Unit 7: Oscillations

This unit focuses on analyzing systems that exhibit periodic motion, particularly those that undergo simple harmonic motion (SHM). Students learn how restoring forces, energy conservation, and calculus-based kinematic models govern oscillating systems like mass-spring setups and pendulums. The unit introduces critical concepts such as period, frequency, amplitude, and angular frequency, and it emphasizes how displacement, velocity, and acceleration vary over time. Connections to circular motion and Newton’s Second Law deepen students’ understanding of oscillatory behavior. By the end of this unit, students will be able to solve both conceptual and quantitative problems involving mechanical oscillations using calculus.

Periodic Motion Overview

Periodic motion refers to any type of motion that repeats itself in a regular, repeating cycle after equal time intervals. This includes circular motion, pendulums, springs, and even complex oscillations like sound waves or planetary orbits. The key feature of periodic motion is that the system returns to the same position and state after each complete cycle, allowing us to define measurable quantities like period and frequency.
The two most fundamental quantities in periodic motion are the period \( T \) and the frequency \( f \). The period is the time for one full cycle (measured in seconds), while frequency is the number of cycles per second: \( f = \frac{1}{T} \). These quantities are inverses of each other and form the foundation of analyzing any repeating system, including both mechanical and wave-based oscillations.
Not all periodic motion is smooth or ideal — it can be irregular, nonlinear, or driven by complex forces. For example, a planet orbiting the Sun follows periodic motion, but its speed and radius change continuously due to gravity. This contrasts with simpler systems like springs or pendulums under ideal conditions, which move in predictable, symmetrical patterns.
Simple Harmonic Motion (SHM) is a special case of periodic motion in which the restoring force is directly proportional to displacement and always points toward equilibrium. Mathematically, this is expressed as \( F = -kx \), leading to sinusoidal motion described by functions like \( x(t) = A \cos(\omega t + \phi) \). SHM allows us to model idealized systems with precise equations, making it central to AP Physics C.
In this unit, we will focus on systems where SHM applies: mass-spring systems, pendulums, and physical oscillators. These systems exhibit predictable relationships between position, velocity, acceleration, force, and energy. By understanding periodic motion, we gain a powerful framework to describe countless physical systems that oscillate, resonate, or vibrate in nature.

Kinematics of Simple Harmonic Motion (SHM)

Simple Harmonic Motion (SHM) is defined by a restoring force that is proportional to and opposite the displacement from equilibrium: \( F = -kx \). This leads to the differential equation \( a = -\frac{k}{m}x \), which has solutions that are sinusoidal in time. The resulting motion repeats in a smooth, regular cycle and can be fully described using trigonometric functions.
The position of an object in SHM as a function of time is typically written as \( x(t) = A \cos(\omega t + \phi) \), where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( \phi \) is the phase constant. The cosine function ensures that the object starts at maximum displacement when \( \phi = 0 \), but sine can also be used based on initial conditions. This expression gives the location of the object at any time \( t \), relative to equilibrium.
The velocity in SHM is the time derivative of position: \[ v(t) = \frac{dx}{dt} = -A\omega \sin(\omega t + \phi) \] The object moves fastest as it passes through equilibrium, where \( x = 0 \), and momentarily stops at the endpoints \( x = \pm A \). The direction of the velocity switches at the turning points, creating the back-and-forth motion characteristic of oscillations.
The acceleration is the derivative of velocity or the second derivative of position: \[ a(t) = \frac{dv}{dt} = -A\omega^2 \cos(\omega t + \phi) \] Since \( a(t) = -\omega^2 x(t) \), acceleration is always directed opposite to displacement. This directly reflects the restoring nature of the force and shows how SHM is a self-correcting system that oscillates about equilibrium.
The angular frequency \( \omega \) determines how rapidly the object oscillates, and it is related to physical properties of the system. For a spring-mass system, \( \omega = \sqrt{\frac{k}{m}} \), and for a simple pendulum, \( \omega = \sqrt{\frac{g}{L}} \) (for small angles). A higher \( \omega \) means a shorter period and faster oscillations, while a lower \( \omega \) means slower, more drawn-out motion.
The period \( T \) and frequency \( f \) are related to angular frequency by the equations: \[ T = \frac{2\pi}{\omega}, \quad f = \frac{1}{T} = \frac{\omega}{2\pi} \] These quantities determine the timing of the cycle and are essential for describing how long the object takes to complete one full oscillation. They are constant in ideal SHM and independent of amplitude.

