Conservation of Energy Study Notes

Conservation of Energy Overview

• Definition of Conservation of Energy: The principle stating that the total energy in a closed system remains constant over time.

Formulas Recap

• Mechanical Work (W)

  • Notation: Capital W
    W = F imes d imes ext{cos}( heta)
  • Where:
    • F = Force applied
    • d = Displacement of the object
    • θ = Angle between the force and displacement directions

• Kinetic Energy (K.E.)

  • Notation: Capital K.E.
    K.E. = rac{1}{2} m v^2
  • Where:
    • m = Mass of the object
    • v = Velocity of the object

• Work-Energy Theorem

  • States that the work done on an object is equal to the change in its kinetic energy:
    W = riangle K.E. = K.E._ ext{final} - K.E._ ext{initial}

• Potential Energy (P.E.) :

  • Gravitational Potential Energy (G.P.E.)

  • Formula:
    G.P.E. = mgh

    • Where:
    • m = mass
    • g = acceleration due to gravity (approximately 9.8 m/s² or 10 m/s²)
    • h = height

  • Elastic Potential Energy

  • Notation: Capital P.E. (elastic)

  • Formula:
    P.E. = rac{1}{2} k x^2

    • Where:
    • k = spring constant (elastic constant)
    • x = displacement from the equilibrium position of the spring

• Total Mechanical Energy (T.E.)

  • Notation: Capital E
    E = K.E. + G.P.E. + P.E.
  • Formula:
    E = rac{1}{2} mv^2 + mgy + rac{1}{2} kx^2
  • Represents the total mechanical energy of a system which is conserved in the absence of non-conservative forces (like friction).

Conservation Principles

• Total mechanical energy is conserved in the absence of non-conservative forces:

  • Formula 7 (No Friction): E1 = E2
    • Where E1 is the total mechanical energy at the initial situation and E2 at the final situation.

• Total mechanical energy does not equal final energy when non-conservative forces like friction are present:

  • Formula 8 (With Friction): E1 + W{ ext{friction}} = E_2
    • Where W_friction is the work done by friction, which is typically negative.

Problem-Solving Strategy

1. Draw the Situation - Visual representation helps clarify the problem.
2. Identify Initial and Final Situations - Label these as Situation 1 and Situation 2 (or A and B).
3. Determine Zero Level for Gravitational Potential Energy - This simplifies calculations (often at ground level).
4. List Known and Unknown Quantities - Identify values you have and the quantities you need to solve for.
5. Select the Appropriate Equation - Use equations based on whether friction is present or not.
6. Solve for the Unknown - This involves rearranging the equation based on your known values.

Example Problems

• Incline Problem (No Friction):

  • Given an object sliding down an incline without friction, calculate its final speed.
  • Initiate by defining initial and final states, ensuring clarity on heights and speeds.
  • Use conservation of energy and solve for the unknown final speed using the derived formulas. For example, solving from a height of 10m results in:
    v_2 = ext{sqrt}(2gy) = ext{sqrt}(2 imes 10 imes 10) = 14.14 ext{ m/s}

• Dropped Object Problem:

  • An object dropped from 100m, with gravity at 10m/s².
  • Determine speed before it hits the ground:
    • Potential energy at top transforms to kinetic energy before impact, simplifying yields:
      vb = ext{sqrt}(2g ya) = ext{sqrt}(2 imes 10 imes 100) = 44.72 ext{ m/s}

• Spring Collision Problem: An object colliding with a spring to find spring constant.

  • Velocity before collision provides kinetic energy that transforms to elastic potential energy:
  • Set kinetic energy equal to potential energy, rearrange for k:
    k = rac{mv^2}{x^2}
  • Parameters include mass, initial velocity, displacement from equilibrium.

Conclusion

• Understanding and applying the conservation of energy principle simplifies problem-solving in mechanics by allowing the use of energy methods instead of force-based approaches. Following a clear strategy makes it feasible to solve both simple and complex problems efficiently.