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
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 |
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
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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
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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.