Study Notes on Conservation of Energy

Physics 1111: Conservation of Energy

Upcoming Assessments

  • Pre-class quizzes scheduled on:
    • 10/15 at 12 PM
    • 10/20 at 12 PM
  • Grades for Exam 2 will be posted before Wednesday's class.

Plan for Midterm 3

  • Week 10: discussion of Conservation of Energy and Power.
  • Week 11: discussion of Conservation of Momentum and Collisions.
  • Week 12: further examination of Collisions and Fall Break.
  • Week 13: will include review and Midterm Exam 3 scheduled for 11/7.
  • Note: Fewer pre-class quizzes during this portion due to overlaps in topics.

Today's Topics

  • Conservative Forces and Potential Energy
  • Conservation of Mechanical Energy (Approx. 50% weight on Midterm #3 grade)

Definitions and Theorems

Work

  • Definition of Work (W):
    • The net effect of a force F acting on an object as it undergoes a displacement A ext{r} is defined as the work done on that object.
    • Formula:
      W = F ullet A ext{r} = F imes |A ext{r}| imes ext{cos}( heta)
    • Where:
      • F = Magnitude of the force
      • A ext{r} = Magnitude of the displacement
      • heta = Angle between the force and the displacement vector.

Total Work

  • Total Work (Wtot):
    • For multiple forces acting on an object, the total work can be calculated by summing the individual work done by each force. This can also be expressed as:
    • Formula:
      W{ ext{tot}} = W1 + W2 + W3 + … = ext{Σ} W
    • An alternative representation using net force:
      W{ ext{tot}} = F{ ext{tot}} (A ext{r}) imes ext{cos}( heta)

Kinetic Energy (K)

  • Definition:
    • Often referred to as “moving energy” or “energy of moving objects.”
    • Variable:
      K denotes kinetic energy; m indicates mass of the object; v indicates speed of the object.
    • Key Concept: Kinetic energy is a scalar quantity without components.

Work-Kinetic Energy Theorem

  • This theorem postulates that the net work done on an object by one or more forces results in a change in the object’s kinetic energy.
  • Formula:
    • W{ ext{tot}} = riangle K = rac{1}{2} mvf^2 - rac{1}{2} mv_i^2
    • Where K = rac{1}{2} mv^2.

Work Done by Springs

  • The work done by a spring is dependent only on the starting and ending lengths of the spring, independent of the path taken.
  • Spring Work Formula:
    W{ ext{sp}} = - rac{1}{2} k xf^2 + rac{1}{2} k x_i^2
  • Springs are classified as conservative forces.

Conservative Forces and Potential Energy

  • Definition:
    • A force is considered conservative if the work done by the force depends only on the initial and final positions, not on the path taken.
  • For conservative forces, it is always possible to define potential energy.
  • Concept of Potential Energy:
    • Potential energy (U) is defined as a difference or change in energy.
    • The change in potential energy is equal to the negative of the work done by a conservative force:
      riangle U = -W_{ ext{cons}}.

Work Done by Gravity (Wg)

  • The work done by gravity depends on the vertical displacement (height) between starting and ending points.
  • Formula for Work Done by Gravity:
    • W_g = -mg riangle y
    • Positive work when moving downward:
    • W_g = +mg riangle y
    • Negative work when moving upward.

Potential Energy

  • Definition: The difference in potential energy is significant; we have the freedom to select the zero reference point as desired.
  • Types of Potential Energy:
    • Gravitational potential energy:
      riangle U = -W_g = mg riangle y
    • Choose U = 0 when y = 0:
      U_g = mgy
    • Spring potential energy:
      riangle U1 = -W1 = rac{1}{2} kxf^2 - rac{1}{2} kxi^2
    • Choose U = 0 at equilibrium (x = 0):
      U_s = rac{1}{2} kx^2.

Conservation of Mechanical Energy

  • Definition:
    • The sum of kinetic and potential energy in a system is termed mechanical energy.
  • Formula for Mechanical Energy: E_{ ext{mech}} = K + U
    • Here, K represents total kinetic energy and U symbolizes the potential energy in the system.

Conservation Principle

  • Kinetic and potential energy can convert into one another; however, if only conservative forces perform work, mechanical energy remains constant:
    E_{ ext{mech}} = ext{constant}.
  • The relation is:
    riangle E_{ ext{mech}} = riangle K + riangle U = 0
  • Equations can be set up for initial and final conditions:
    E{ ext{mech}} ext{ before} = E{ ext{mech}} ext{ after} \ K{ ext{before}} + U{ ext{before}} = K{ ext{after}} + U{ ext{after}}.

Problem Solving with Energy Conservation

  1. Identify the system (draw a picture).
  2. Identify the objects of interest (which are in motion).
  3. Identify the forces acting within the system.
  4. Determine if those forces are conservative.
    • Work done by a conservative force can be expressed as a change in potential energy: riangle U = -W.
  5. If only conservative forces are doing work, then: riangle E_{ ext{mech}} = 0 ightarrow riangle K + riangle U = 0
    • This can be expressed as:
      Kf + Uf = Ki + Ui
  6. Prospect initial and final conditions; isolate an important event or interval.
  7. Identify information, then solve.
  8. Check step: Does the change in kinetic energy riangle K logically align with expectations (i.e. should riangle K > 0, < 0, or = 0)?

Example: Ball and Ramp

  • In the example, an object starts down a frictionless ramp with an initial speed v_0. Upon reaching the bottom, the curvature of the ramp redirects the object upward to a maximum height of 4.00 m above the ground.
  • Given data includes:
    • Height reached: 4.00 m
    • Initial speed at top of ramp: v_0.
    • Additional context needed: Determine v_0 based on energy principles accounting for height change and conversion between potential and kinetic energy.