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Energy

Energy is one of the most important concepts in science, but it is hard to define in just a few words. What is energy? We say that small children "have a lot of energy." They are always running instead of walking, jumping instead of stepping. We also say that we need to eat to gain energy after spending a day working, moving heavy objects. We also can say that the blocks in the walls of our high school have energy.

We can summarize all these examples of energy by saying an object has energy if it can produce a change in itself or in its surroundings.

Energy
the ability to do work (this definition does not describe all kinds of energy, but will suffice for our study of mechanical energy.) Energy, like work, is a scalar quantity. The SI unit of energy is Joules.

Work and energy are interrelated. If you do work, you get something as a result of your efforts. If net work is done on a particle, its speed is changed. We relate the motion of the particle to the net work done using a property called kinetic energy. It takes work to make an object move and a moving object can do work. If positive net work is done on an object, its speed increases. If negative net work is exerted on an object, its speed decreases.

Kinetic energy (usually Ek but I use KE)
energy due to the motion of an object; if an object has speed, it has kinetic energy. We define this as translational kinetic energy (or energy of an object moving in the xy plane.
KE = 1/2 m v2
where m is mass in kilograms and v is the object's speed

If an object's mass is doubled, its kinetic energy is doubled, but if its speed is doubled, its kinetic energy is quadrupled!

Work-Energy Theorem
work causes a change in an object's kinetic energy; if you want to change the kinetic energy of an object, you must do work on it
W = DKE = KEf - KEi
W = 1/2 m vf2 - 1/2 m vi2

Derivation of Work-Energy Principle

A constant net force Fnet acts parallel on an object of mass m that is moving with initial speed v1. It accelerates to speed v2 over a distance d. The work done on the object is Wnet = Fnet d. Applying Newton's second law and substituting for Fnet gives Wnet = m a d. Knowing that vf2 = vi2 + 2 ad, solve for acceleration a and substitute into the expression for Wnet. This gives the expression for the Work/Energy Principle.

Sometimes an unbalanced force causes acceleration, but without doing work and thus no loss in energy. An example would be the moon revolving about the earth. The moon is centripetally accelerated by the gravitational force between it and the earth. The speed of the moon as it revolves about the earth is constant. No work is done (there is no change in kinetic energy) because the force is perpendicular to the direction of the moon's movement.

The work/energy theorem also predicts the increase in stopping distance for a car as its speed increases. The work/energy theorem shows that the stopping distance of a car is proportional to the sqare of its speed. Doubling its initial speed quadruples its stopping distance.

Potential energy (usually U but I use PE)
Energy due to an object's position. Energy can be stored for later use in the form of potential energy. The mathematical formula for the potential energy depends upon the type of force involved.

Gravitational potential energy
energy of an object due to its position in a gravitational field
PEgrav = m g h
where m is mass in kilograms, g is the acceleration due to gravity, and h is the object's height. (Thus, mg is equivalent to the object's weight.)

Base level
position where the potential energy of an object is defined to be zero. The higher an object is above base level, the greater its gravitational potential energy.

Notice that gravitational potential energy depends upon an object's vertical height above base level. The work done by the gravitational force only depends upon the vertical distance between two points, not the path taken to get from one point to another. The gravitational force is an example of a conservative force. For a conservative force, the net work done depends upon the intial and final positions, not the path taken to get from the initial to the final position.

Advanced look at potential energy Since potential energy is based upon conservative forces, not all forces have potential energy. For example, friction is a nonconservative force. The amount of work done by friction is dependent upon the path taken between the initial and final points. Thus, friction is an example of a force which has no potential energy assiciated with it.

An advanced look at the work/energy theorem  Conservative and nonconservative forces act on a body. The net work done is the sum of the work done by the conservative forces and by the nonconservative forces. Remember that the net work is equal to the change in kinetic energy and that work done by a conservative force results in a loss of potential energy. This yields the following, where the work done by the nonconservative forces is equal to the sum of the changes in the potential and kinetic energies.

