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# 3: Energy

• 3.1: Work
We define work as the product of the component of the force in the direction of the displacement, times the displacement itself.  We calculate this component by projecting the force vector on the displacement vector, using the dot product.
• 3.2: Kinetic Energy
If you want to start moving something that is initially at rest, you’ll need to accelerate it, and Newton’s second law tells you that this requires a force - and moving something means that you’re displacing it. Therefore, there is work involved in getting something moving. We define the kinetic energy (K) of a moving object to be equal to the work required to bring the object from rest to that speed, or equivalently, from that speed to rest.
• 3.3: Potential Energy
The work done by a conservative force is (by definition) path-independent; that means that in particular the work done when moving along any closed path must be zero. For a conservative force, we can thus define a potential energy difference between points 1 and 2 as the work necessary to move an object from point 1 to point 2.
• 3.4: Conservation of Energy
Conservation of energy means that the total energy of a system cannot change, but of course the potential and kinetic energy can - and by conservation of total energy we know that they get converted directly into one another. Exploiting this fact will allow us to analyze and easily solve many problems in classical mechanics - this conservation law is an immensely useful tool.
• 3.5: Energy Landscapes
Typically, the potential energy is a function of your position in space. When we plot it as a function of spatial coordinates, we get an energy landscape, measuring an amount of energy on the vertical axis.
• 3.E: Energy (Exercises)