Simple Pendulum

A child swaying on a swing set, a wrecking ball dangling at the end of a crane, a person’s arm swinging to toss a bowling ball: what do all these scenarios have in common? The periodic motion of a swinging mass is a core component of simple harmonic motion and oscillations. In this article, we’ll walk through the definition of a simple pendulum, the formulas we use to solve simple pendulum problems and how we find these, some real-world applications, and example problems. Let’s get started!

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    Defining the Simple Pendulum

    During your physics studies so far, you’ve likely encountered the concept of a pendulum before — let’s review what we mean by this.

    A pendulum is a weight that hangs from a fixed point and freely swings under the force of gravity.

    Simple Pendulum Classic clocks like grandfather clocks obey simple harmonic motion StudySmarter

    An old example of a pendulum is the swing of hanging weight inside a wall clock, Wikimedia Commons CC BY-SA 4.0

    Pendulums are generally fixed using a rigid rod, such as the weight suspended in the clock above. The weight swings back and forth, or oscillates, between two maximum points. So, we know what a pendulum is — what about a simple pendulum?

    A simple pendulum is an ideal pendulum where we consider all the mass to be at a point on the end. The line connecting the point mass to the axis of rotation cannot stretch and has no mass.

    In other words, a simple pendulum is composed of a mass concentrated on the end of an inelastic and massless string. Of course, this is idealized — massless objects cannot exist in real life, but many examples of pendulums can be modeled as simple pendulums to a sufficient degree of accuracy.

    The Motion of Simple Pendulums

    For many different topics, it’s often useful to identify a simplified version of a more complex form of motion in order to both understand the system and solve problems. So, how can we analyze the motion of a simple pendulum? Relying on what we know from kinematics alone would be difficult. However, if we can show that simple pendulums exhibit simple harmonic motion, we can apply the same tools we’ve learned from simple harmonic motion.

    Lets start by considering the case that a simple pendulum exhibits simple harmonic motion. This means that the restoring forceFrmust be proportional to the displacementd. Mathematically, we can write this relationship as:


    Frd.

    A simple pendulum exhibits rotational motion. In our case, this means the displacement is equal to the angle, orθtimes the radius. For our simple pendulum, the radius is just the length of the stringL:

    FrLθ.

    However, because the length of the string is constant, we’re only interested in the theta dependence of our pendulum. Therefore, we can drop the term for length from our proportionality, since we aren’t solving an equation just yet:

    Frθ.

    Now, we should have enough information to draw a free-body diagram and determine if our restoring force shows this relationship with the angle. The only forces we have acting on our simple pendulum are the force of gravity acting downwards, and the tension in the string keeping our pendulum held in its rotational motion.

    Simple Pendulum Free-body diagram of idealized pendulum under the forces of tension and gravity StudySmarterA simple pendulum free-body diagram, StudySmarter Originals

    We can split up our force of gravity into its x and y-components to get our restoring force. Gravity supplies the restoring force because, again, the tension in the string maintains the rotational motion of our mass.

    Simple Pendulum Free body diagram with x and y components of gravitational force and theta component of tension StudySmarterA pendulum free-body diagram with the individual components of the involved forces sketched out, StudySmarter Originals

    Here, the gravity in the x-direction is the restoring force, the force that acts against its direction of motion. Note here that no matter where our mass is in the pendulum’s arc, the tension is perpendicular to the motion. This is why it’s gravity that provides the restoring force and not the tension. Next, we need to solve forFgxin terms ofθ:

    sin(θ)=FgxFgFgx=Fgsin(θ)Fgx=-mgsin(θ).

    Here,mis the mass andgis the acceleration due to gravity. Thus, our restoring force is proportional tosin(θ), notθ! However, we have a powerful mathematical tool at our hands to simplify this relationship for us.

    The small-angle approximation is a rule stating that for small enough angles (in radians), the value ofsin(θ)is approximately equal to the angle, or:

    sin(θ)θ.

    In other words, this means we don’t need to evaluate thesin(θ)in a calculation as long asθis small. Therefore, for small angles, a simple pendulum indeed exhibits simple harmonic motion! Let’s walk through an example using the small-angle approximation to see this tool in action.

