Energy - California State University, Northridge



Work and Energy

Energy is one of the most useful concepts in physics, but it is hard to define. All of us have an intuition of what is energy, even if we cannot touch it or see it. If it is so abstract, why bother with it? The concept of energy is one of the most powerful tools in physics! This is true because:

1) Energy is a scalar (just a number)

2) Energy is conserved (it cannot be created nor destroyed)

Energy has many forms (kinetic, potential, chemical, thermal, nuclear, etc.). The key is that although we cannot create or destroy it, it can change form! Before we see this in action, we merit some definition of energy:

Energy = a measure of an object’s ability to do work

Work = force times displacement (in the direction of the force) = FΔdcosθ

Power = the rate at which work is done = W/Δt

Note that work is force times displacement, this might be against your intuition, since we use work differently in everyday language. Holding a book on the palm of your hand feels like doing work because you get tired doing it, but no work is being done on the book because there is no displacement! Confused? What you feel is your muscles being slightly displaced internally, so your muscles are indeed doing work even if no work is done on the book. Who is doing work in the following cases?

To see the power of thinking in terms of energy, let’s start by introducing two forms of it: kinetic energy and potential energy. Kinetic energy is the energy of motion. It is proportional to the mass of the object times its velocity squared. This has profound implications. Decreasing speed limits from 75 mph to 55 mph, for example, reduced fuel consumption and fatalities in car accidents by a factor of two! (you should convince yourself that the kinetic energy is reduced by half)

KE = ½ mv2 (energy of motion)

Potential energy is stored energy. There are many ways to store energy, we will consider gravitational and elastic potential energy. Gravitational potential energy is stored by changing an object’s height. Think about tossing a ball up in the air. You initially do some work on the ball by applying a force over a distance, this work is converted into kinetic energy. Initially the kinetic energy is high because the velocity is high, but it goes down as the ball slows down on its way up. At its highest point the velocity is zero, so the kinetic energy is also zero. Where did the energy go? It is stored as gravitational potential energy!

Energy can also be stored in springs by compressing them (toys often make use of this), this is called elastic potential energy. It is proportional to a spring constant k (each spring has a particular value, the stiffer the spring the higher it is) times the compressed or stretched distance squared. Note that the force provided by a spring is given by Fs = -kx (Hooke’s law).

PEg = mgΔh (gravitational) Fg = mg (gravitational)

PEe = ½ kx2 (springs) Fs = - kx (springs)

Conservation of energy states that the total energy in a system remains constant. This along with the definitions of kinetic and potential energy allow us to solve problems that are otherwise too hard to solve using Newton’s laws – note that this approach, however, it’s not something new but a more novel perspective. The concept that energy can move from place to place and change forms help us to understand many things. The rate of energy flow (power) is also quite useful. Together these concepts form the basis for understanding practically everything, from chemistry and biology to geology and engineering.

In this class, we will be mainly be concerned with mechanical energy. Mechanical energy is the sum of kinetic plus potential energy. Energy is always conserved, but mechanical energy is only conserved in the absence of friction. When there is friction energy is lost because it is dissipated as heat. If there is no friction or it is small, using conservation of mechanical energy help us solve many problems.

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Mechanical energy = Kinetic energy + Potential energy (conserved if there is no friction)

Energy = Kinetic energy + Potential energy + Other forms of energy (always conserved)

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