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:
    $W<em>{ ext{tot}} = W</em>1 + W<em>2 + W</em>3 + … = ext{Σ} W$
  • An alternative representation using net force:
    $W<em>{ ext{tot}} = F</em>{ ext{tot}} (A ext{r}) imes ext{cos}( heta)$

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:
  $E<em>{ ext{mech}} ext{ before} = E</em>{ ext{mech}} ext{ after} \ K<em>{ ext{before}} + U</em>{ ext{before}} = K<em>{ ext{after}} + U</em>{ ext{after}}$.

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:
     $K<em>f + U</em>f = K<em>i + U</em>i$
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.