    Consider a pendulum with an angle of10degrees. Calculate the percent difference between the actual value of the restoring force and the estimated value using the small-angle approximation.

    To solve this, we can need to compare our approximate restoring force to our actual restoring force. Starting with our actual restoring force, we won’t approximate:

    Fgx=-mgsin(θ)Fgx=-mgsin(10°)Fgx=-mgsinπ18Fgx=-mg·0.1736

    Remember that we need to convert our angle into radians for this calculation. Next, we need to solve for our approximate restoring force, this time using the small-angle approximation:

    Fgx=-mg(10°)Fgx=-mg(π18)Fgx=-mg·0.1745.

    We’re tasked with finding the percent difference between these two calculations. The percent error is given by the following formula:

    %error=approximate value - exact valueexact value*100.

    Therefore, by plugging in our approximate and exact values for the restoring force, we find:

    %error=-mg·(0.1745) - (-mg)·0.1736-mg·0.1736*100%error=0.1745 - 0.17360.1736*100%error=0.1745 - 0.17360.1736*100%error=0.5184%.

    The percent error is only about half a percent. Conventionally, the small-angle approximation forsin(θ)holds for angles less than about fifteen degrees.

    Let’s take a look at the most important formulas we need to know for working with simple pendulums.

    Simple Pendulum Formulas

    Now that we’ve shown the small-angle approximation holds for simple pendulums as long as the angle isnt too large, we can relate simple pendulums to simple harmonic motion by exploring some of their properties. Lets first consider our new restoring force assuming the small-angle approximation:

    Fgx-mgθFgx-mgxLFgx-mgLx.

    Recall that simple harmonic motion obeys Hookes law, the theory that states a linear proportionality between the displacement and a constant factor of proportionality (such as stiffness) with the force:

    F=-kx.

    Thus, comparing the equation for the restoring force to Hookes law, we can see that our force constantkis:

    k=mgL.

    Now, we can apply all of our formulas for simple harmonic motion to a simple pendulum. Let’s start by considering the period, the time it takes to complete a cycle of motion. The periodTis given by:

    T=2πmk.

    The period is measured in units of time, most commonly seconds,s. Plugging in our value for the force constantkyields the period for a simple pendulum:

    T=2πmmgLT=2πLg.

    The frequency, or the number of occurrences of some periodic event per unit time, is simply the inverse of the period:

    ƒ=1Tƒ=12πgL.

    The frequency is measured in units of inverse time, most commonly inverse seconds,1s, also known as Hertz,Hz. Now, we can use the formula for the period of a pendulum to solve for the angular frequencyω, the frequency of a periodic event. Recall that the formula for the period can be given by:

    T=2πω.

    Thus, we find the angular frequency to be:

    ω=2πTω=2π2πLgω=gL.

    Angular frequency is usually measured in radians per second,rads. One interesting thing to note is that none of these formulas rely on the displacement angle or the mass of our pendulum. Within the bounds of the small-angle approximation, no matter how large of an amplitude we give our pendulums, the period and frequency remain the same, as long as the pendulum length remains constant!

    To summarize, here are the formulas you should be ready to use to tackle simple pendulum problems:

    • We can find the period of a simple pendulum using the formulaT=2πLg.
    • We find the frequency of a simple pendulum using the inverse of the previous formula,ƒm=12πgL(or more easily, the inverse of the answer you find for the period).
    • We can find the angular frequency for a simple pendulum using the formulaω=gL.

    Simple Pendulum Applications

    Applications of simple pendulums are more common in everyday life than you might think! We’ve already recognized that old clocks, particularly “grandfather clocks”, are a classic application of a pendulum in action. We also briefly considered that the back-and-forth motion on a swingset, a wrecking ball, or the throwing of a bowling ball can be approximated as simple pendulums. Each of these motions involves an approximately rigid attachment to a fixed point and a repetitive swing.

