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Physics 1111: Conservation of Energy Study Notes

Definition of Work

  • The net effect of a force extbf{F} on an object as it undergoes a displacement extbf{A} is called the work, W, done on the object.

  • For constant force, work is defined as: W = extbf{F} ullet extbf{A} = F A ext{ cos}( heta)

    • Where:

    • extbf{F} = magnitude of the force

    • extbf{A} = magnitude of the displacement

    • heta = angle between extbf{F} and extbf{A}.

Total Work

  • For multiple forces acting on an object:
    W{ ext{total}} = W1 + W2 + W3 + … = extstyle extsum W

  • Alternatively, this can be expressed using the net force:
    W{ ext{total}} = extbf{F}{ ext{total}} ullet extbf{A} ext{ cos}( heta)

Kinetic Energy

  • Defined as the energy of moving objects:

    • K = rac{1}{2}mv^2

    • Where:

    • m = mass of the object

    • v = speed of the object.

  • Important note: Kinetic energy is a scalar quantity; there are no components.

Work-Kinetic Energy Theorem

  • States that when forces act on an object that displaces it, the net work done results in a change in kinetic energy:
    W{ ext{total}} = riangle K = rac{1}{2}mvf^2 - rac{1}{2}mv_i^2

Work Done by Springs

  • Work done by a spring is determined solely by the starting and ending positions (spring lengths) and is path-independent.

  • Formula: W{ ext{sp}} = - rac{1}{2}k xf^2 + rac{1}{2}k x_i^2

  • Springs represent a conservative force.

Conservative Forces and Potential Energy

  • A force is considered conservative if the work done depends only on initial and final positions, independent of path.

  • For conservative forces, it is possible to define a potential energy (PE):

    • Change in potential energy equals the negative work done by the force:
      riangle U = -W_{ ext{cons}}

Work Done by Gravity (Wg)

  • The work done by gravity is defined by the height difference between starting and ending points: W_g = -mg riangle y

    • Positive when moving downward, negative when moving upward.

    • Work is independent of path taken.

Potential Energy

  • The only defined difference in potential energy allows the zero level to be chosen arbitrarily.

  • Types of potential energy:

    • Gravitational Potential Energy:

    • Formula: U_g = mg riangle y

    • Set U = 0 where y = 0.

    • Spring Potential Energy:

    • Formula: U_s = rac{1}{2}kx^2 (with x as the displacement from equilibrium).

Conservation of Mechanical Energy

  • The total mechanical energy of a system is the sum of kinetic and potential energy:
    E_{ ext{mech}} = K + U

  • This includes:

    • Gravitational potential energy (U_g)

    • Spring potential energy (U_s)

  • Energy can transition between kinetic and potential forms but remains constant if only conservative forces do work: E_{ ext{mech}} = ext{constant}

    • Changes in energy:
      riangle E_{ ext{mech}} = riangle K + riangle U = 0

    • Equally states:
      K{ ext{before}} + U{ ext{before}} = K{ ext{after}} + U{ ext{after}}

Solving Problems using Energy Conservation

  1. Identify the system (draw a picture).

  2. Identify the object(s) in motion.

  3. Identify forces acting in the system.

  4. Determine if the forces are conservative.

    • The work done can reveal changes in potential energy: riangle U = -W.

  5. Conclude with mechanical energy conservation conditions:

    • riangle E_{ ext{mech}} = 0 = riangle K + riangle U.

  6. Set initial and final conditions for analysis.

  7. Gather information and proceed to solve the problem.

  8. Validation step: Confirm that the changes in kinetic energy align with expectations.

Example Scenario: Ball and Ramp

  • A frictionless ramp scenario:

    • Starts with initial speed v_0; reaches maximum height of 4.00m.

    • At the bottom, the path curves, changing direction upward.

  • Determine initial speed v_0 given the height of rise.