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Simple pendulum review

Overview of key terms, equations, and skills for simple pendulums, including how to analyze the forces on the mass. 

Key terms

TermMeaning
Simple pendulumA mass suspended from a light string that can oscillate when displaced from its rest position.

Equations

EquationSymbolsMeaning in words
Tp=2πlgTp is period, l is pendulum length, and g is the acceleration due to gravityThe pendulum’s period is proportional to the square root of the pendulum’s length and inversely proportional to the square root of g

Analyzing the forces on a simple pendulum

An object is a simple harmonic oscillator when the restoring force is directly proportional to displacement.
Figure 1: A simple pendulum with length l, mass m, and displacement angle θ has a net restoring force of mgsinθ.
For the pendulum in Figure 1, we can use Newton's second law to write an equation for the forces on the pendulum. The only force responsible for the oscillating motion of the pendulum is the x-component of the weight, so the restoring force on a pendulum is:
F=mgsinθ
For angles under about 15°, we can approximate sinθ as θ and the restoring force simplifies to:
Fmgθ
Thus, simple pendulums are simple harmonic oscillators for small displacement angles.

Common mistakes and misconceptions

Sometimes people think that a pendulum’s period depends on the displacement or the mass. Increasing the amplitude means that there is a larger distance to travel, but the restoring force also increases, which proportionally increases the acceleration. This means the mass can travel a greater distance at a greater speed. These attributes cancel each other, so amplitude has no effect on period. The pendulum’s inertia resists the change in direction, but it’s also the source of the restoring force. As a result, the mass cancels out too.

Learn more

For deeper explanations, see our video introducing pendulums.
To check your understanding and work toward mastering these concepts, check out the exercise on period and frequency of simple pendulums.

Want to join the conversation?

  • starky sapling style avatar for user Armin
    Hi! I don't really know why you're using Pi 2 in the equation. What does pi have to do with that? I really want to know. Although im 13 i still love physics and math. Greetings from Armin.
    (10 votes)
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    • leafers sapling style avatar for user Yanissa Amadis
      The reason Pi is used is because if you think about the shape a pendulum makes as it swings back and forth, it makes a shape similar to a circular arc. When dealing with circles, you mainly deal with the value of Pi, as what my physics teacher explained to my class.
      (11 votes)
  • piceratops ultimate style avatar for user Saketh Koona
    In the equation for the period of the pendulum, is the value g always constant or can it change
    (2 votes)
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  • blobby green style avatar for user sooji8322
    Hello. I don't understand why
    this equation "F≈−mgθF" makes simple pendulums to be simple harmonic oscillators
    (2 votes)
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  • blobby green style avatar for user Lumbini
    In figure 1 isn't there a y-component to gravity as well? Which's responsible to balance tension out, which actually is added over by centrifugal force as well?
    (2 votes)
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    • hopper cool style avatar for user obiwan kenobi
      First of all, centrifugal force is not a thing. It is just an illusion that is due to the objects inertia. It is not an actual force. Second of all, the diagram is a free-body diagram. So, the solid lines show the forces but not the component of the forces. The dotted line shows the component of gravity that causes the motion. The thing is that, the force of tension is not balanced out. If it were, the ball would not travel in a circular path, it would just go in a straight line. The tension force is larger than the component of gravity that goes in the same direction, thus there is a net centripetal force causing circular motion. Hope this helps!
      (2 votes)
  • blobby green style avatar for user preetie.kotur
    Is the motion of the simple pendulum uniform or non-uniform
    (2 votes)
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  • blobby green style avatar for user Aboagye Leonard
    How do you find w,a and v
    (2 votes)
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  • blobby green style avatar for user nicole_muravsky122
    Can someone please explain this part? I don't quite understand it yet... "The pendulum’s inertia resists the change in direction, but it’s also the source of the restoring force. As a result, the mass cancels out too."
    (2 votes)
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  • blobby green style avatar for user kamchan.yim
    why can I write Tcos θ = mg?
    (1 vote)
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  • piceratops ultimate style avatar for user jerryma2004
    Hi,

    Is there a relationship that links the amplitude of the pendulum, theta (in radians), to the percentage difference in the period of a pendulum due to small-angle approximation?

    Thank you very much.
    (1 vote)
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  • blobby green style avatar for user pini.katia
    Hi,
    I am comparing the video analysis of a pendulum slowing down (with a small angle) and an electric fan also slowing down. Looking at the graphs I can say that the first is uniform circular motion (angular velocity is constant), while the second is non-uniform. However I do not get why. In both cases I do have a tangential acceleration, because I have a net force in the pendulum that is not the centripetal one. So what should be the difference between the two object moving?
    (1 vote)
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