Chapter 6 Study Notes: Work and Energy

Chapter 6: Work and Energy

6-1 Work Done by a Constant Force

• Definition of Work: The work done by a constant force is defined as the product of the distance moved and the component of the force in the direction of displacement:
  $W = F imes d imes ext{cos}( heta)$
  where (W) is work, (F) is the force, (d) is the displacement, and ( heta) is the angle between the force and the displacement vector.

• SI Units of Work: The SI unit for work is joules (J), defined as:
  $1 ext{ J} = 1 ext{ N} imes ext{m}$

• Example Scenario: When a person carries a bag of groceries without lifting or lowering it, he does no work on the groceries because the force exerted (upwards) has no component in the direction of motion (horizontal).

6-1 Work Done by a Constant Force: Problem Solving Steps

1. Draw a Free-Body Diagram: Visual representation of all forces acting on the object.

2. Choose a Coordinate System: Define axes to measure forces and displacements.

3. Apply Newton’s Laws: Determine any unknown forces acting on the object.

4. Calculate Work Done by a Specific Force: Use the work formula mentioned above.

5. Find Net Work: This can be achieved either by determining the net force and calculating the work it does, or by calculating the work done by each individual force and summing those values.

• Note: Work done by forces opposing the direction of motion (e.g., friction) will be negative, while centripetal forces do no work as they are always perpendicular to the direction of motion.

6-2 Work Done by a Varying Force

• For varying forces, work done can be approximated by dividing the path into small segments. The work done over each segment is calculated and summed.

• As the distance segments become infinitesimally small, the work done by a varying force is defined as the area under the force vs. distance curve.

6-3 Kinetic Energy and the Work-Energy Principle

• Traditional Definition of Energy: Energy has often been defined as the ability to do work; however, not all forces can do work.

• Mechanical Energy Context: The chapters focus on mechanical energy which adheres to the definition.

• Definitions and Relationships:

  • Kinetic Energy: Defined through work done, where the relationship can be derived using acceleration in terms of velocity and distance, resulting in:
    $KE = rac{1}{2} mv^2$

  • The Work-Energy Principle tells us that the work done is equal to the change in kinetic energy:
    $W = riangle KE$

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

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

• Units: Since work and kinetic energy can be equated, they share the same units; kinetic energy is also measured in joules.

6-4 Potential Energy

• Definition of Potential Energy: An object can have potential energy due to its position or configuration in a system.

• Common Examples of Potential Energy:

  • A wound-up spring

  • A stretched elastic band

  • An object at height (h) above the ground.

• Work Done in Raising a Mass: When raising a mass (m) to a height (h), the work done can be formulated as:
  $W = mgh$

• Gravitational Potential Energy: Defined as: $PE_G = mgy$

  • Consistency in Reference Points: The position where (y = 0) can be chosen arbitrarily, as only differences in potential energy matter, not absolute values.

• Conversion of Potential to Kinetic Energy: Potential energy stored can be converted to kinetic energy upon release (e.g., when an object falls).

• Elastic Potential Energy in Springs: The force to compress or extend a spring is dictated by Hooke's Law:
  $F = -kx$
  where (k) is the spring constant and (x) is the displacement from the equilibrium position.

• Potential Energy Stored in Springs: Can be expressed as:
  $PE_{spring} = rac{1}{2} kx^2$

6-5 Conservative and Nonconservative Forces

• Distinction Between Forces: If friction is present, the work done depends on the path taken, categorizing it as a nonconservative force.

• Table 6-1: Examples of Forces

  • Conservative Forces: Gravitational, Elastic, Electric

  • Nonconservative Forces: Friction, Air Resistance, Tension in Cords.

• Potential energy can only be defined in the context of conservative forces.

• Work Done by Nonconservative Forces: The work done is equal to the total change in kinetic and potential energy:
  $W_{nc} = riangle KE + riangle PE$

6-6 Mechanical Energy and Its Conservation

• Mechanical Energy Conservation: If no nonconservative forces are acting on the system, the sum of changes in kinetic and potential energy equals zero; thus, yields:
  $riangle KE + riangle PE = 0$

• Total Mechanical Energy Definition:
  $E = KE + PE$

• Conservation Equation for Mechanical Energy:
  If energies are conserved, we have:
  $E<em>{ ext{initial}} = E</em>{ ext{final}}$

6-7 Problem Solving Using Conservation of Mechanical Energy

• Total Mechanical Energy Formula:
  $E = KE + PE = rac{1}{2} mv^2 + mgy$

• Energy Transitions: Energy conservation principles allow one to calculate how energy transforms from potential to kinetic under certain conditions (like free fall).

• Frictionless Scenarios: In absence of friction, the speed of objects (e.g., roller coasters) relies solely on their height.

• Energy Formula in Elastic Forces: For elastic potential energy:
  $rac{1}{2} kx^2 + mv^2 = rac{1}{2} kx<em>f^2 + mv</em>f^2$

6-8 Other Forms of Energy and Energy Transformations; the Law of Conservation of Energy

• Types of Energy: Additional forms include electric energy, nuclear energy, thermal energy, chemical energy.

• Energy Transfer: Work is defined as the transition of energy between objects.

• Conservation Principle: In a closed system, the total energy remains constant, hence energy is conserved overall.

6-9 Energy Conservation with Dissipative Processes; Solving Problems

• Deductions with Friction: Nonconservative forces like friction transform kinetic and potential energy into heat. Actual temperature change can be quantified in terms of energy.

• Problem Solving Steps:

  1. Draw a picture.

  2. Identify the system for which energy conservation will apply.

  3. Specify what is being calculated; define initial and final positions.

  4. Establish a reference frame.

  5. Apply conservation of energy principles.

  6. Solve.

6-10 Power

• Definition of Power: The rate at which work is performed.

• Units of Power in SI: Expressed in watts (W); where
  $1 ext{ W} = 1 ext{ J/s}$

• Example: The distinction between walking versus running illustrates different power outputs despite similar changes in gravitational potential energy.

• Power's Role in Dynamics: Power is also critical for acceleration and in opposing gravitational force.

• Average Power Formula: Can be expressed with respect to force and velocity:
  $P = F imes v$

Summary of Chapter 6

• Work: $W = Fd ext{cos}( heta)$

• Kinetic energy: $KE = rac{1}{2} mv^2$

• Potential energy: Energy resulting from position or configuration.

• Net work done correlates with kinetic energy change.

• Conservation: If only conservative forces are present, mechanical energy is conserved.

• Power as a measurement of work rate.