Topic 7: Oscillations

An oscillation is a repeating back-and-forth motion about a stable equilibrium point. When the force pulling an object back is proportional to how far it has been displaced, the result is a clean, predictable motion called simple harmonic motion. This unit builds the tools to describe that motion in time, predict how fast it repeats, and track how energy shifts between kinetic and potential forms.

Simple Harmonic Motion and the Restoring Force

Simple harmonic motion (SHM) occurs whenever a system has a restoring force that always points back toward equilibrium and grows in proportion to the displacement from that equilibrium. Mathematically this is the relationship F = -k x, where x is the displacement and the minus sign signals that the force opposes the displacement. Push the object right and the force points left; push it left and the force points right. Because force and displacement are linked this way, the acceleration is also proportional to displacement and oppositely directed: the further out the object goes, the harder it is pulled back, which is exactly the condition that produces smooth, sinusoidal motion. Any system that obeys this proportional restoring-force rule, at least for small displacements, will oscillate with SHM.

Amplitude, Period, Frequency, and Angular Frequency

Four quantities describe the rhythm and size of an oscillation. The amplitude (A) is the maximum displacement from equilibrium, measured in m; it sets how big the swing is but, importantly, does not affect how long each cycle takes in ideal SHM. The period (T) is the time for one complete cycle, measured in s. The frequency (f) is the number of cycles per second, measured in Hz, and it is simply the reciprocal of the period: f = 1 / T. The angular frequency (w) packages the rate in radians per second and connects to the others through w = 2 * pi * f = 2 * pi / T. Angular frequency is what appears inside the sine and cosine functions that model the motion, so it is the most natural rate to work with when writing position as a function of time.

The Mass-Spring System and Its Period

A block of mass m attached to a spring of stiffness k is the textbook SHM system. The spring supplies the restoring force F = -k x directly, so the motion is simple harmonic for any displacement within the spring's elastic range. The period of a mass-spring oscillator is T = 2 * pi * sqrt(m / k). Read this carefully: a heavier mass increases the period because more inertia means slower acceleration, while a stiffer spring (larger k) decreases the period because it pulls the mass back more forcefully. Notice that amplitude does not appear in the formula, so stretching the spring further does not change the timing. The orientation does not matter either; a horizontal spring and a vertical spring (where gravity just shifts the equilibrium point) share the same period formula.

The Simple Pendulum and Its Period

A simple pendulum is a small mass on a light string of length L swinging under gravity. For small swing angles (roughly under 15 degrees), the component of gravity along the arc acts as a restoring force proportional to displacement, so the motion approximates SHM. Its period is T = 2 * pi * sqrt(L / g), where g is the acceleration due to gravity. Two surprises live in this formula. First, the mass of the bob does not appear, so a heavy and a light pendulum of equal length keep the same time. Second, amplitude again drops out for small angles, which is why pendulums make reliable clocks. A longer string lengthens the period, and a weaker gravitational field (such as on the Moon) also lengthens it.

Worked Example: Calculating a Period

Suppose a 0.50 m string holds a small bob and swings as a simple pendulum on Earth, where g = 9.8 m/s^2. To find the period, use T = 2 * pi * sqrt(L / g). First compute the ratio inside the root: L / g = 0.50 / 9.8 = 0.051 s^2. Take the square root: sqrt(0.051) = 0.226 s. Multiply by 2 * pi: T = 2 * 3.14159 * 0.226 = 1.42 s. So one full swing back and forth takes about 1.4 s. As a check, the frequency would be f = 1 / T = 1 / 1.42 = 0.70 Hz, meaning the pendulum completes a little under one cycle each second. The same step pattern works for a mass-spring system: plug into T = 2 * pi * sqrt(m / k) instead.

Energy in Simple Harmonic Motion

An ideal oscillator continually trades energy between two forms while keeping the total constant. At the extremes of the motion (displacement equal to the amplitude) the object is momentarily at rest, so its kinetic energy is zero and all the energy is stored as potential energy. As it rushes back through equilibrium, the potential energy falls to zero and the kinetic energy reaches its maximum, which is why the object moves fastest at the center. For a spring the stored energy is U = (1/2) k x^2, and the kinetic energy is KE = (1/2) m v^2. The sum E = KE + U stays the same throughout the cycle when friction is absent. Because total energy depends on amplitude (the maximum spring energy is (1/2) k A^2), a larger amplitude oscillation carries more energy even though it shares the same period.

Graphs of Displacement, Velocity, and Acceleration vs Time

Plotting SHM against time produces three linked sine-shaped curves. If displacement follows a cosine starting at the maximum, then velocity is a negative sine and acceleration is a negative cosine. The key feature is the phase relationship. Velocity is one quarter cycle out of step with displacement: when displacement is at a maximum, velocity is zero, and when displacement passes through zero at equilibrium, velocity peaks. Acceleration, on the other hand, is always exactly opposite to displacement, mirroring it across the time axis, which directly reflects the restoring relationship a = -(k/m) x. Reading these graphs, the slope of the displacement curve gives velocity, and the slope of the velocity curve gives acceleration, so the three curves are connected by successive slopes.