Restoring Forces and Hooke’s Law

Simple Harmonic Motion arises from a restoring force — a force that always acts to bring the system back to its equilibrium position. The most common example is a spring, where the force is proportional to how far the spring is stretched or compressed from equilibrium. This restoring behavior leads to oscillatory motion when no energy is lost to friction or external forces.
Hooke’s Law mathematically defines the restoring force of a spring as \( F = -kx \), where \( x \) is the displacement from equilibrium and \( k \) is the spring constant. The negative sign indicates that the force always acts in the opposite direction of the displacement. A stiffer spring has a larger value of \( k \), meaning it requires more force to stretch or compress by the same amount.
This linear relationship between force and displacement is what makes SHM predictable and solvable using calculus. Since \( F = ma \), substituting Hooke’s Law gives \( a = -\frac{k}{m}x \), which is a second-order differential equation whose solutions are sinusoidal. This is the mathematical foundation for the position, velocity, and acceleration equations in SHM.
Not all restoring forces obey Hooke’s Law, but many systems approximate it for small displacements. For example, pendulums, molecular bonds, and some rotational systems act approximately like springs when disturbed from equilibrium. Understanding Hooke’s Law helps you recognize and model any system where the force increases proportionally with displacement in the opposite direction.
On AP problems, always check if the restoring force is proportional to \( x \); if it is, SHM methods can be applied. Be careful with units — the spring constant \( k \) is measured in newtons per meter (N/m), and \( x \) must be in meters. Hooke’s Law is also the foundation for potential energy in a spring, which links directly to energy conservation in oscillating systems.

Energy in Simple Harmonic Motion

In SHM, the total mechanical energy of the system remains constant (in the absence of non-conservative forces), and it shifts back and forth between potential and kinetic forms. At maximum displacement \( x = \pm A \), all the energy is stored as potential energy, and the object momentarily stops moving. At equilibrium \( x = 0 \), all the energy is kinetic, and the object moves at its maximum speed.
The potential energy stored in a spring is given by \( U = \frac{1}{2}kx^2 \), where \( x \) is the instantaneous displacement from equilibrium. This quadratic relationship shows that the further the object is stretched or compressed, the more energy is stored. This energy is always positive and becomes zero only when the object passes through the equilibrium position.
The kinetic energy at any point in the motion is \( K = \frac{1}{2}mv^2 \), where \( v \) is the instantaneous velocity of the object. Since \( v \) varies sinusoidally, so does the kinetic energy — it reaches a maximum at equilibrium and is zero at the turning points. These energy exchanges make SHM predictable and symmetrical over each cycle.
The total mechanical energy in SHM is constant and equal to the maximum potential or kinetic energy: \[ E_{\text{total}} = \frac{1}{2}kA^2 = \frac{1}{2}mv_{\text{max}}^2 \] This tells us that the amplitude \( A \) and mass \( m \) fully determine how much energy is in the system. Since there is no friction or damping in ideal SHM, this energy never changes over time.
At any point in time, the sum of kinetic and potential energy remains constant: \[ E = K + U = \frac{1}{2}mv^2 + \frac{1}{2}kx^2 \] This expression allows you to calculate unknown values (such as velocity at a given position) using energy conservation rather than kinematics. It’s especially useful when dealing with oscillators released from rest or reaching specific displacements.
Energy graphs in SHM typically show kinetic energy as a sine-squared curve and potential energy as a cosine-squared curve. These two curves are out of phase: when one increases, the other decreases, and they intersect at half the total energy. These graphs help visualize the energy transformations and reinforce the cyclical nature of SHM.

