It may be called the simple pendulum but deriving the equation for its motion isn't easy.
The trick is to realise that, for the pendulum to continue swinging as it moves through the equililbrium position, the tension in the string is greater than the weight of the mass.
Key Concepts
If a hanging ball is displaced to one side and released, it will swing backwards and forwards. This is because there is a restoring force that always points back towards the equilibrium position. If the swings are small, this force is proportional to the displacement, thus obeying simple harmonic motion.
Consider the forces acting on a pendulum bob. We can show that, for small swings, the acceleration is proportional to displacement and always directed towards the equilibrium position.
Use quizzes to practise application of theory.
START QUIZ!
Simple harmonic motion is when:
Specifically small amplitude swings
Which of the following are SHM
B has small amplitude so is appproximately SHM
Which pendulum has the biggest frequency?
C completes most cycles per unit time
Which pendulum has the longest time period?
D takes the longest time to complete a full cycle
Which pendulum has the largest amplitude?
The distance from the equilibrium position to maximum displacement is largest in D
In which pendulum will the tension in the strinmg reach the largest value?
This is a tricky one.
The tension is greatest at the bottom of the swing and depends velocity2/radius. D has the highest velocity but also the longest radius. However the higher velocity outweighs the larger radius so D has the biggest tension.
The simplest way to answer this is to use your intuition, which would be the most difficult to hold?
The pendulum below is swinging to the left
The direction of the resultant force is
The force will act towards the equilibrium position
The pendulum below is at the furthest point to the left
The resultant force is
Restoring force is proportional to displacement and directed towards the equilibrium position.
The pendulum below is moving to the right
The resultant force is
The pendulum bob is moving along the arc of a circle so there is a force towards the centre.
The pendulum bob below is at rest
The resultant force is
Forces are balanced
If the pendulum amplitude is very small we can assume that
In deriving the equation we assume that T = mg at the equilibrium position
The equation for the time period of a simple pendulum is
If the time period of a pendulum is 0.5s, The length of the string is increased by a factor 4 the time period will be
T is proportional to √4 so will be 2x bigger
The equation for the time period of a simple pendulum is
The time period of a pendulum is 0.5s, if the pendulum is taken to the moon the time period will be
g on the moon is about 1/6 that on Earth
T is proportional to 1/√g so T = 0.5 x √6
The equation for the time period of a simple pendulum is
Ihe time period of a pendulum is 0.5s, if the mass is doubled the time period will be
The time period does not depend on mass.
The equation for the time period of a simple pendulum is
The frequency of a pendulum is 1 Hz The length of the string is increased by a factor 4 the frequency will be
f= 1/T so f is proportional to 1/√L so if L is x4, f will be x 1/2
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