Simple harmonic motion (SHM) is a type of repetitive motion in which an object moves back and forth around an equilibrium position under the influence of a restoring force proportional to its displacement and directed toward that equilibrium. It describes the idealized oscillations of systems such as a mass attached to a spring and, under certain conditions, a swinging pendulum.
Simple harmonic motion is important in physics because it provides a straightforward way to understand oscillations, waves, vibrations, and many other natural phenomena. Its mathematical relationships connect an object’s position, velocity, acceleration, and the forces acting on it.
What is simple harmonic motion?
Simple harmonic motion occurs when an object repeatedly moves away from an equilibrium position and returns toward it, following a predictable pattern. The defining feature is that the restoring force—the force that pulls or pushes the object back toward equilibrium—is directly proportional to how far the object has moved from that position.
For example, imagine pulling a spring attached to a wall and then releasing it. If the spring behaves ideally, it pulls the attached object back toward its resting position. The object passes through equilibrium, stretches or compresses the spring on the other side, and reverses direction. This cycle repeats.
The farther the object moves from equilibrium, the stronger the restoring force becomes. At the equilibrium position, the displacement is zero, so the restoring force is also zero in the ideal spring model. However, the object can still be moving rapidly as it passes through that point.
Simple harmonic motion is a specific kind of periodic motion, meaning motion that repeats after a regular interval. Not all periodic motion is simple harmonic motion. For SHM, the restoring force must have a linear relationship with displacement and point toward equilibrium.
The defining equation of simple harmonic motion
The central equation of simple harmonic motion describes the relationship between the restoring force and displacement:
F = −kx
Here:
- F is the restoring force, measured in newtons (N).
- k is the spring constant, which measures the stiffness of the spring and is measured in newtons per meter (N/m).
- x is the displacement from equilibrium, measured in meters (m).
- The negative sign indicates that the force acts in the direction opposite to the displacement.
This relationship is known as Hooke’s law when applied to an ideal spring.
Suppose an object attached to a spring is displaced 0.10 meter from equilibrium. If the spring constant is 20 N/m, the restoring force has a magnitude of 2 newtons and acts toward equilibrium.
If the displacement doubles while the spring remains within its elastic range, the restoring force doubles as well. This proportional relationship produces the characteristic oscillation of simple harmonic motion.
Hooke’s law describes the force for an ideal, linearly elastic spring. Other systems can exhibit SHM when their restoring force follows the same mathematical relationship, even if the force does not come from a physical spring.
The equation of motion
Newton’s second law states that the net force on an object equals its mass multiplied by its acceleration:
F = ma
Combining this law with the restoring-force equation gives:
ma = −kx
Dividing both sides by the mass produces the equation of motion:
a = −(k/m)x
This equation is fundamental to simple harmonic motion. It shows that acceleration is proportional to displacement and directed toward equilibrium.
The equation can also be written as:
a = −ω²x
Here, ω (the Greek letter omega) is the angular frequency, measured in radians per second. For a mass attached to an ideal spring,
ω = √(k/m)
Angular frequency describes how quickly the oscillation progresses through its cycle. A larger spring constant produces a higher angular frequency, while a larger mass produces a lower angular frequency.
The negative sign in the acceleration equation is essential. When an object is displaced to one side of equilibrium, its acceleration points back toward the center. When it passes to the other side, the acceleration reverses direction.
Position, velocity, and acceleration in simple harmonic motion
Simple harmonic motion has a sinusoidal pattern. Its position, velocity, and acceleration change continuously as the object moves through each cycle.
The position of an object undergoing SHM can be expressed as:
x(t) = A cos(ωt + φ)
In this equation:
- x(t) is the displacement at time t.
- A is the amplitude, or maximum displacement from equilibrium.
- ω is the angular frequency.
- t is the elapsed time.
- φ (phi) is the phase constant, which specifies the object’s starting point in its cycle.
The phase constant depends on the initial conditions, such as the object’s position and velocity when the clock starts. A cosine function is a convenient choice when the object begins at maximum displacement, but a sine function can describe the same motion with a different phase.
The velocity is the rate at which position changes with time:
v(t) = −Aω sin(ωt + φ)
The acceleration is the rate at which velocity changes with time:
a(t) = −Aω² cos(ωt + φ)
Because the position is A cos(ωt + φ), the acceleration equation can also be written as:
a(t) = −ω²x(t)
These equations reveal a consistent pattern. The object’s speed is greatest as it passes through equilibrium, where its displacement and acceleration are zero. At either endpoint of its motion, its velocity is momentarily zero, while the magnitude of its acceleration is greatest.