Frequency, Period, and Amplitude

The period \( T \) is the time it takes for an oscillating system to complete one full cycle of motion, and it is measured in seconds. The frequency \( f \) is the number of complete cycles per second, measured in hertz (Hz), and is the inverse of the period: \( f = \frac{1}{T} \). These values are constant for any ideal simple harmonic oscillator and describe the timing of the system’s oscillations.
The amplitude \( A \) is the maximum displacement of the object from its equilibrium position. It determines the total energy of the system but does not affect the period or frequency in ideal SHM. This is a key feature of SHM: the time it takes to complete a cycle is independent of how far the object is displaced, a property known as isochronism.
In a mass-spring system, the period depends only on the mass \( m \) and spring constant \( k \), given by: \[ T = 2\pi \sqrt{\frac{m}{k}}, \quad \omega = \sqrt{\frac{k}{m}} \] A larger mass slows down the oscillation (increases \( T \)), while a stiffer spring speeds it up (decreases \( T \)). The amplitude does not appear in this formula, reinforcing that it doesn’t affect timing.
For a simple pendulum of length \( L \) and small angle of oscillation, the period is: \[ T = 2\pi \sqrt{\frac{L}{g}}, \quad \omega = \sqrt{\frac{g}{L}} \] The pendulum’s period depends only on the length and gravity, not the mass or amplitude (as long as the angle stays small). This is another example of isochronous behavior in SHM.
Angular frequency \( \omega \) is related to the period and frequency by: \[ \omega = 2\pi f = \frac{2\pi}{T} \] It represents how rapidly the system moves through each cycle in radians per second. In equations for position, velocity, and acceleration, \( \omega \) controls how fast the sine or cosine functions oscillate.

The Spring-Mass System

The spring-mass system is the classic example of simple harmonic motion where a mass attached to an ideal spring oscillates horizontally or vertically. The restoring force is provided by the spring and follows Hooke’s Law: \( F = -kx \), where \( x \) is the displacement from equilibrium. The motion is sinusoidal, and the object oscillates symmetrically about the equilibrium position.
The angular frequency \( \omega \) and period \( T \) of the spring-mass system depend only on the mass and the spring constant: \[ \omega = \sqrt{\frac{k}{m}}, \quad T = 2\pi \sqrt{\frac{m}{k}} \] These relationships show that a more massive object oscillates more slowly, and a stiffer spring increases the oscillation frequency. The amplitude does not affect \( T \) or \( \omega \) in ideal SHM.
The position as a function of time is described by \( x(t) = A \cos(\omega t + \phi) \), where \( A \) is the amplitude and \( \phi \) is the phase constant. Velocity and acceleration are also sinusoidal and are 90° and 180° out of phase with displacement, respectively. These equations allow you to model the object’s full motion at any moment using calculus.
In vertical spring systems, the equilibrium position is shifted downward by the weight of the mass: \( mg = kx_0 \), where \( x_0 \) is the static stretch. The motion still obeys SHM but oscillates around the new equilibrium point. Once you account for this shift, all equations for horizontal motion apply identically to vertical motion.
Energy in the spring-mass system oscillates between potential and kinetic forms, with total energy constant throughout the motion: \[ E = \frac{1}{2}kA^2 = \frac{1}{2}mv^2 + \frac{1}{2}kx^2 \] This makes energy conservation a powerful tool for solving problems, especially when the position or speed of the mass is unknown. Many AP questions involve solving for energy, speed at a given location, or equilibrium extensions in vertical setups.

The Simple Pendulum

A simple pendulum consists of a small mass (called a bob) suspended from a string of fixed length and negligible mass, swinging back and forth under the influence of gravity. When displaced from equilibrium and released, the pendulum undergoes periodic motion due to the restoring force of gravity. If the angle of displacement is small (typically less than 15°), the restoring force is approximately proportional to displacement, making the motion simple harmonic.
The restoring force for a pendulum is the component of gravitational force acting tangentially to the arc: \( F = -mg \sin\theta \). For small angles, \( \sin\theta \approx \theta \) (in radians), so this becomes \( F \approx -mg\theta \), which mimics Hooke’s Law. This approximation allows us to treat the pendulum as a simple harmonic oscillator for small angular displacements.
The angular frequency and period of a simple pendulum depend only on the length of the string and the local acceleration due to gravity: \[ \omega = \sqrt{\frac{g}{L}}, \quad T = 2\pi \sqrt{\frac{L}{g}} \] The period is independent of mass and amplitude (under the small-angle approximation), which makes pendulums useful for timekeeping and demonstrating SHM principles.
The motion of the pendulum is sinusoidal in terms of angular displacement \( \theta(t) \), where \( \theta(t) = \theta_0 \cos(\omega t + \phi) \). The linear displacement of the bob along the arc is related to the angular displacement by \( s = L\theta \). This links angular and linear descriptions of motion and allows you to convert between them when needed in problem-solving.
Energy in the pendulum system shifts between gravitational potential energy and kinetic energy. At the maximum angle \( \theta = \theta_0 \), all energy is potential, and at the bottom \( \theta = 0 \), all energy is kinetic. As with spring systems, total mechanical energy remains constant in the absence of damping.

