Unit 7: Oscillations

Introduction to Periodic Motion

• Periodic motion refers to motion that repeats in regular intervals of time, such as the swinging of a pendulum or the vibration of a guitar string. Each full cycle of motion takes the same amount of time, called the period (\(T\)). The frequency (\(f\)) is the number of cycles per second, and it is related to the period by \(f = 1/T\).
• Displacement, velocity, and acceleration in periodic motion often follow sinusoidal patterns, making trigonometric functions essential for describing oscillations. These sinusoidal variations explain why oscillations are smooth and predictable rather than abrupt.
• Understanding periodic motion is crucial because many natural and engineered systems — from molecules vibrating in chemistry to electrical circuits oscillating in physics — can be modeled as oscillatory systems.

Simple Harmonic Motion (SHM)

• Simple harmonic motion occurs when the restoring force on an object is directly proportional to its displacement from equilibrium and directed toward that equilibrium point. The mathematical condition is \(F = -kx\) for springs, where the negative sign indicates the force opposes displacement.
• The acceleration of an object in SHM is not constant but instead varies with position. Since \(a = F/m = -\frac{k}{m}x\), the object accelerates more strongly when farther from equilibrium. This is why oscillations speed up near the edges and slow down near the center.
• Examples of SHM include masses on springs, pendulums at small angles, and even molecules vibrating in chemical bonds. Recognizing SHM allows us to predict motion using standard equations for period, velocity, and energy.

Spring-Mass Systems

• For a mass on a spring, Hooke’s Law applies: \(F = -kx\), where \(k\) is the spring constant and \(x\) is displacement from equilibrium. The system oscillates with a period given by \(T = 2\pi\sqrt{\frac{m}{k}}\), which shows that heavier masses oscillate more slowly, while stiffer springs oscillate more quickly.
• The energy in a spring-mass system alternates between kinetic and potential energy. At maximum displacement, all energy is stored as spring potential energy \(\frac{1}{2}kx^2\). At equilibrium, all the energy is kinetic (\(\frac{1}{2}mv^2\)), ensuring total mechanical energy remains constant if no friction acts.
• Spring-mass systems highlight connections between Newton’s laws and energy conservation. The restoring force is a direct consequence of Newton’s Second Law, while the smooth transfer between kinetic and potential energy reflects the conservation of mechanical energy.

Factors Affecting the Period of a Mass-Spring System

• If the mass of the object in a mass-spring system is increased, the period increases. This is because the formula \( T = 2\pi\sqrt{\frac{m}{k}} \) shows that period depends on the square root of mass. A larger mass increases inertia, causing the system to oscillate more slowly since it is harder to accelerate.
• If the spring constant \(k\) of the spring is increased, the period decreases. A stiffer spring produces a stronger restoring force for the same displacement, which increases the system’s acceleration. As a result, the system completes cycles more quickly, reducing the period.
• If the amplitude of oscillation is changed, the period remains the same. The period depends only on mass and spring constant, not amplitude, as long as the system stays within the simple harmonic motion approximation. This is why both small and large oscillations of a mass-spring system take the same amount of time per cycle.
• If the gravitational field changes, the period of a horizontal mass-spring system remains the same. Gravity does not affect the restoring force of the spring, since the spring force is independent of vertical weight in a horizontal setup. The period continues to depend only on mass and spring constant, making the system unaffected by changes in gravitational strength.

Pendulums

• A simple pendulum consists of a mass (the bob) suspended by a string or rod of length \(L\). When displaced slightly and released, it oscillates back and forth under the influence of gravity. For small angles (\(<15^\circ\)), the motion can be approximated as simple harmonic motion because the restoring force is proportional to displacement.
• The period of a simple pendulum is given by \(T = 2\pi\sqrt{\frac{L}{g}}\), which shows that the period depends only on the length of the pendulum and the acceleration due to gravity. This surprising result means the mass of the bob does not affect the period, as long as air resistance is negligible.
• Pendulums provide an important connection between oscillations and gravitational force. They demonstrate how restoring forces derived from weight components can produce predictable, periodic motion, a concept that connects back to Newton’s Second Law.

