Welcome to Periodic Motion!

In this chapter, we are going to explore things that repeat. From the rhythmic swing of a grandfather clock to the steady orbit of the Moon around the Earth, the universe is full of patterns. Understanding periodic motion helps us design safer cars, more stable buildings, and even better musical instruments. Don't worry if it seems a bit "maths-heavy" at first—we'll break it down into simple steps and relate it to things you see every day!


1. Circular Motion: Moving in Circles

When an object moves in a circular path at a constant speed, something interesting happens: it is constantly changing direction. In physics, a change in direction is a type of acceleration, even if the speedometer stays the same!

Angular Speed (\(\omega\))

Instead of just measuring how many meters an object travels, we often measure how many radians (angles) it turns through per second. This is called angular speed.

The Formula: \(\omega = \frac{v}{r} = 2\pi f\)

Think of it like this: If you are on a merry-go-round, your linear speed (\(v\)) is how fast you'd fly off if you let go, but your angular speed (\(\omega\)) is how quickly you complete a full circle.

Centripetal Acceleration and Force

To keep an object moving in a circle, there must be a force pulling it toward the center. We call this the centripetal force. Without it, the object would just fly off in a straight line!

  • Centripetal Acceleration: \(a = \frac{v^2}{r} = \omega^2 r\)
  • Centripetal Force: \(F = \frac{mv^2}{r} = m\omega^2 r\)

Did you know? Centripetal force isn't a "new" type of force. It's just a label for whatever force is doing the pulling. For a planet, it’s gravity; for a car turning a corner, it’s friction; for a ball on a string, it’s tension.

Common Mistake to Avoid: Never draw "centrifugal force" (pushing outward) on your diagrams. In your A-Level exams, the only real force is the centripetal force acting toward the center.

Key Takeaway: Any object moving in a circle is accelerating toward the center, requiring a resultant force called centripetal force.


2. Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a special type of periodic motion. Think of a weight hanging on a spring. If you pull it down and let go, it bounces up and down. This is SHM.

The Golden Rule of SHM

For motion to be "Simple Harmonic," it must follow one strict rule: The acceleration must be proportional to the displacement and act in the opposite direction.

The Defining Equation: \(a = -\omega^2 x\)

What does the minus sign mean? It means the object is always being "pushed" back toward its starting (equilibrium) position. If you pull the spring down (\(+x\)), the acceleration is upward (\(-a\)).

The Graphs of SHM

Students often find the graphs tricky, but here is the secret: they are all related by gradients!

  • Displacement-time (\(x-t\)): Usually a cosine or sine wave.
  • Velocity-time (\(v-t\)): This is the gradient of the \(x-t\) graph. When displacement is zero, velocity is at its maximum!
  • Acceleration-time (\(a-t\)): This is the gradient of the \(v-t\) graph. It is always the "upside-down" version of the displacement graph.

Important SHM Formulas:

  • Displacement: \(x = A \cos(\omega t)\) (where \(A\) is the maximum displacement, called amplitude).
  • Velocity: \(v = \pm \omega \sqrt{A^2 - x^2}\)
  • Max Speed: \(v_{max} = \omega A\)
  • Max Acceleration: \(a_{max} = \omega^2 A\)

Quick Review Box: In SHM, at the center (equilibrium), velocity is maximum and acceleration is zero. At the edges (amplitude), velocity is zero and acceleration is maximum.


3. Simple Harmonic Systems

In your exams, you'll mainly look at two systems: the mass-spring system and the simple pendulum.

Mass-Spring System

The time it takes for one full oscillation (the period, \(T\)) depends on the mass and the stiffness of the spring (\(k\)).

Formula: \(T = 2\pi \sqrt{\frac{m}{k}}\)

Simple Pendulum

Surprisingly, the mass of the pendulum bob doesn't matter! Only the length of the string (\(l\)) and gravity (\(g\)) affect the time.

Formula: \(T = 2\pi \sqrt{\frac{l}{g}}\)

Note: This formula only works for small angles (usually less than 10 degrees).

Energy in SHM

Energy is constantly swapped between Kinetic Energy (\(E_k\)) and Potential Energy (\(E_p\)).

  • At the center, all energy is Kinetic.
  • At the maximum displacement, all energy is Potential.
  • The Total Energy stays the same throughout (assuming no friction).

Key Takeaway: The period of a pendulum depends only on its length, while the period of a spring depends on the mass and the spring constant.


4. Damping, Forced Vibrations, and Resonance

In the real world, things don't vibrate forever. Friction and air resistance eventually stop the motion.

Damping

Damping is when energy is taken out of an oscillating system. This reduces the amplitude over time.

  • Light damping: The amplitude gradually decreases over many cycles (like a pendulum in air).
  • Heavy damping: The system returns to equilibrium slowly without oscillating much.
  • Critical damping: The system returns to equilibrium in the shortest time possible without overshooting (used in car suspension!).

Free vs. Forced Vibrations

  • Free Vibration: You pluck a guitar string and let it ring at its natural frequency.
  • Forced Vibration: You keep "driving" the system with an external periodic force (like pushing a child on a swing repeatedly).

Resonance

Resonance happens when the frequency of the "driving force" matches the natural frequency of the system. When this happens, the amplitude of the vibrations increases dramatically!

Example: A singer breaking a wine glass with their voice. They sing at the exact natural frequency of the glass, causing the vibrations to become so large the glass shatters.

The Effect of Damping on Resonance: If you add damping to a resonating system, the "peak" on a graph becomes flatter and wider, and the maximum amplitude decreases.

Key Takeaway: Resonance occurs when the driving frequency equals the natural frequency, leading to maximum energy transfer and large amplitudes.


Summary of Periodic Motion

1. Circular motion requires a centripetal force (\(F = m\omega^2 r\)).
2. SHM is defined by acceleration being proportional to negative displacement (\(a = -\omega^2 x\)).
3. Systems like pendulums and springs have specific time period formulas.
4. Resonance is the "sweet spot" where a driving force creates the biggest vibrations.