This relationship explains why an oscillating object slows as it approaches an endpoint, stops momentarily, and then reverses direction.
Amplitude, period, frequency, and angular frequency
Four quantities are especially useful for describing simple harmonic motion: amplitude, period, frequency, and angular frequency.
Amplitude is the maximum distance the object moves from equilibrium. A larger amplitude means the object travels farther from the center before reversing direction.
Period is the time required to complete one full oscillation. It is represented by T and measured in seconds. One full oscillation means returning to the same position while moving in the same direction.
Frequency is the number of complete oscillations per second. It is represented by f and measured in hertz (Hz). One hertz equals one cycle per second.
The period and frequency are reciprocals:
f = 1/T
T = 1/f
An oscillator with a period of 2 seconds completes one cycle every 2 seconds, giving it a frequency of 0.5 Hz.
Angular frequency describes the rate at which the oscillation advances through its cycle, expressed in radians per second. It is related to ordinary frequency by:
ω = 2πf = 2π/T
One complete cycle corresponds to 2π radians, which explains the factor of 2π.
Although these quantities are related, they describe different aspects of the motion. Amplitude describes the extent of the movement, period describes the time for a cycle, frequency describes how often cycles occur, and angular frequency expresses the cycle rate in angular units.
Examples of simple harmonic motion
Simple harmonic motion appears in several familiar systems. Some closely approximate ideal SHM, while others exhibit it only under particular conditions.
A mass attached to a spring
A mass attached to an ideal spring is one of the clearest examples of simple harmonic motion. Pulling the mass away from equilibrium stretches or compresses the spring, creating a restoring force.
When released, the mass accelerates toward equilibrium. Its speed increases as it approaches the center, then decreases after it passes through the center and moves toward the opposite endpoint.
For an ideal mass-spring system, the period is:
T = 2π√(m/k)
Here, m is the mass and k is the spring constant.
This equation shows that increasing the mass increases the period, making the oscillation slower. Increasing the spring constant decreases the period, making the oscillation faster.
For an ideal linear spring, the period does not depend on the amplitude. A small oscillation and a larger oscillation therefore take the same amount of time per cycle, provided the spring remains within its linear range and other ideal assumptions hold.
A swinging pendulum
A pendulum consists of a mass suspended from a fixed point so that it can swing under gravity. When the pendulum is displaced from its lowest position, gravity produces a restoring component of force that pulls it back toward equilibrium.
A pendulum does not exhibit exact simple harmonic motion at every angle. Its restoring force depends on the sine of its angular displacement rather than directly on the angle itself.
For small angles, however, the sine of the angle in radians is approximately equal to the angle. This approximation makes the restoring effect nearly proportional to displacement, allowing the pendulum to behave approximately like a simple harmonic oscillator.
For a simple pendulum with small oscillations, the period is:
T = 2π√(L/g)
Here, L is the pendulum’s length and g is the acceleration due to gravity.
A longer pendulum has a longer period and swings more slowly. A shorter pendulum has a shorter period and swings more quickly. Under the small-angle approximation, the period does not depend on the pendulum bob’s mass.
At larger angles, the motion departs more noticeably from ideal SHM, and the period becomes somewhat longer than the small-angle formula predicts.
Vibrating tuning forks
A tuning fork produces sound when its prongs vibrate after being struck. The prongs move back and forth around their equilibrium positions, and their vibrations can be approximated as simple harmonic motion when the displacements are small.
The vibrating metal transfers energy to the surrounding air, creating pressure variations that travel as sound waves. The vibration frequency determines the pitch of the sound: higher frequencies generally correspond to higher pitches.
Real tuning forks lose energy through sound production and other dissipative effects, so their vibration amplitude gradually decreases. Nevertheless, their motion can closely approximate SHM over much of the vibration.
Vibrations in molecules
Atoms in a molecule can vibrate around their equilibrium positions. When the atoms move slightly away from these positions, the forces between them can act approximately like springs, pulling them back toward equilibrium.
For small displacements, many molecular vibrations can be modeled as harmonic oscillators. This approximation helps scientists understand vibrational energy and how molecules interact with infrared radiation.
Actual molecular motion can be more complicated because atoms interact with one another, and the forces do not remain perfectly linear at larger displacements. Quantum mechanics is also necessary for a complete description of molecular vibrational energy.
Energy in simple harmonic motion
Simple harmonic motion involves a continuous exchange between kinetic energy and potential energy. In an ideal, undamped oscillator, the total mechanical energy remains constant.
