Week 8 Pre-work for Work, Energy, and Conservation of Energy

Work and Energy

• Lecture PowerPoints are based on Chapter 7 and 8 of "Physics for Scientists and Engineers, with Modern Physics, 4th Edition" by Giancoli, Copyright © 2009 Pearson Education, Inc.

• These materials are protected by copyright and are intended for use by instructors in teaching their courses and assessing student learning.

• Dissemination or sale of this work is not permitted.

  

Work Done by a Constant Force

• Work done by a constant force is defined as the distance moved multiplied by the component of the force in the direction of displacement.

Scalar or Dot Product

• Definition of the scalar, or dot, product allows us to write:

  

General Definition of Work

• The general definition of work is:
  $W = \int_{a}^{b} \vec{F} \cdot d\vec{l}$ where $\vec{F}$ is the force vector and $d\vec{l}$ is the infinitesimal displacement vector.

  

Work Done by a Spring

• The force exerted by a spring is given by:
  $F_s = -kx$
  where:

  • $F_s$ is the force exerted by the spring.

  • $k$ is the spring constant.

  • $x$ is the displacement from the equilibrium position ($x = 0$).

• The figure illustrates:

  • (a) Unstretched spring.

  • (b) Stretched spring.

  • (c) Compressed spring.

    

Work Done on a Spring

• Work done on a spring is calculated by integrating the force over the displacement:

  • The work done is equal to the change in potential energy stored in the spring.

    

Kinetic Energy

• Kinetic energy is defined as: $K = \frac{1}{2} mv^2$ where:

  • $K$ is the kinetic energy.

  • $m$ is the mass of the object.

  • $v$ is the speed of the object.

Work-Energy Principle

• If the net work is positive, the kinetic energy increases.

• If the net work is negative, the kinetic energy decreases.

• The net work done on an object is equal to the change in its kinetic energy:

  

Conservative and Nonconservative Forces

• A force is conservative if:

  • The work done by the force on an object moving from one point to another depends only on the initial and final positions of the object.

  • It is independent of the particular path taken.

• If friction is present, the work done depends not only on the starting and ending points but also on the path taken.

• Friction is a nonconservative force.

• Potential energy can only be defined for conservative forces.

  

Potential Energy

• In raising a mass $m$ to a height $h$, the work done by the external force is $W = mgh$.

• Gravitational potential energy at a height $y$ above some reference point is:
  $U_{grav} = mgy$

• This potential energy can become kinetic energy if the object is dropped.

• Potential energy is a property of a system as a whole, not just of the object because it depends on external forces.

• Only changes in potential energy can be measured; the absolute value is not meaningful.

• 

Elastic Potential Energy

• A spring has potential energy, called elastic potential energy, when it is compressed or stretched.

• The potential energy is: $U_{el}(x) = \frac{1}{2} kx^2$ where:

  • $k$ is the spring constant.

  • $x$ is the displacement from the equilibrium position.
    

    

Conservation of Energy

• If there are no nonconservative forces, the sum of the changes in the kinetic energy and in the potential energy is zero

• The kinetic and potential energy changes are equal but opposite in sign.

• The total mechanical energy is:

• Conservation of mechanical energy:

• If only conservative forces are doing work, the total mechanical energy of a system neither increases nor decreases in any process. It stays constant—it is conserved.

• 

Nonconservative Forces

• Nonconservative, or dissipative, forces include:

  • Friction

  • Heat

  • Electrical energy

  • Chemical energy

• These forces do not conserve mechanical energy.

• However, when these forces are taken into account, the total energy is still conserved.

  

Law of Conservation of Energy

• The law of conservation of energy is one of the most important principles in physics.

• The total energy is neither increased nor decreased in any process.

• Energy can be transformed from one form to another and transferred from one object to another, but the total amount remains constant. Mathematically expressed as: $\Delta E<em>{total} = 0$, where $E</em>{total}$ represents the total energy of an isolated system.

•