A simple pendulum is an idealized system consisting of a small mass suspended from a fixed point by a lightweight, inextensible string. When the mass is pulled away from its resting position and released, gravity causes it to swing back and forth in a repeating motion. This motion provides a clear example of oscillation, a fundamental concept in physics.
A simple pendulum helps explain several important ideas in mechanics, including period, frequency, restoring force, and simple harmonic motion. Its behavior also illustrates how the length of a pendulum and the strength of gravity determine the timing of its swing.
What is a simple pendulum?
A simple pendulum consists of three essential elements: a suspended mass, a string or cord, and a fixed support. The suspended mass is commonly called the bob. In the ideal model, the bob is treated as a point mass, the string has no mass and does not stretch, and the support remains stationary.
When the pendulum hangs vertically, the bob occupies its equilibrium position, where the forces acting on it produce no net force along the direction of motion. If the bob is displaced to one side and released, it begins to swing toward the equilibrium position.
Gravity provides the main driving mechanism for this motion. The bob’s weight acts vertically downward, while the string’s tension pulls toward the support. When the string is at an angle to the vertical, part of gravity acts along the arc of the swing, pulling the bob toward its lowest point. This component is called the restoring force because it tends to return the system to equilibrium.
As the bob moves downward, gravitational potential energy is converted into kinetic energy. At the lowest point, the bob reaches its greatest speed in an ideal swing. It then rises on the opposite side, converting kinetic energy back into gravitational potential energy. At the highest point on that side, its instantaneous speed becomes zero before it reverses direction.
In a real pendulum, air resistance and friction at the support gradually remove mechanical energy, causing the swing to become smaller. The ideal simple pendulum ignores these losses to make its motion easier to describe mathematically.
How a simple pendulum moves
The motion of a simple pendulum is constrained to an arc of a circle centered on its support. The string’s length determines the radius of that arc, while the bob’s position can be described by its angular displacement from the vertical.
Three forces and effects are especially important to understanding the motion. Gravity pulls the bob downward, tension keeps it at a fixed distance from the support, and the component of gravity along the arc accelerates it toward equilibrium. Tension acts perpendicular to the bob’s instantaneous direction of motion, so it does not directly change the bob’s speed in the ideal model.
The pendulum’s acceleration changes throughout a swing. At either endpoint, the bob momentarily stops, but the restoring force is greatest in magnitude for a given maximum angular displacement. As the bob moves toward the center, its speed increases. At the equilibrium position, the restoring force along the arc is zero, but the bob has its greatest speed and experiences centripetal acceleration directed toward the support.
After passing the equilibrium position, the restoring force points against the direction of travel, slowing the bob as it climbs the opposite side. The same sequence then repeats in reverse.
A complete oscillation occurs when the bob returns to its starting position with the same direction of motion. One trip from one side to the other is only half an oscillation.
What is the period of a simple pendulum?
The period is the time required for one complete oscillation. It is represented by the symbol T and is measured in seconds.
For a simple pendulum swinging through a small angle, the period is approximately
T = 2π√(L/g)
Here, L is the pendulum’s length, measured from the support to the bob’s center of mass, and g is the local acceleration due to gravity. Near Earth’s surface, g is approximately 9.81 meters per second squared.
This equation reveals two important relationships. First, the period increases with the square root of the pendulum’s length. A longer pendulum takes more time to complete a swing because the bob travels along a larger arc and has a lower characteristic acceleration for the same angular displacement.
Second, the period decreases as gravitational acceleration increases. Stronger gravity produces a stronger restoring acceleration, allowing the pendulum to complete its oscillations more quickly.
The square-root relationship means that doubling the length does not double the period. Instead, the new period is √2 times the original period, or about 1.41 times as long. To double the period, the length must be increased by a factor of four, assuming gravity remains constant and the small-angle approximation still applies.
For example, a pendulum 1 meter long has an approximate period of 2.01 seconds under standard near-Earth gravity. A complete oscillation therefore takes about two seconds, including the motion to one side and back again.
The period formula is independent of the bob’s mass. In the ideal model, a heavy bob and a light bob suspended from strings of equal length have the same period when they swing through small angles under the same gravitational conditions.