Energy in Simple Harmonic Motion

• In SHM, total mechanical energy remains constant as long as no non-conservative forces (like friction) act. Energy continuously shifts between potential and kinetic forms. For a spring-mass system, potential energy is \(U = \frac{1}{2}kx^2\), while kinetic energy is \(K = \frac{1}{2}mv^2\).
• At maximum displacement (amplitude), the velocity is zero, so all energy is potential. At equilibrium, displacement is zero, so all energy is kinetic. Between these extremes, the system smoothly exchanges energy, but the total remains constant.
• This principle connects directly to earlier units on energy conservation. Recognizing that oscillatory systems conserve energy provides powerful shortcuts in problem solving, since students can often use energy equations instead of solving complicated differential equations.

Graphical Analysis of SHM

• Graphs of displacement, velocity, and acceleration in SHM are sinusoidal but shifted in phase relative to one another. Displacement follows a cosine or sine curve, while velocity leads displacement by 90 degrees, and acceleration leads velocity by another 90 degrees. This phase relationship explains why maximum speed occurs at equilibrium while maximum acceleration occurs at the extremes.
• The amplitude of displacement graphs corresponds to the system’s maximum displacement, while the amplitude of velocity and acceleration graphs represent the maximum values of those quantities. These graphs make the motion more intuitive by showing how the system cycles through its motion smoothly.
• Graphical analysis ties back to kinematics, where students first encountered sinusoidal motion. By interpreting graphs, students can confirm whether energy conservation holds and predict system behavior at any point in time.

Damped and Driven Oscillations

• In real systems, oscillations often lose energy over time due to damping forces such as friction or air resistance. This causes the amplitude to decrease gradually, though the period remains nearly the same for weak damping. The rate of decay depends on the strength of the damping force.
• Driven oscillations occur when an external periodic force is applied to a system, supplying energy to counteract damping. At certain frequencies (called resonance frequencies), the amplitude of oscillation can become very large, sometimes causing dramatic effects like bridge collapses or shattering glass.
• Damped and driven oscillations connect oscillation theory to real-world engineering and safety concerns. Engineers must account for resonance in designs like skyscrapers, suspension bridges, and even musical instruments to ensure systems remain stable under external vibrations.

Frequency and Period of SHM

• The period (\(T\)) is the time required for one complete cycle of oscillation, while the frequency (\(f\)) is the number of cycles per second. They are related by \( f = \frac{1}{T} \). These quantities allow us to predict how quickly an oscillating system repeats its motion.
• For a spring-mass system, the period is given by \( T = 2\pi \sqrt{\frac{m}{k}} \), where \(m\) is mass and \(k\) is the spring constant. For a simple pendulum, the period is \( T = 2\pi \sqrt{\frac{L}{g}} \), showing that length and gravitational acceleration determine the timing of oscillations.
• These formulas highlight important connections: in spring systems, inertia (mass) slows oscillations while stiffness speeds them up. In pendulums, gravitational acceleration governs the period, which ties back to our earlier studies of free-fall and motion under gravity.
• The angular frequency of SHM is defined as \( \omega = \frac{2\pi}{T} = 2\pi f \), linking the motion’s timing to its sinusoidal description. This means that once we know the period or frequency, we can fully describe how fast the system oscillates using trigonometric functions.
• For the spring-mass system, increasing the spring constant makes the system oscillate faster because the restoring force is stronger. On the other hand, increasing the mass slows the oscillation because the system has more inertia to overcome. These effects mirror Newton’s Second Law, where acceleration depends on the ratio of force to mass.
• In pendulums, the period increases as length increases because the restoring torque is weaker for a longer pendulum. This provides a practical way of measuring gravitational acceleration by timing pendulum swings, connecting SHM directly to gravitational physics studied earlier.

Representing and Analyzing SHM

• Simple harmonic motion can be represented mathematically using trigonometric functions. A common form is \( x(t) = A \cos(\omega t + \phi) \), where \(A\) is amplitude, \(\omega\) is angular frequency, and \(\phi\) is the phase constant. This representation captures both the shape and timing of the oscillations.
• Velocity and acceleration can also be derived from this equation: \( v(t) = -A\omega \sin(\omega t + \phi) \) and \( a(t) = -A\omega^2 \cos(\omega t + \phi) \). These relationships show the sinusoidal nature of SHM and how displacement, velocity, and acceleration are phase-shifted by 90° from one another.
• Analyzing SHM through equations and graphs provides powerful tools for solving AP problems. Students can determine maximum velocity (\(v_{max} = A\omega\)) and maximum acceleration (\(a_{max} = A\omega^2\)), which connect directly to the energy and force relationships studied in earlier units.
• The phase constant \( \phi \) in the equation \( x(t) = A \cos(\omega t + \phi) \) accounts for where the oscillator is at time \(t = 0\). This allows us to describe any initial conditions, such as starting from maximum displacement or beginning at equilibrium with maximum velocity.
• By differentiating displacement with respect to time, we can show that velocity and acceleration are also sinusoidal but out of phase with displacement. This explains why a pendulum is fastest at the bottom and slowest at the extremes, and why acceleration always points back toward equilibrium.
• Graphical analysis of SHM helps confirm energy conservation: when displacement graphs show a maximum, velocity graphs cross zero, matching the idea that potential energy is maximum when kinetic energy is zero. This reinforces the connection between graphical data and energy principles from earlier units.

