Another way to look at nonconservative forces doing work, is that energy is being exchanged with the environment. Energy is being lost from the system and added to the environment or taken from the environment and added to the system. In this context, environment means everything that is not defined as the system.

It is not always possible to calculate the energy gained or lost from nonconservative forces. However, if it is known or can be calculated, then the concept of conservation of energy can still be useful.

### Solution

1. Alice is working on a roof. She is at the top of the roof, which slants at an angle of 24 degrees with respect to the horizontal. She is a vertical distance of 3 m above the position where the roof ends. She has a 50 kg box of shingles that she no longer needs, so with a little push, gives them a 1/2 m/s velocity down the roof. If the coefficient of friction for the box of shingles in 0.35, what is the speed of the box when it falls off the edge of the roof?