What is the frequency of a simple pendulum?
The frequency of a pendulum is the number of complete oscillations it makes per unit of time. Frequency is represented by f and is measured in hertz (Hz), where one hertz equals one oscillation per second.
Frequency and period are reciprocals of each other:
f = 1/T
A pendulum with a period of 2 seconds has a frequency of 0.5 Hz, meaning it completes half an oscillation per second on average, or one complete oscillation every two seconds.
Combining the frequency relationship with the small-angle period equation gives
f = (1/2π)√(g/L)
This expression shows that a longer pendulum has a lower frequency, while stronger gravity produces a higher frequency. The same relationships appear in the period equation, expressed from the perspective of how often the motion repeats rather than how long each cycle takes.
Period and frequency describe the same repeating behavior in different ways. Period answers the question, “How long does one oscillation take?” Frequency answers, “How many oscillations occur each second?”
The equations of motion for a simple pendulum
The equations of motion describe how a pendulum’s angular position and angular velocity change with time. To derive them, consider a bob of mass m attached to a string of length L, displaced by an angle θ from the vertical.
The component of gravitational force acting along the arc is directed toward equilibrium and has magnitude proportional to mg sin θ. Taking displacement to one side as positive, the tangential equation of motion is
mL(d²θ/dt²) = −mg sin θ
Dividing by mL gives the angular equation of motion:
d²θ/dt² + (g/L) sin θ = 0
The negative sign in the original equation expresses the restoring nature of gravity: the angular acceleration always points toward the equilibrium position for displacements within the ordinary swinging range.
This equation is nonlinear because sin θ is not generally equal to θ. Consequently, the exact motion of a pendulum at large angles is more complicated than simple harmonic motion.
For small angular displacements measured in radians, however, sin θ ≈ θ. The equation then becomes
d²θ/dt² + (g/L)θ = 0
This is the standard differential equation for simple harmonic motion. It describes a system whose acceleration is proportional to its displacement from equilibrium and directed in the opposite direction.
The corresponding angular frequency of oscillation, denoted by ω, is
ω = √(g/L)
Angular frequency is measured in radians per second. It is related to ordinary frequency by ω = 2πf and to the period by T = 2π/ω.
The small-angle solution for the pendulum’s angular position is
θ(t) = θ₀ cos(ωt + φ)
In this expression, θ₀ is the maximum angular displacement, t is time, and φ is the phase constant, which specifies the initial state of the motion. The cosine form is convenient when the bob is released from rest at its maximum displacement at time zero and φ is set to zero.
For that particular initial condition, the angular velocity is
dθ/dt = −θ₀ω sin(ωt)
Angular velocity describes how quickly the angle changes, including its direction. It is zero at the endpoints and reaches its greatest magnitude as the bob passes through equilibrium.
The bob’s tangential speed is related to angular velocity by v = L(dθ/dt). Its magnitude therefore varies throughout the oscillation, reaching its maximum near the bottom of the swing and falling to zero at the turning points.
These equations are approximate because they rely on the small-angle assumption. They are nevertheless extremely useful because they provide a simple mathematical description of the pendulum’s repeating motion.
Why the small-angle approximation matters
The familiar period equation works best when the pendulum swings through relatively small angles. The approximation sin θ ≈ θ becomes increasingly accurate as θ approaches zero, provided the angle is measured in radians.
At larger amplitudes, the restoring force no longer increases in direct proportion to angular displacement. The motion therefore departs from ideal simple harmonic motion, and the period becomes slightly longer than the small-angle formula predicts.
For an ideal pendulum released from rest at a maximum angle θ₀, a more accurate expression for the period can be written as a correction to the small-angle result:
T ≈ 2π√(L/g)(1 + θ₀²/16 + 11θ₀⁴/3072 + …)
The angle θ₀ must be expressed in radians. The additional terms account for the effect of increasing amplitude. This expansion is useful for moderate angles, but it is not a substitute for the full mathematical treatment at every possible amplitude.
The practical lesson is that the period of a simple pendulum is approximately independent of amplitude only when the swing is small. At larger angles, the amplitude matters, even if the pendulum’s length and the local gravitational acceleration remain unchanged.
