Unit 6: Simple Harmonic Motion

📚Study Guide: Simple Harmonic Motion

Unit 6: Simple Harmonic Motion

Simple Harmonic Motion (SHM) is the repetitive, oscillatory motion exhibited by systems where the restoring force is directly proportional to the displacement from equilibrium and oppositely directed. This unit focuses on two canonical examples: a mass attached to an ideal spring and a simple pendulum swinging through small angles. For the spring, Hooke's Law (F = -kx) provides the linear restoring force that drives the oscillation. The resulting motion is sinusoidal, described by x(t) = A cos(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase constant determined by initial conditions. A crucial insight is that the period of a spring-mass system depends on the mass and the spring constant but is completely independent of the amplitude. This isochronism means a mass oscillating with a large amplitude takes exactly the same time per cycle as one with a small amplitude, assuming small enough displacement that the spring remains ideal. For the simple pendulum, the restoring force is a component of gravity, and the period T = 2π√(L/g) depends only on length and gravitational acceleration—not on the mass of the bob. However, this formula is only valid for small angles (typically θ < 15°) where sinθ ≈ θ in radians. Energy analysis in SHM reveals a continuous exchange between kinetic and potential energy. At maximum displacement, all energy is potential; at equilibrium, all energy is kinetic. The total mechanical energy E = ½kA² is constant if damping is negligible. The AP Exam frequently tests your ability to sketch x-t, v-t, and a-t graphs, to identify points of maximum speed or acceleration, and to use energy conservation to find speed at any position. You will also encounter the concepts of frequency, period, and angular frequency, and you must keep them distinct: ω = 2πf = 2π/T. Resonance—the dramatic increase in amplitude when a system is driven at its natural frequency—appears conceptually, explaining phenomena from bridge collapses to molecular vibrations.

Key Concepts

• Restoring Force Proportional to Displacement: The defining condition for SHM is F = -kx. The negative sign indicates the force always pushes the object back toward equilibrium.
• Amplitude, Period, and Frequency: Amplitude A is the maximum displacement from equilibrium. Period T is the time for one full cycle. Frequency f is the number of cycles per second.
• Spring-Mass System: The angular frequency is ω = √(k/m) and the period is T = 2π√(m/k). Notice mass matters, but amplitude does not.
• Simple Pendulum: For small angles, T = 2π√(L/g). The mass of the bob does not appear in the formula, but the small-angle approximation is essential.
• Energy in Oscillation: Total energy E = ½kA² is constant. At x = ±A, KE = 0 and PE = ½kA². At x = 0, KE = ½kA² and PE = 0.
• Phase and Initial Conditions: The phase constant φ is determined by where the oscillator starts at t = 0. Starting at maximum displacement gives φ = 0; starting at equilibrium with positive velocity gives φ = -π/2.
• Resonance: When a periodically driven system is driven at its natural frequency, energy transfer is maximized and amplitude grows dramatically.

Vocabulary

• Simple Harmonic Motion (SHM): Oscillatory motion in which the restoring force is directly proportional to displacement.
• Amplitude (A): The maximum displacement from the equilibrium position.
• Period (T): The time required to complete one full cycle of motion.
• Frequency (f): The number of complete cycles per unit time. Unit: hertz (Hz).
• Angular Frequency (ω): Related to frequency by ω = 2πf. Unit: radians per second.
• Restoring Force: A force that acts to bring a system back toward equilibrium.
• Equilibrium Position: The location where the net force on the oscillating object is zero.
• Resonance: The phenomenon of enhanced amplitude when a system is driven at its natural frequency.

Essential Formulas

• F = -k*x (Hooke's Law)
• a = -(k/m)*x
• x(t) = A*cos(ω*t + φ)
• ω = sqrt(k/m) (spring)
• T = 2π*sqrt(m/k) (spring)
• T = 2π*sqrt(L/g) (simple pendulum, small angles)
• E = ½*k*A² (total mechanical energy)
• v_max = A*ω

Common Mistakes

• Using the Wrong Formula for Pendulum Mass: Students often try to insert mass into T = 2π√(L/g). For a simple pendulum, mass cancels out and does not affect the period.
• Confusing Frequency and Angular Frequency: f is in Hz (cycles per second); ω is in rad/s. They differ by a factor of 2π. Do not substitute f where ω belongs.
• Forgetting the Small-Angle Approximation for Pendulums: The period formula T = 2π√(L/g) is only accurate when θ_max is small (under about 15°). Large-angle swings have longer periods.
• Thinking Amplitude Affects Period for Springs: The period of a spring-mass system is independent of amplitude. Do not include A in the period formula.

AP Exam Strategies

• Identify the Restoring Force First: Ask what pushes the object back to equilibrium. If it is linear (F ∝ -x), you have SHM and can use the standard formulas.
• Use Energy Conservation for Speed at Any Position: Instead of solving differential equations, set ½kA² = ½kx² + ½mv² and solve for v. This is faster and avoids phase headaches.
• Sketch the Graphs: Practice drawing x-t, v-t, and a-t curves. Remember v is the slope of x-t, and a is the slope of v-t. At maximum displacement, velocity is zero and acceleration is maximum.
• Remember Period Is Independent of Amplitude for Springs: If a question asks how period changes when amplitude doubles, the answer for a spring is "no change."

Real-World Applications

• Car Shock Absorbers: Springs and dampers in vehicle suspensions are tuned to oscillate at frequencies that isolate passengers from road bumps.
• Clock Pendulums: Grandfather clocks use the constant period of a pendulum to keep accurate time.
• Molecular Vibrations: Atoms in a molecule vibrate like simple harmonic oscillators, and spectroscopy measures these frequencies to identify substances.