General Method for Solving Energy Problems

(Physics 151 and 160)

Center for Academic Program Support, University of New Mexico

  1. Draw a Free Body Diagram (FBD). You may need it!!
  2. Write energy conservation equation. This is simply
    E(final) = E(initial)
  3. Identify the different forms of energy you'll be dealing with in your problem. The three forms you've encountered so far are:
    1. Kinetic Energy: ½ mv2
    2. Gravitational Potential Energy: mgh
    3. Spring Potential Energy: ½ kx2
      If ALL forms of energy are present in your problem, your statement of energy conservation, E(initial) = E(final) , becomes
      ½ mvi2 + mghi + ½ kxi2 = ½ mvf2 + mghf + ½ kxf2
      where vi, vf are initial, final velocity; hi, hf are initial, final height, and xi, xf are initial, final displacement of spring from equilibrium respectively, (and m is mass of object, of course)
  4. Use information from your specific problem to determine which of these quantities, if any, are zero.
  5. Friction: If friction is relevant to your problem, you need to adjust the law of energy conservation accordingly. How? Physically, we know object LOSE energy due to friction. So, instead of the final energy being equal to the initial energy (as it is in the case of no friction), it is now LESS than the initial energy:
    E(initial) - E(friction) = E(final) ,
    or, E(initial) = E(final) + E(friction) .

    [Note the mathematical equivalence of these 2 equations, and make sure the sign on the E(friction) term makes sense to you in both cases.]
  6. Finding E(friction). The work-energy theorem states that W = - DE. So, finding the work done BY friction is equivalent to finding the energy spent ON friction. Recall
    W = Fd cosq


    (note: 'F' and 'd' refer to the magnitudes of the force and distance, respectively).

    Frictional forces always oppose motion, or equivalently, they oppose the displacement vector, so the angle involved is typically 180°. Since cos 180° = -1, the work done by friction is a NEGATIVE quantity. Therefore, by the negative sign in the work-energy theorem, energy spent on friction is a POSITIVE quantity. This agrees with our formulas-we subtract a positive number from the initial energy, or add a positive number to the final energy. So...
    E(friction) = - W(friction) = - (m N) (d) cos q.


    You may need an FBD to find N, the normal force!!! (This is why the FBD should be your first step). Again, the cosine term comes out to be negative, so the overall quantity E(friction) is positive. …
  7. Solve the energy conservation equation, which now looks like
    ½ mvi2 + mghi + ½ kxi2 = ½ mvf2 + mghf + ½ kxf2 + E(friction)

    for your desired variable.



Created by Kate Smith