Does the mass of a pendulum affect its motion?
In the ideal simple-pendulum model, the bob’s mass does not affect the period. This follows directly from the equation of motion.
The gravitational force along the arc is proportional to mass, and the bob’s inertia is also proportional to mass. When the equation is divided by the mass, the mass cancels. The resulting equation depends on the pendulum’s length and gravitational acceleration rather than on the bob’s mass.
This cancellation is consistent with the broader principle that objects in the same gravitational field undergo the same gravitational acceleration when other relevant forces are absent.
Real pendulums may show small differences due to effects excluded from the ideal model. Air resistance depends on factors such as the bob’s size, shape, speed, and surface properties. The distribution of mass also matters if the bob cannot be treated as a point mass. Such effects can influence the motion, but they do not change the basic prediction of the ideal simple-pendulum equations.
How energy changes during an oscillation
Energy provides another way to understand a pendulum’s motion. In an ideal system without friction or air resistance, the total mechanical energy remains constant, although energy shifts continuously between gravitational potential energy and kinetic energy.
At a turning point, the bob reaches its greatest height and has zero instantaneous speed. Its kinetic energy is zero, while its gravitational potential energy is greatest relative to the lowest point.
As the bob descends, gravitational potential energy decreases and kinetic energy increases. At the bottom of the arc, the bob has its maximum speed and maximum kinetic energy. It then rises again, slowing as kinetic energy is converted back into potential energy.
For a bob of mass m raised by a vertical height h, its gravitational potential energy relative to a chosen reference level is
U = mgh
Its kinetic energy is
K = ½mv²
In the ideal model, the sum of these energies remains constant:
E = mgh + ½mv²
The height depends on the angle of the pendulum. If the bob’s vertical position is measured relative to its lowest point, then h = L(1 − cos θ), giving the energy relation
E = mgL(1 − cos θ) + ½mL²(dθ/dt)²
This expression describes how the pendulum’s energy depends on its angular position and angular velocity. It also explains why the bob speeds up as it descends and slows down as it rises.
In a real pendulum, friction and air resistance gradually transfer some mechanical energy into thermal energy and other forms. As a result, the bob reaches progressively lower heights and the oscillations become smaller. This gradual reduction in amplitude is called damping.
How to calculate a pendulum’s period and frequency
The period and frequency equations can be used to predict the behavior of a pendulum when its length and local gravitational acceleration are known.
Suppose a pendulum is 0.50 meter long and swings through a small angle near Earth’s surface. Using g = 9.81 m/s², its approximate period is
T = 2π√(0.50/9.81) ≈ 1.42 s
The frequency is the reciprocal of the period:
f = 1/1.42 ≈ 0.70 Hz
The pendulum therefore completes approximately 0.70 full oscillations per second.
If the length is increased to 2 meters, the period becomes approximately 2.84 seconds, and the frequency falls to about 0.35 Hz. Quadrupling the length has doubled the period and halved the frequency.
These calculations assume small oscillations, a fixed support, a lightweight inextensible string, a point-like bob, and negligible energy loss. They are suitable for illustrating the basic relationships, although a real pendulum may differ slightly from the prediction.
What determines the motion of a real pendulum?
A real pendulum’s motion can be influenced by several factors beyond the ideal equations. Air resistance opposes motion and removes mechanical energy. Friction at the pivot can also dissipate energy, while a flexible or stretching string can alter the system’s effective behavior.
The location matters because gravitational acceleration varies slightly with altitude, latitude, and local differences in Earth’s mass distribution. Since the period is proportional to 1/√g, even modest changes in gravitational acceleration produce corresponding changes in the period.
The distribution of mass matters as well. A rigid object swinging around a pivot is generally described as a physical pendulum, not a simple pendulum. Its motion depends on its moment of inertia and the location of its center of mass, rather than on a single string length alone.
Despite these differences, the simple pendulum remains a valuable model. It captures the central mechanism of gravitational restoring motion and gives accurate predictions under well-defined conditions. By connecting forces, acceleration, energy, period, and frequency, it provides a foundation for understanding oscillations in many other physical systems.