Energy of Simple Harmonic Oscillators

• In a simple harmonic oscillator, total mechanical energy remains constant and is shared between kinetic and potential energy. The total energy is given by \( E = \frac{1}{2}kA^2 \), where \(A\) is the amplitude. This expression shows that the energy depends only on amplitude, not on frequency or phase.
• Potential energy is maximum at maximum displacement (\(x = A\)), while kinetic energy is maximum at equilibrium (\(x = 0\)). This smooth exchange between energy forms ensures oscillations continue indefinitely in an ideal frictionless system.
• This concept ties SHM directly to energy conservation, a theme that has run through earlier units. By recognizing that amplitude controls the total energy, students can solve complex problems without tracking velocity or acceleration at every moment.
• The total energy of a simple harmonic oscillator can also be expressed in terms of angular frequency: \( E = \frac{1}{2}m\omega^2A^2 \). This shows how both the amplitude and angular frequency contribute to the system’s total energy, linking motion characteristics to energy conservation.
• In real systems, damping reduces total energy over time. Although the amplitude shrinks, the period often remains nearly the same for weak damping, which is why a swinging pendulum in air slows down gradually but still maintains regular timing for several cycles.
• SHM energy analysis provides a useful shortcut for problem solving. Instead of solving trigonometric equations for position or velocity, students can calculate system behavior by equating total energy to the sum of kinetic and potential energies at a given displacement.
• The total mechanical energy of an object in simple harmonic motion is the sum of its kinetic and potential energies. Kinetic energy depends on the object’s velocity, while potential energy depends on its displacement from equilibrium. Together, these two forms of energy account for the entire energy budget of the system.
• Due to conservation of mechanical energy, the total mechanical energy of an isolated system in SHM remains constant, assuming no energy is lost to friction or air resistance. This means that as one form of energy decreases, the other increases by the exact same amount. This principle ensures the motion is perfectly periodic in an ideal system.
• When the system has its maximum kinetic energy, its potential energy is at a minimum. This happens as the object passes through the equilibrium position, where velocity is greatest and displacement is zero. At this instant, nearly all the system’s energy is stored as kinetic energy.
• When the system has its maximum potential energy, its kinetic energy is zero. This occurs at the turning points of the oscillation, where the object is farthest from equilibrium. At these points, the object momentarily stops before reversing direction, so all energy is stored in potential form.

Position of an Object in Simple Harmonic Motion

• The position of an object in SHM is usually described by a sinusoidal function: \[ x(t) = A \cos(\omega t + \phi) \] or \[ x(t) = A \sin(\omega t + \phi) \] where \(A\) is amplitude, \(\omega\) is angular frequency, and \(\phi\) is the phase constant. This formula allows us to predict where the object is at any moment in time.
• The amplitude \(A\) represents the maximum displacement from equilibrium. At these points, the object changes direction, and its velocity is momentarily zero. This connects directly to the energy of the system, since potential energy is maximum at maximum displacement.
• The phase constant \(\phi\) determines the starting position of the object at time \(t = 0\). For example, if \(\phi = 0\), the object begins at maximum displacement. Changing \(\phi\) allows us to model systems that don’t start their motion at the same point in the cycle.
• Understanding position in SHM is crucial because it links to both velocity and acceleration. Velocity is the derivative of position, and acceleration is the derivative of velocity, which explains why all three quantities are sinusoidal but shifted in phase. This reinforces how the motion is smooth and predictable.

Example Problem 1: Spring-Mass System Released from Rest

Context: A 0.50 kg block is attached to a horizontal spring with a spring constant \( k = 80\, \text{N/m} \). The block is pulled to the right, stretching the spring by 0.15 m, and then released from rest. Assume a frictionless surface.

