Crash Course — Unit 7: Oscillations

• Equilibrium Position
  €” The point where an object would rest, with zero net force acting on it. This is the "home base" for any oscillation.
• Restoring Force
  €” The force that always pulls or pushes an object back towards its equilibrium position. It's what drives the oscillation.
• Simple Harmonic Motion (SHM)
  €” A specific type of oscillation where the restoring force is directly proportional to the displacement from equilibrium (like a spring, F = -kx).
• Amplitude (A)
  €” The maximum displacement from the equilibrium position. It's the "size" of the oscillation.
• Period (T)
  €” The time it takes to complete one full cycle of motion, measured in seconds. Think: from the far right, to the far left, and back to the far right.
• Frequency (f)
  €” The number of cycles completed per second, measured in Hertz (Hz). It's the inverse of the period (f = 1/T).
• Period's Independence
  €” For ideal springs and pendulums, the period does not depend on the amplitude. Pulling a swing back farther doesn't change the time it takes to complete a swing.
• Position, Velocity, & Acceleration
  €” These three are always out of sync. When the object is at its maximum displacement (amplitude), velocity is zero and acceleration is maximum. At equilibrium, acceleration is zero and velocity is maximum.
• Energy Conservation in SHM
  €” In an ideal system, the total mechanical energy (Kinetic + Potential) is constant. Energy just transforms back and forth between the two forms.
• Energy at Key Points
  €” Energy is all potential at the endpoints (amplitude) and all kinetic at the equilibrium position.

Key Formulas / Terms

• Period of a Mass-Spring System
  T_s = 2π√(m/k)

  • m is the mass, k is the spring constant.
• Period of a Simple Pendulum
  T_p = 2π√(L/g)

  • L is the length of the pendulum, g is the acceleration due to gravity.
• Period and Frequency Relationship
  T = 1/f
• Spring (Restoring) Force (Hooke's Law)
  F_s = -kx

  • k is the spring constant, x is the displacement from equilibrium.
• Elastic Potential Energy (Spring)
  U_s = ½kx²

Exam Traps

• Trap
  Believing amplitude affects the period. The exam loves asking, "If you double the amplitude, what happens to the period?"

  • Counter: Memorize this: for the systems in this course, period is independent of amplitude. Look at the period formulas for a spring (m, k) and pendulum (L, g)—there is no amplitude (A or x_max) in them. The period stays the same.
• Trap
  Mixing up the conditions for max/min velocity and acceleration. It's easy to think everything is max at the endpoints.

  • Counter: Use a playground swing as your mental model. You are fastest at the bottom (equilibrium), so v is max at x=0. You momentarily stop at the highest points (endpoints), so v=0 at x=A. The restoring force is strongest when you're farthest from the center, so acceleration is max at x=A.
• Trap
  Forgetting that a pendulum's period is independent of its mass. They'll ask you to compare the period of a heavy object and a light object on identical strings.

  • Counter: Look at the pendulum formula: T_p = 2π√(L/g). Mass (m) is not in the equation. As long as the length (L) is the same, a bowling ball and a baseball will have the same period.
• Trap
  Confusing total energy with potential or kinetic energy. A question might ask you to compare the total mechanical energy at the endpoint vs. the equilibrium point.

  • Counter: In an ideal system, total energy is always conserved. It's the same value at every point in the oscillation. Don't get tricked into thinking it's zero at equilibrium. At equilibrium, potential energy is zero, but kinetic energy is max, and their sum is the same total energy you had at the start.
• Trap
  Using the wrong period formula. In a time crunch, it's easy to grab the pendulum formula for a spring problem.

  • Counter: Before you write, say it in your head: "Springs have m and k. Pendulums have L and g." This simple check prevents you from using T_p = 2π√(L/g) for a block bouncing on a spring.