    What are some other applications? Consider the classic amusement park ride of a swinging pirate ship:

    Simple Pendulum Swinging boat ride at amusement park acts as a simple pendulum StudySmarter

    The swinging boat ride exhibits the swinging motion of a simple pendulum, Public Domain

    In this ride, the boat displays simple harmonic motion as it swings back and forth between two extreme heights about its rigid metal attachment to a central support beam. Metronomes, the tool musicians use to keep a precise rhythm and time as they play an instrument, also obey the behavior of a simple pendulum, with gravity acting on counterweights for the restoring force.

    Examples of Simple Pendulums

    Let’s work through an example solving for some variables of periodic motion given the length of a pendulum.

    Find the period and frequency of a simple pendulum2.0 mlong.

    In this problem, we just have to apply our formulas for the period and frequency of a pendulum. Lets start with the period:

    T=2πLgT=2π2.0 m9.8 ms2T=2.8 s.

    Knowing that frequency is just the inverse period, we find:

    𝑓=1T𝑓=12.8 s𝑓=0.36 Hz.

    Our units for frequency are Hertz, an inverse second, which describes the number of cycles the pendulum oscillates per second.

    Let’s walk through another example of a simple pendulum, this time examining how this type of motion differs in different surface gravity.

    Consider two different simple pendulums: one on the moon and one on Earth. Suppose the length of the pendulum on Earth is twice as long as the length of the pendulum on the moon. Find the frequency of the pendulum on the moon in terms of the frequency of the pendulum on Earth. Let the acceleration due to gravity on the moon be0.17times the acceleration due to gravity on Earth.

    To begin this problem, lets first consider the pendulum on Earth, with its motion governed by the lengthLEand the acceleration due to gravity on the surface of the Earthg. Using the formula for the period of a pendulumTand the relationship between period and frequency, we find:

    TE=2πLEgfE=1TEfE=12πgLE.

    The same equation applies to our pendulum on the moon, so we can simply take our previous result and plug in what we know:

    ƒm=12πgmLm.

    Next, we need to rewrite this in terms of the frequency of our pendulum on Earth. Recall the information we’re given about the lengths:

    LE=2LmLm=12LE.

    Similarly, we know the acceleration due to gravity on the moon expressed in multiples of Earth’s gravityg:

    gm=0.17·g.

    Let’s put all the information we’ve gathered so far together to solve forfmin terms offe:

    𝑓m=12π0.17g12LE𝑓m=12π0.34gLE𝑓m=12π0.34gLE𝑓m=12π0.34gLE𝑓m=0.58·12πgLE𝑓m=0.58𝑓E.

    Therefore, the frequency of the pendulum on the moon is0.58times the frequency of the pendulum on Earth.

    Simple pendulums are just another form of simple harmonic motion, a periodic motion that can be found in many different everyday scenarios. Although we make approximations to analyze these systems, our simplified calculations are still valuable for understanding this type of motion.

    Simple Pendulum - Key takeaways

    • A simple pendulum can be analyzed using simple harmonic motion if the angle of the pendulum swing is small enough.
    • Gravity provides the restoring force for a simple pendulum.
    • All the formulas applicable to simple harmonic motion are applicable to simple pendulums.
    • The period and frequency of a simple pendulum are independent of the mass and initial amplitude of the pendulum.
    Frequently Asked Questions about Simple Pendulum

    What is a simple pendulum?

    A simple pendulum is an idealized pendulum, or hanging mass with periodic motion, where we consider all mass to be concentrated at a point on the end of a massless, rigid, inelastic string or rod.

    Who invented the simple pendulum?

    The simple pendulum was not invented by a single person, though several applications of simple pendulums have been used. Astronomer and physicist Christiaan Huygens first patented the pendulum clock to improve timekeeping accuracy. Galileo theorized a pendulum clock prior but did not complete it. 

    What does a simple pendulum consist of?

    A simple pendulum consists of a hanging mass m, a string or rod with a constant length L, and a fixed pivot point P. A simple pendulum in motion displaces by a maximum angle θ restored by the force of gravity g.

    How to find the length of a simple pendulum?

    To find the length of a simple pendulum, rearrange the formula for the period T= 2π×√(L/g) of the pendulum and solve for L. The period must be a known quantity to find the length of the simple pendulum.

    What is the law of a simple pendulum?

    The law of a simple pendulum states that the period of a simple pendulum T is directly proportional to the square root of the length L.

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