Wnet = DKE + DPE

Elastic Potential Energy

A simple spring has potential energy when compressed (or stretched). When it is released, it is capable of doing work. Hooke's Law describes the "restoring force" of a spring.
Fs = - kx
where k represents the spring constant and x represents the distance stretched
In order to calculate the potential energy of a spring, you must calculate the work done to stretch it. The force used to stretch the spring varies over distance, so W = Fd is not valid. Use the average force to stretch the spring which is given by Fav = 1/2 kx. Substituting into W = Favd, yields the formula for the work done to stretch a spring, or the formula for elastic potential energy,
elastic PE = 1/2 kx2

Mechanical energy
the sum of the potential and the kinetic energies of an object (conservative forces only). Mechanical energy is also conserved.

Law of Conservation of Energy
Energy can be transformed from one form to another in an isolated system, but it cannot be created or destroyed. The total energy of the system is constant. The total energy is neither increased nor decreased in any process. Energy can be transformed from one form to another, and transferred from one body to another, but the total energy remains constant.

energy bank

Principle of Conservation of Mechanical Energy
the total mechanical energy of an object remains constant as an object moves provided that no work is done by forces other than gravity.
Etot = KE + PE

General Form of Work/Energy Principle

The work done by a nonconservative force acting on an object is equal to the total change in kinetic and potential energy.

Mechanical Energy and its Conservation

If only conservative forces are acting on a system, the sum of the kinetic and potential energies at any moment equal the total mechanical energy of the system. Thus, the total mechanical energy of a system remains constant if only conservative forces are acting.

Dissipative Forces

Because frictional forces decrease the total mechanical energy of a system, they are called dissipative forces. If heat is considered a form of thermal energy, then energy is conserved in any process. The increase in thermal energy due to frictional forces is equal to the work done by the frictional forces. The initial mechanical energy of an object upon which frictional forces act equals the final mechanical energy of the object plus the energy transformed by friction into thermal energy.
1/2 mvi2 + mghi = 1/2 mvf2 + mghf + Ffd

Potential/Kinetic Energy Virtual Laboratory
This applet shows the relation between potential energy and kinetic energy and energy loss. Balls can be given different initial energies and masses and therefore dropped from different heights. They bounce off a surface which can absorb an amount of energy that is set by a parameter tag. The impact velocity is recorded. The general relation should be readily deduced by using this applet.

AP Multiple Choice Questions on Energy

  1. Be able to recognize the relationship between potential energy and kinetic energy for oscillating objects. The energies are equal, but occur at different times. When is potential energy a maximum? When is kinetic energy a maximum? The total mechanical energy (the sum of the KE and the U at each point) is always a constant value.
  2. Be able to graphically show how kinetic energy changes. For example, you roll a ball off the table. If you graph KE vs time, the graph will be parabolic (since the object is accelerating). KE increases with time, but not linearly. Also, in this example, initial KE is not zero. Since the ball is rolling, it already had speed, so it already had KE. Remember, if the object is accelerating, it's KE vs time graph is parabolic, not linear!
  3. Know the formula for elastic potential energy.
  4. Be able to manipulate other formulas to solve for KE. An example, think of the centripetal force formula (F=mv2/r) and the KE formula (KE=1/2 mv2). See how you can manipulate both to get an expression for KE for an object in circular motion?
  5. Be able to perform simple KE and U calculations.

AP Free Response Questions on Energy

  1. Know the work/energy theorem. The change in an object's KE is equal to the work done. This can be used to find, for example, the distance that an object slides before coming to rest. Use the work/energy theorem to solve problems that can't be solved by acceleration formulas.
  2. Energy isn't usually asked seperately, but is incorporated into problems. For example, an object rolls off a table. You are asked to find its kinetic energy the instant before it hits the floor. Remember, total KE = KE x + KEy. You must include the KE it has before due to its horizontal speed. You must find its vertical velocity the instant before it hits the floor and use that to find its vertical KE.
  3. If friction is ignored, KE equals U. For example, a pendulum - you can find its initial potential energy and use that to find its speed at the bottom of its swing.
  4. In a pendulum problem, think how you can use trig to find how far the bob is above base level when all you know is the length of the string and the angle relative to the vertical the string is pulled back.
  5. Be able to draw total mechanical energy vs time, KE vs time, and U vs time graphs for objects in SHM (pendulums, springs, etc.).
  6. Be able to read a graph of experimental data (U vs x, for example) for an object in SHM to determine how to find an object's potential energy at any position.

Work Notes

AP Objectives-Energy

Work Sample Problems

Energy Sample Problems

Work Homework

Energy Homework

AP Work & Energy Class Problems