Q4.8 Conservative forces (K)

  1. Explain what is meant by a conservative force.
  2. A particle (free to move in only one dimension) experiences a conservative force F, a function of its coordinate x. Write down an equation defining the potential energy U(x), and express it in words.
  3. Write down an expression for the potential energy U(x) when
    1. the force is F(x)=-kx
    2. the force is F(x)=-λx3
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Hint

Look at the potential energy section of the course handbook (Key Point 3.10).
 

Solution

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Solution

  1. A conservative force is one for which the work done in moving an object between two points is independent of the path taken between the two points.
  2. A potential energy can only be defined for a conservative force. The potential energy is only defined up to a constant, so that only changes in potential energy are meaningful. The defining equation for the potential energy of our 1D system is (compare eqn Key Point 3.10)

    \[ \Delta U = - \int_{\rm start}^{\rm finish} F(x) dx \]

    Here F(x) is the conservative force and the ΔU notation emphasises that we only obtain the difference in potential energy between two points. We are free to choose the zero of our potential energy as we wish. Very often you will find books talking about potential energies U(x) without reference to the zero point, which is assumed. As long as everyone knows where the zero is, this causes no problems.

    In words: The potential energy difference of a body between two points is the negative of the work done by the conservative force in moving the body between the two points.

  3. The potential energy U(x) (dropping the Δ, and assuming the zero is at x=0) is
    1. For linear force:
      \[ U(x) = -\int F(x) dx = \int kx dx = \frac{1}{2} k x^2 \]
    2. For a cubic force
      \[ U(x) = -\int F(x) dx = \int \lambda x^3 dx = \frac{1}{4} \lambda x^4 \]