• (a) Calculate the period and frequency of the oscillation.
  The period is given by \( T = 2\pi\sqrt{\frac{m}{k}} = 2\pi\sqrt{\frac{0.50}{80}} ≈ 0.50\, \text{s} \)
  Frequency: \( f = \frac{1}{T} = 2.00\, \text{Hz} \)
• (b) Write the equation for the position \( x(t) \) of the block assuming it starts at maximum displacement at \( t = 0 \).
  Since the block is released from maximum displacement: \[ x(t) = 0.15\cos(2\pi ft) = 0.15\cos(4\pi t) \]
• (c) What is the maximum speed of the block during the motion?
  Maximum speed occurs at equilibrium: \[ v_{\text{max}} = A\omega = 0.15 \cdot \sqrt{\frac{80}{0.5}} = 0.15 \cdot \sqrt{160} ≈ 1.90\, \text{m/s} \]
• (d) Determine the total mechanical energy of the system and show it remains constant throughout the motion.
  Since energy is conserved, we can calculate total energy at max potential: \[ E = \frac{1}{2}kA^2 = \frac{1}{2}(80)(0.15)^2 = 0.90\, \text{J} \]
  This energy remains the same as KE and PE trade off during motion.
• (e) At what position is the kinetic energy equal to potential energy?
  In SHM, KE = PE when \( x = \frac{A}{\sqrt{2}} \) \[ x = \frac{0.15}{\sqrt{2}} ≈ 0.106\, \text{m} \]
  This occurs twice per cycle — once on each side of equilibrium.

Example Problem 2: Pendulum and Energy at Various Points

Context: A simple pendulum is made from a 1.2 kg mass hanging from a 1.5 m string. It is pulled to a small angle (so SHM applies) and released from rest. Assume \(g = 9.8\, \text{m/s}^2\).

• (a) Calculate the period and frequency of the pendulum’s motion.
  \[ T = 2\pi\sqrt{\frac{L}{g}} = 2\pi\sqrt{\frac{1.5}{9.8}} ≈ 2.46\, \text{s}, \quad f = \frac{1}{T} ≈ 0.41\, \text{Hz} \]
• (b) If the pendulum is pulled to a height of 0.20 m above equilibrium, what is the maximum speed during its swing?
  Use energy conservation: \( mgh = \frac{1}{2}mv^2 \Rightarrow v = \sqrt{2gh} = \sqrt{2(9.8)(0.20)} ≈ 1.98\, \text{m/s} \)
• (c) What is the angular displacement \( \theta \) (in radians) if the vertical height is 0.20 m?
  Use arc geometry: \( h = L(1 - \cos\theta) \Rightarrow \cos\theta = 1 - \frac{h}{L} = 1 - \frac{0.20}{1.5} ≈ 0.867 \Rightarrow \theta ≈ \cos^{-1}(0.867) ≈ 30^\circ ≈ 0.52\, \text{rad} \)
• (d) At what point in the swing is the tension in the string greatest? Justify.
  The tension is greatest at the lowest point of the swing because both gravitational and centripetal forces are acting upward. At that point, the string must supply enough force to support the weight and provide the necessary centripetal acceleration.

Common Misconceptions in Oscillations

• Many students mistakenly believe that the amplitude affects the period of a mass-spring system or a pendulum. In reality, the period of simple harmonic motion is independent of amplitude as long as the motion stays within the small-angle or linear approximation. This can be counterintuitive because we often expect larger swings to take longer, but for ideal SHM, that’s not the case.
• It’s a common error to think that gravity affects the period of a mass-spring system. The vertical position of the system may shift due to gravity, but it does not change the restoring force of the spring or alter the time it takes to complete a cycle. Only the mass and spring constant determine the period.
• Students often believe that velocity is greatest at maximum displacement, but this is false. In SHM, the object is momentarily at rest at the turning points (maximum displacement), and velocity is greatest when passing through equilibrium. This misunderstanding leads to incorrect energy or graph interpretations.
• Some students confuse when kinetic or potential energy is at a maximum. In fact, potential energy is maximum when the object is farthest from equilibrium, and kinetic energy is maximum at equilibrium. Forgetting this can lead to incorrect use of energy conservation formulas and misinterpretation of energy diagrams.
• In pendulum motion, students sometimes incorrectly assume that mass affects the period. While mass does influence energy and tension, the period of a pendulum is independent of mass and depends only on the length of the string and gravitational acceleration. This is a classic misconception that contradicts the experimental behavior of real pendulums.