Angular SHM and Physical Pendulums

Angular Simple Harmonic Motion occurs when an object rotates back and forth around a fixed axis under the influence of a restoring torque. The angular version of Hooke’s Law is \( \tau = -k_\theta \theta \), where \( k_\theta \) is the angular spring constant and \( \theta \) is the angular displacement from equilibrium. Just like in linear SHM, this torque produces oscillatory motion governed by sinusoidal functions.
In terms of rotational inertia, Newton’s Second Law becomes \( \tau = I\alpha \), and combining this with the restoring torque gives: \[ I\alpha = -k_\theta \theta \quad \Rightarrow \quad \alpha = -\frac{k_\theta}{I}\theta \] This differential equation leads to angular SHM with angular frequency \( \omega = \sqrt{\frac{k_\theta}{I}} \), showing how mass distribution affects oscillation speed.
A physical pendulum is an extended rigid object (like a rod or a board) that swings about a pivot point, unlike a simple pendulum which assumes all mass is concentrated at one point. The restoring torque comes from the component of gravitational force acting on the center of mass: \( \tau = -mgd \sin\theta \), where \( d \) is the distance from the pivot to the center of mass. For small angles, \( \sin\theta \approx \theta \), so the motion becomes SHM.
The angular frequency and period of a physical pendulum are: \[ \omega = \sqrt{\frac{mgd}{I}}, \quad T = 2\pi \sqrt{\frac{I}{mgd}} \] These depend on the mass \( m \), distance \( d \), and moment of inertia \( I \) about the pivot point. Compared to a simple pendulum, the extended mass and shape of the object must be factored in when calculating its motion.
Physical pendulums demonstrate how real-world oscillators behave when mass is not concentrated at a single point. They help bridge the gap between rotational motion, torque, and oscillatory behavior, making them valuable for both conceptual and quantitative analysis. Many AP problems involve identifying the center of mass and computing \( I \) to determine the period of motion.

Damped and Driven Harmonic Motion

Damped harmonic motion occurs when energy is gradually lost from an oscillating system due to forces like friction or air resistance. The damping force usually depends on velocity and acts in the opposite direction, often modeled as \( F_{\text{damp}} = -bv \), where \( b \) is the damping coefficient. Damping causes the amplitude of oscillation to decrease over time, though the motion may remain periodic depending on the strength of the damping.
There are three main types of damping: underdamped, critically damped, and overdamped. In underdamped systems, the object continues to oscillate while gradually losing energy. Critically damped and overdamped systems return to equilibrium without oscillating, with critical damping being the fastest possible return without overshooting.
In driven harmonic motion, an external force is applied periodically to the system, usually in the form \( F_{\text{drive}} = F_0 \cos(\omega_{\text{drive}} t) \). If the driving frequency is close to the natural frequency of the system, resonance can occur, causing the amplitude to grow dramatically. The presence of damping limits this growth and prevents the amplitude from becoming infinite.
Resonance is a phenomenon where a system responds with maximum amplitude when the driving frequency matches its natural frequency. This is why systems like bridges, buildings, or musical instruments must be designed with resonance in mind. A small periodic input can lead to large oscillations if resonance is not controlled, potentially resulting in structural failure.
Damped and driven systems are not ideal SHM, but they still obey differential equations that can be solved using calculus. These systems model real-world behavior more accurately than ideal SHM because they account for energy loss and external input. Understanding these behaviors is essential for analyzing engineering systems and oscillations in nature.

