Lecture 5

Chapter 5: Work and Energy

• Introduction by Dr. Val G. Rousseau - Xavier University of Louisiana

Work: The Scientific Definition

• Definition differs from everyday meaning but is closely related.

• Classification of "workers":

  • Those producing nothing (e.g., ex-French president François Hollande).

  • Those producing something useful (e.g., engineers, firemen, cashiers).

• In science, forces act as "workers" where some produce nothing and some produce something.

Enlightening Example

• Moving a frictionless cart:

  • A force is applied making an angle θ with the horizontal.

  • Free-body diagram includes three forces: weight, normal force, and applied force.

  • Forces can be resolved into:

    • Parallel components (in motion direction).

    • Perpendicular components (no effect on motion).

• According to Newton's second law, normal force, weight, and perpendicular components don’t affect motion; only the parallel component does.

Work Done by a Force

• Work is defined as:

  • Proportional to the parallel component of the applied force.

  • Proportional to the distance moved (d).

• Equation for work done (W):

  • W = F * d * cos(θ), where θ is angle between force direction and displacement.

  • Positive if force points in direction of motion (-90° < θ < 90°), negative if in opposite direction (90° < θ < 270°).

• S.I. unit of work:

  • Newton-meter (N·m), called Joule (J).

Energy

• Definition of Energy:

  • A measure of a system's ability to do work.

• Kinetic Energy:

  • Defined through mass m moving under a net force in the same direction as displacement:

  • Uses kinematic equations for constant acceleration.

  • Work can alter the kinetic energy of the mass.

Work-Energy Theorem

• The work-energy theorem states that:

  • The work done on an object by a net force changes its kinetic energy.

  • Energy is conserved in the universe, meaning it is neither created nor destroyed.

Conservative and Non-Conservative Forces

• Conservative force: Work depends only on starting and ending points (e.g., gravitational force).

• Non-conservative force: Work depends on the path taken (e.g., kinetic friction force).

• Energy from conservative forces can be recovered; energy from non-conservative forces is often transformed into heat.

Net Work

• Total work done by all forces acting on an object.

• Work-energy theorem is applicable here, enunciating that net work leads to changes in kinetic energy.

Hooke’s Law (Spring Force)

• Hooke’s law describes how spring force relates to spring compression/extension:

  • Fs = -kx (where k = spring constant, x = displacement from equilibrium).

Potential Energy

• Defined in terms of gravitational force as:

  • W = mgh (work done by gravity when falling from height h).

• Potential energy is the stored energy based on an object's position or configuration.

  • When falling, gravitational potential energy transforms to kinetic energy.

Mechanical Energy

• Mechanical energy of a system is the total of kinetic and potential energy.

• Conservation of mechanical energy holds for systems subjected only to conservative forces.

Conservation of Total Energy

• When both conservative and non-conservative forces act:

  • Change in mechanical energy equals work done by non-conservative forces.

• Total energy remains constant.

Power

• Power defined as:

  • The rate at which work is done (P = W/t).

• S.I. unit of power is the watt (W), equivalent to J/s.

• Other unit includes horsepower; crucial for contexts like engines or power outputs.