PhysicsLAB Resource Lesson
Energy Conservation in Simple Pendulums

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Remember the formula used to calculate the gravitational potential energy of a mass given its mass and height above an arbitrary zero level is
PEgravity = mgh
When a pendulum is pulled back from equilibrium through an angle θ, its height is calculated with the formula
pendulumheight.gif (3647 bytes)  pendulum.gif (3775 bytes)
h = L - L cos θ
where θ is the angular displacement
The formula used to calculate the kinetic energy of a massive particle is
KE = ½ mv2
In the absence of non-conservative forces, such as friction or applied, external forces, the mechanical energy in a system is conserved. That is
 During the swing of a simple pendulum, when does the bob possess maximum PE?

 During the swing of a simple pendulum, when does the bob possess maximum KE?

 During the swing of a simple pendulum, what is the magnitude of the bob's maximum velocity?

Another way of looking at conservation of energy is with the following energy diagram. As you can see,
  • the "purple" curve represents the pendulum bob's KE which during each cycle begins with an initial value of zero, increases to a maximum value, and then returns to zero
  • the "green" curve represents the PE of the bob which begins each cycle at a maximum value, then becomes zero as the bob passes through its equilibrium position, and returns to its maximum value
  • the "brown" line represents the total energy of the pendulum bob that always remains constant

 If a pendulum is initially released at an angle of 37º, at what angle will its PE and the KE be equal?

Refer to the following information for the next question.

At any intermediate position during the oscillation, the pendulum bob would have both PE and KE.
PEmax = PEintermediate + KEintermediate = KEmax
If the pendulum was released at point A, derive an expression for the pendulum's instantaneous velocity at point B, an intermediate position in its swing.

See the related lesson on vertical circles if you are asked to calculate the tension of the string during the pendulum's oscillation. Remember that a pendulum is merely the bottom half of a vertical circle! These conservation of energy methods are the easiest way to determine an object's speed so that tensions can be calculated.

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