Kinetic vs. Potential Energy

Kinetic vs. Potential Energy: Physics 111 Notes

Theory & Concepts

• Mechanical Energy: The total energy (kinetic + potential) in a system.

  • Kinetic Energy (KE): Energy associated with motion.

  • Parameters:

    • Velocity ($v$)

    • Mass ($m$)

  • Formula:
    KE = rac{1}{2} mv^2

  • Potential Energy (PE): Energy stored based on an object's position.

  • Parameters:

    • Height ($h$) for gravitational potential energy.

    • Spring compression for elastic potential energy.

  • Types of Potential Energy:

    • Gravitational Potential Energy (GPE):

    • Formula:
      PE_g = mgh

    • Elastic Potential Energy (SPE):

    • Stored in springs.

    • Formula: PE_s = rac{1}{2} kx^2

      • Where $k$ is the spring constant, $x$ is the displacement from the equilibrium position.

    • Electric Potential Energy: Related to positions of charged particles.

• Work-Energy Theorem: Connects work done on an object to its kinetic energy change.

  • Formula:
    W{net} = riangle KE = KEf - KE_i

• Non-conservative Forces: Forces like friction which transform mechanical energy into other forms (e.g., heat).

  • Include non-conservative work in the theorem:
    W{net} + W{c} = riangle KE

  • Where $W_c$ is work done by non-conservative forces.
    } mv^2

Work-Energy Theorem

• Scalar Quantity:

  • Kinetic energy is measured in Joules (J).

  • Work-Energy Theorem Statement:

  • The net work on an object is equal to the change in the object's kinetic energy.

  • Important to consider nonconservative forces:
    W{net} + W{c} = riangle KE

  • When friction is present, adjust the equation accordingly:
    W{net} = KEf - KE_i

Gravitational Potential Energy

• Conceptualization:

  • Potential energy (PE) as a system property, especially in relation to gravitational forces acting on objects.

  • Update the work-energy theorem to include gravitational potential energy:

  • When dealing with gravitational work:
    Wg = F ext{cos}( heta) = mg(yi - y_f) ext{cos}(0°)

  • Incorporate modifications when applying non-conservative forces leading to overall mechanical energy changes:
    W{net} = W{net} + W_g = riangle KE

Conservation of Mechanical Energy

• Conservative Force Definition: Pressure/forces that behave independently of the path taken.

• Key Concepts:

  • Work done through a closed path leads to zero change in energy:
    riangle MME = 0

  • Mechanical energy conservation relationship:
    W_{net} = riangle KE + riangle PE

  • Where total mechanical energy $ riangle MME = KEf + PEf - (KEi + PEi)$ remains constant.

Elastic Potential Energy

• Spring Mechanics:

  • The strength of the spring's bounce-back relates to its compression, signifying increased potential energy.

  • Hooke's Law: States that restoring force of the spring, $F_s$, is proportional to the displacement from its rest position:

  • F_s = -kx

  • Work-Energy Relation:

  • Average spring force impacts work done over a distance:
    W_s = rac{1}{2}kx^2

  • The potential energy in the spring is defined as:
    PE_s = rac{1}{2} kx^2 $$

Problem-Solving Approach

• Steps:

  1. Read and elucidate the question.

  2. Define starting conditions and set a point of reference for conservation of energy.

  3. Write the conservation equation while identifying unknowns.

  4. Solve for the unknown value(s).