Graphical Analysis of SHM

In Simple Harmonic Motion, the position-time graph is a sinusoidal wave that shows the object oscillating between \( +A \) and \( -A \), where \( A \) is the amplitude. The graph is typically described by \( x(t) = A \cos(\omega t + \phi) \) or \( x(t) = A \sin(\omega t + \phi) \), depending on initial conditions. This graph helps visualize the repeating nature and symmetry of the motion over time.
The velocity-time graph is also sinusoidal but is shifted 90° out of phase with the position graph. Velocity is zero at the turning points (maximum \( x \)) and greatest at the equilibrium position \( x = 0 \). The expression \( v(t) = -A\omega \sin(\omega t + \phi) \) reflects this phase difference and helps interpret motion direction and speed at any time.
The acceleration-time graph is another sinusoidal curve, but it is 180° out of phase with the position graph. When position is positive, acceleration is negative, and vice versa, as seen in \( a(t) = -A\omega^2 \cos(\omega t + \phi) \). This anti-phase relationship confirms that the acceleration always acts as a restoring force directed toward equilibrium.
Plotting energy vs. time reveals that kinetic and potential energy vary continuously but their sum remains constant. Potential energy \( U = \frac{1}{2}kx^2 \) reaches a maximum at maximum displacement, while kinetic energy \( K = \frac{1}{2}mv^2 \) peaks at equilibrium. These energy curves are always out of phase and cross at the points where energy is equally shared between kinetic and potential forms.
Understanding SHM graphs helps students connect motion equations to real physical behavior. These graphs provide insight into when the object moves fastest, changes direction, or experiences maximum force. They also make it easier to interpret AP free-response and multiple-choice questions involving qualitative analysis of motion.

Connection to Circular Motion and Calculus

Simple Harmonic Motion can be understood as the projection of uniform circular motion onto one axis. If a particle moves at constant speed in a circle, its horizontal or vertical position as a function of time traces out a sine or cosine wave. This geometric relationship explains why SHM equations involve trigonometric functions like \( x(t) = A \cos(\omega t + \phi) \).
The angular frequency \( \omega \) in circular motion corresponds directly to the angular frequency in SHM, both measured in radians per second. In the circular model, the particle rotates with constant angular velocity, and its shadow on a wall (if illuminated) oscillates just like a mass on a spring. This model makes it easier to visualize phase relationships between position, velocity, and acceleration.
Calculus is essential for analyzing SHM because position, velocity, and acceleration are all connected through derivatives. The velocity is the derivative of position: \( v(t) = \frac{dx}{dt} \), and the acceleration is the derivative of velocity: \( a(t) = \frac{d^2x}{dt^2} \). These relationships reinforce that SHM is governed by a second-order differential equation: \( a = -\omega^2 x \).
Integrating acceleration over time gives velocity, and integrating velocity gives position, which matches the antiderivatives of sine and cosine. This symmetry helps students apply calculus techniques to solve oscillation problems, including those involving initial conditions and phase constants. AP Physics C frequently asks students to use derivatives and integrals to analyze motion in SHM.
Connecting SHM to circular motion and calculus gives a deeper understanding of the mathematics behind oscillating systems. It shows that sinusoidal motion is not just empirical but arises from fundamental geometric and differential principles. This perspective bridges the gap between physics and mathematics and prepares students for more advanced applications in electromagnetism and wave physics.

Practice Problem 1: Spring-Mass Oscillator

A block of mass \( m = 2.0 \, \text{kg} \) is attached to a spring with spring constant \( k = 50 \, \text{N/m} \) and placed on a frictionless horizontal surface. The block is pulled \( 0.30 \, \text{m} \) from the equilibrium position and released from rest at time \( t = 0 \).

(a)Derive the position function \( x(t) \) for the block. Then find the angular frequency and period of the motion.