Kinetic energy is the energy associated with motion. For an object of mass m moving at speed v, it is:
KE = ½mv²
Potential energy is stored energy associated with the object’s position. For an ideal spring, the elastic potential energy is:
PE = ½kx²
The total mechanical energy is the sum of kinetic and potential energy:
E = ½mv² + ½kx²
For an ideal mass-spring oscillator, the total energy can also be expressed in terms of amplitude:
E = ½kA²
At maximum displacement, the object momentarily stops, so its kinetic energy is zero. The spring’s potential energy is greatest because the displacement has reached its maximum magnitude.
At equilibrium, the spring’s potential energy is zero when measured relative to its equilibrium state. The object’s speed and kinetic energy are greatest at this point.
Between these positions, both forms of energy are present. As the object moves toward equilibrium, potential energy decreases while kinetic energy increases. As it moves away from equilibrium, kinetic energy decreases while potential energy increases.
In a real oscillator, friction, air resistance, or internal material losses gradually transfer mechanical energy into other forms, usually including heat and sound. The oscillation amplitude then decreases unless an external source supplies energy.
What determines the period of simple harmonic motion?
The factors that determine an oscillator’s period depend on the physical system.
For an ideal mass-spring system, the period depends on the mass and spring constant:
T = 2π√(m/k)
A heavier object oscillates more slowly because the same restoring force produces less acceleration. A stiffer spring produces a stronger restoring force for a given displacement, allowing the system to oscillate more quickly.
For a simple pendulum undergoing small oscillations, the period depends on its length and the local acceleration due to gravity:
T = 2π√(L/g)
The period increases with length and decreases as gravitational acceleration increases.
Amplitude behaves differently in these two examples. For an ideal linear spring, the period is independent of amplitude. For a pendulum, this independence is only approximately true at small angles. At larger amplitudes, the period increases.
These differences demonstrate why it is important to identify the physical system before applying a formula. There is no single period equation that applies to every oscillator under all conditions.
Simple harmonic motion versus periodic motion
Periodic motion is any motion that repeats after a fixed interval. Simple harmonic motion is a more specific category with a restoring force proportional to displacement and directed toward equilibrium.
A rotating wheel, for example, can complete identical rotations at regular intervals, making its rotation periodic. Its angular position does not generally behave like the position of an object undergoing SHM.
However, the projection of a point moving uniformly around a circle onto a diameter follows simple harmonic motion. If the point moves around the circle at constant angular speed, its projected position changes sinusoidally.
This connection between circular motion and SHM is useful because it provides a geometric way to understand oscillations. The circular motion has a constant angular speed, while its projection repeatedly speeds up, slows down, and reverses direction along a straight line.
Not every repeating movement is harmonic. A bouncing ball, for instance, has a repeating pattern under some conditions, but its motion is not SHM because impacts interrupt its motion and the restoring force does not remain proportional to displacement.
When simple harmonic motion is only an approximation
Simple harmonic motion is an ideal model, and real systems do not always follow it exactly.
The model works best when the restoring force is approximately proportional to displacement, the oscillations are small enough to remain within the system’s linear range, and energy losses are negligible or can be ignored.
Springs can become nonlinear when stretched or compressed too far. Pendulums deviate from SHM at larger swing angles. Vibrating structures may have complicated modes of motion, and friction or air resistance can reduce their amplitude over time.
A damped oscillator loses energy as it moves. If the damping is weak, the object may continue to oscillate with a gradually decreasing amplitude. In this case, the motion resembles SHM but is not the ideal, constant-amplitude form.
Some systems are driven by an external periodic force. Their motion depends on both the system’s natural properties and the applied force. When the driving frequency approaches a natural frequency, the response can become especially large under suitable conditions, a phenomenon known as resonance.
Despite these limitations, SHM remains a powerful model. It captures the essential behavior of many oscillating systems and often provides an accurate starting point for understanding more complicated motion.
Why simple harmonic motion matters in physics
Simple harmonic motion provides a foundation for studying vibrations, sound, wave behavior, and the dynamics of mechanical systems. Its equations show how forces produce acceleration, how energy moves between kinetic and potential forms, and how mass and stiffness determine an oscillation’s timing.
The same mathematical structure appears in many different physical settings. A spring, a small-angle pendulum, a vibrating tuning fork, and a molecule’s small vibrations may look very different, but each can be approximated by the same basic equations when its restoring behavior is sufficiently close to linear.
The essential idea is that an object displaced from equilibrium experiences a restoring influence that pulls it back, with greater displacement producing a proportionally stronger response. That relationship creates the repeating, sinusoidal motion that defines simple harmonic motion and makes it one of the most useful models in classical physics.
