Work and Energy

• The Work done by a force is defined as the energy transferred by that force on an object along a displacement. More formally for a constant force we have

  • The work is, effectively, a measure of how similar the direction of the force is compared to the object’s displacement.
  • In the general form, when force is not constant The total work is given using the following line integral. Denote the path taken as .

• Energy pertains to an abstract conserved quantity that is transferred through work.

• The Kinetic Energy of an object is the energy it possesses due to its motion. It is defined as follows

• The Work Energy Theorem states that the total work done is equal to the change in Kinetic Energy. That is

  Proof (Average Case): Using Newton’s Second Law and the definition of Work.

  • Viewed in this way, the kinetic energy equals the amount of work you must do to induce to accelerate the object from rest to speed .

  • The total kinetic energy can still change even when there is no work being done by anything outside the system

  • Proof (Integral Case): Observe that . We then get the following using the definition of kinetic energy and work

    The theorem follows from using the fundamental theorem of calculus and setting

    This is different from the Impulse-Momentum Theorem: work depends on distance, whereas impulse depends on time.

• Power is the time rate at which work is done. That is,

  • Using the definition of power and work, we can also derive
  • By the Work-Energy Theorem the rate of change of kinetic energy of an object is equal to the power expended by the forces acting on it.

• The Potential Energy is the energy held by an object because of work done on by a force that is path independent ¹

  • Because of path independence, the formulae for the potential energy of an object due to a force applies regardless of what path the object took.

  • Gravitational Potential Energy involves Gravity acting on an object near the Earth’s surface. It is given by

    • Another way to write this is with the center of mass

    • Another way to write this is using Newton’s Law of Gravitation. Let be the mass of the attracting object and the mass of the object in study

  • Elastic Potential Energy involves the force of a spring that has been compressed or stretched with displacement . For an ideal spring, it is given by

    Here if the string is stretched and if compressed. is the force constant of the spring.

• For a given potential function dependent on position ,the force can be calculated as ²

  • Any minimum in the potential function is a stable equilibrium
  • Any maximum in the potential function is an unstable equilibrium.

• The work done by all forces other than any conservative force equals the change in total mechanical energy of the system

  • When the only forces that act on the object are conservative, the total mechanical energy is constant.

• The Internal Energy of a system is the energy contained within it. It is the amount of energy needed to bring the system from its standard state to its current state. We denote this with

• The Law of Conservation of Energy states that energy is never created nor destroyed, only transformed. That is

  • In its most general form, the conservation of energy holds because the laws of physics are time invariant .They are the same regardless of time.

Links §

• University Physics with Modern Physics 15th Edition By Freedman - Ch. 6, 7
• Kinds of Forces

Footnotes §

1. See Integral Calculus. This is connected to the concept of conservative gradient fields and potential functions. ↩

2. The potential function in calculus does not have a negative sign. Here, however, we want the negative sign since we interpret conservative forces as pushing the system towards lower potential energy. ↩