Since the block is released from rest at maximum displacement, its position function is: \[ x(t) = A \cos(\omega t), \quad A = 0.30 \, \text{m} \] The angular frequency is: \[ \omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{50}{2.0}} = \sqrt{25} = 5 \, \text{rad/s} \] So the position function is \( x(t) = 0.30 \cos(5t) \). The period is: \[ T = \frac{2\pi}{\omega} = \frac{2\pi}{5} \approx 1.26 \, \text{s} \]

Solution:

(b)At what time does the block first pass through the equilibrium position? What is the block’s speed at that moment?
The block passes through equilibrium when \( x = 0 \), which occurs when: \[ \cos(5t) = 0 \Rightarrow 5t = \frac{\pi}{2} \Rightarrow t = \frac{\pi}{10} \approx 0.314 \, \text{s} \] The speed at that moment is the maximum speed: \[ v_{\text{max}} = A \omega = 0.30 \cdot 5 = 1.5 \, \text{m/s} \]

Practice Problem 2: Physical Pendulum

A uniform rod of length \( L = 0.80 \, \text{m} \) and mass \( M = 3.0 \, \text{kg} \) is pivoted at one end and swings freely in a vertical plane. The angle of displacement is small enough to assume simple harmonic motion.

(a)Derive an expression for the period of oscillation of the rod and calculate its numerical value.
The moment of inertia of a rod about one end is: \[ I = \frac{1}{3}ML^2 = \frac{1}{3}(3.0)(0.80)^2 = 0.64 \, \text{kg} \cdot \text{m}^2 \] The distance to the center of mass is \( d = \frac{L}{2} = 0.40 \, \text{m} \), and the period is: \[ T = 2\pi \sqrt{\frac{I}{Mg d}} = 2\pi \sqrt{\frac{0.64}{(3)(9.8)(0.40)}} \approx 2\pi \sqrt{0.0544} \approx 1.47 \, \text{s} \]

(b)Determine the total mechanical energy of the system if the rod is released from rest at an initial angle of \( 10^\circ \).
The mechanical energy is purely potential at the release point: \[ U = Mg d (1 - \cos\theta_0), \quad \theta_0 = 10^\circ = 0.1745 \, \text{rad} \] \[ U = (3)(9.8)(0.40)(1 - \cos(0.1745)) \approx (11.76)(0.0152) \approx 0.179 \, \text{J} \] This is the total mechanical energy of the oscillating system.

Common Misconceptions – Unit 7: Oscillations

Confusing Period and Frequency
Many students incorrectly believe that the period and frequency are the same thing or use them interchangeably. In reality, period \( T \) is the time it takes for one full cycle (measured in seconds), while frequency \( f \) is how many cycles occur per second (measured in Hz). These two are reciprocals: \( T = \frac{1}{f} \). Confusing them can lead to errors when calculating angular frequency or energy. Always check units to avoid this mistake.
Assuming All Oscillations Are Simple Harmonic
Students often apply SHM equations (like \( x(t) = A \cos(\omega t) \)) to situations that aren’t true SHM. For a system to be simple harmonic, the restoring force must be directly proportional to displacement and directed toward equilibrium. If the motion involves damping, nonlinear forces, or large angles (e.g., pendulum >15°), it’s no longer SHM. Misapplying SHM equations in those cases will produce wrong results.
Ignoring Direction in Acceleration and Force
Some students treat acceleration and restoring force as always positive in magnitude, forgetting they are vectors. In SHM, acceleration is always directed opposite to the displacement — this is what makes it “restoring.” For example, if \( x \) is positive, then \( a = -\omega^2 x \) must be negative. Ignoring the direction can lead to incorrect graphs, sign errors in Newton’s laws, and contradictions in energy analysis.
Using Maximum Values at the Wrong Times
It’s common to incorrectly assume that all quantities (position, velocity, acceleration) reach maximum values simultaneously. In reality, velocity is zero when displacement is maximum, and acceleration is maximum at maximum displacement — but opposite in sign. Velocity reaches its maximum at the equilibrium point where displacement is zero. Knowing when each quantity peaks is essential for interpreting graphs and solving time-dependent equations.
Misunderstanding Energy Transfer in SHM
Students may think kinetic and potential energy are constant in SHM, which is false. The total mechanical energy is constant, but \( KE \) and \( PE \) continuously trade off. When the object is at maximum displacement, all energy is potential; when passing through equilibrium, it’s all kinetic. Graphing the changing energies can help students better visualize this dynamic energy transformation.