For the most part so far, we have considered motion that occurs once and does not repeat—a thrown ball, an accelerating car, a pushed crate. In Uniform Circular Motion, we saw our first example of repeating motion: a particle moving in a circular path returns to the starting point and performs the same motion over and over. Here, we will be applying the principles of mechanics to the special case of an oscillating object. From this point of view, this article is based on understanding a new and important situation based on material we have studied in previous articles.

You’ve probably seen a clock pendulum swing back and forth. You would have noticed that every swing followed the same path, and each trip back and forth took the same amount of time. This action is an example of vibrational motion. Other examples include a metal block bobbing up and down on a spring and a vibrating guitar string.

These motions, which all repeat in a regular cycle, are examples of periodic motion. In each example, the object has one position at which the net force on it is zero. At that position, the object is in equilibrium. Whenever the object is pulled away from its equilibrium position, the net force on the system becomes nonzero and pulls the object back toward equilibrium. If the force that restores the object to its equilibrium position is directly proportional to the displacement of the object, the motion that results is called simple harmonic motion.

Two quantities describe simple harmonic motion. The period, T, is the time needed for an object to repeat one complete cycle of the motion, and the amplitude of the motion is the maximum distance that the object moves from equilibrium.

Springs are familiar objects that have many applications, ranging from push-button switches on electronic components, to automobile suspension systems, to mattresses. In use, they can be stretched or compressed.

To stretch or compress a spring, a force must be applied to it. In accord with Newton's third law, the spring exerts an oppositely directed force of equal magnitude. This reaction force is applied by the spring to the agent that does the pulling or pushing. In other words, the reaction force is applied to the object attached to the spring. The reaction force is also called a "restoring force," for a reason that will be clarified shortly.

The back-and-forth motion of the spring then repeats itself, continuing forever, since no friction acts on the object or the spring. When the restoring force has the mathematical form given by $F_x = -kx$, the type of friction-free motion is designated as "simple harmonic motion." By attaching a pen to the object and moving a strip of paper past it at a steady rate, we can record the position of the vibrating object as time passes.

A simple pendulum consists of a particle of mass m, attached to a frictionless pivot P by a cable of length L and negligible mass. When the particle is pulled away from its equilibrium position by an angle $\theta$ and released, it swings back and forth. By attaching a pen to the bottom of the swinging particle and moving a strip of paper beneath it at a steady rate, we can record the position of the particle as time passes. The graphical record reveals a pattern that is similar (but not identical) to the sinusoidal pattern for simple harmonic motion.

Simple harmonic motion can be described in terms of displacememt, velocity, and acceleration, and a model consists of a small ball attached to the top of a rotating turntable is helpful in explaining these characteristics. The ball is moving in uniform circular motion on a path known as the reference circle. As the ball moves, its shadow falls on a strip of film, which is moving upward at a steady rate and recording where the shadow is. A comparison of the film with the paper reveals the same kind of patterns, suggesting that the model is useful.

Displacement

The reference circle (radius= A) indicates how to determine the displacement of the shadow on the film. The ball starts on the x axis at x = +A and moves through the angle $\theta$ in a time t. The circular motion is uniform, so the ball moves with a constant angular speed $\omega$ (in rad/s), and the angle has a value (in rad) of $\theta =\omega t$.The displacement x of the shadow is just the projection of the radius A onto the x axis:

$$x= A \cos \theta = A \cos \omega t$$

Velocity

The reference circle model can also be used to determine the velocity of an object in simple harmonic motion.

$$v_x= -v_T \sin \theta = -A \omega \sin \omega t$$

Acceleration

In simple harmonic motion, the velocity is not constant; consequently, there must be an acceleration. This acceleration can also be determined with the aid of the reference-circle model.

$$a_x= -a_c \cos \theta = -A \omega^2 \cos \omega t$$

Frequency of Vibration

With the aid of Newton's second law, it is possible to determine the frequency at which an object of mass m vibrates on a spring.

In simple harmonic motion, an object oscillates with a constant amplitude, because there is no mechanism for dissipating energy. In reality, however, friction or some other energy-dissipating mechanism is always present. In the presence of energy dissipation, the oscillation amplitude decreases as time passes, and the motion is no longer simple harmonic motion. Instead, it is referred to as damped harmonic motion, the decrease in amplitude being called "damping."

Driven Harmonic Motion

To set an object on an ideal spring into simple harmonic motion, some agent must apply a force that stretches or compresses the spring initially. Suppose that this force is applied at all times, not just for a brief initial moment. The force could be provided, for example, by a person who simply pushes and pulls the object back and forth. The resulting motion is known as driven harmonic motion, because the additional force drives or controls the behavior of the object to a large extent. The additional force is identified as the driving force.

Resonance

Resonance is the condition in which a time-dependent force can transmit large amounts of energy to an oscillating object, leading to a large-amplitude motion. In the absence of damping, resonance occurs when the frequency of the force matches a natural frequency at which the object will oscillate.