Study Notes: Oscillating Systems (3.5.1)
Hello future physicist! Welcome to the world of oscillations. Don't worry if this chapter seems tricky at first—it’s all about things wiggling and wobbling! From the swing of a clock pendulum to the vibrations in a guitar string, oscillations are fundamental to how the universe works. Understanding them is key to mastering waves later on.
Let's dive into the core concepts of periodic motion!
1. Understanding Periodic Motion
An oscillation is simply repetitive movement about a central point. When this movement takes the same amount of time for each complete cycle, we call it periodic motion.
Key Definitions for Oscillations
- Equilibrium Position: The point where the net force acting on the oscillating object is zero. This is the 'rest' position.
- Displacement (\(x\)): The distance of the object from its equilibrium position at any given moment.
- Amplitude (\(A\)): The maximum displacement from the equilibrium position. It tells you how big the swing is.
- Period (\(T\)): The time taken for one complete cycle (e.g., from one side, across the equilibrium, to the other side, and back to the start). Measured in seconds (s).
- Frequency (\(f\)): The number of complete cycles per second. It is the reciprocal of the period: \(f = \frac{1}{T}\). Measured in Hertz (Hz) or \(\text{s}^{-1}\).
Memory Aid: Period (T) is time taken for one, Frequency (f) is how frequent (many) per unit time.
2. Classic Simple Harmonic Systems
Two systems that demonstrate the characteristics of simple harmonic motion (SHM) are required in this chapter. While a full mathematical treatment of SHM acceleration is usually A-level content (Section 3.6.2), you must know the formulae for their periods.
2.1 The Mass-Spring System
Imagine a mass attached to a spring, oscillating horizontally or vertically.
The period of oscillation (\(T\)) for a mass-spring system is given by:
$$T = 2\pi \sqrt{\frac{m}{k}}$$
- \(m\): The mass attached to the spring (in kg).
- \(k\): The spring constant or stiffness (in \(\text{N m}^{-1}\)). This value is determined by the stiffness of the spring (Hooke's Law).
Key Takeaway: The period of a mass-spring system depends on the mass and the stiffness, but not on how far you initially stretch it (the amplitude).
Real-World Example: This principle is used in vehicle suspension systems. If the spring constant (\(k\)) is too low (soft spring), the period is too long, and the car bobs excessively. If \(k\) is too high (stiff spring), the period is short, and the ride is bumpy.
2.2 The Simple Pendulum
This is a mass (called a 'bob') tied to a string of length \(l\), swinging freely.
The period of oscillation (\(T\)) for a simple pendulum (assuming small amplitude swings) is:
$$T = 2\pi \sqrt{\frac{l}{g}}$$
- \(l\): The length of the pendulum string (in m).
- \(g\): The acceleration due to gravity (in \(\text{m s}^{-2}\)).
Important Note: This formula is only valid if the swing angle is small (usually less than 10 degrees). At large angles, the motion is no longer truly SHM, and the period increases.
Key Takeaway: The pendulum period depends on the length and the local gravitational field strength (\(g\)), but not on the mass of the bob.
Mass-Spring: T depends on Mass (m) and Stiffness (k).
Simple Pendulum: T depends on Length (l) and Gravity (g).
3. Energy in Oscillating Systems
In physics, energy is always conserved. In an ideal (undamped) oscillating system, mechanical energy simply transforms back and forth between two forms: Kinetic Energy (\(E_k\)) and Potential Energy (\(E_p\)).
The Total Energy of the system always remains constant:
$$E_{Total} = E_k + E_p = \text{Constant}$$
3.1 Variation of Energy with Displacement (\(x\))
Consider a spring oscillating between \(x = -A\) and \(x = +A\), with equilibrium at \(x=0\).
- At Maximum Displacement (\(x = \pm A\)):
- The object is momentarily stationary (it stops before changing direction). Therefore, \(E_k\) is zero.
- The spring is fully stretched or compressed (maximum force). Therefore, \(E_p\) is maximum. All the energy is stored as potential energy.
- At Equilibrium (\(x = 0\)):
- The object is moving at its fastest speed. Therefore, \(E_k\) is maximum.
- The spring is neither stretched nor compressed (zero force). Therefore, \(E_p\) is minimum (zero). All the energy is kinetic.
Think of a skater on a U-shaped ramp: Highest points = maximum \(E_p\), zero \(E_k\). Bottom point = maximum \(E_k\), minimum \(E_p\).
3.2 Variation of Energy with Time (\(t\))
Both \(E_k\) and \(E_p\) graphs vary periodically with time, but they have a crucial difference from the displacement graph:
- The object reaches maximum speed (maximum \(E_k\)) and maximum displacement (maximum \(E_p\)) twice per period (\(T\)).
- Therefore, both the \(E_k\) and \(E_p\) graphs oscillate at twice the frequency of the actual oscillating system.
- The graph for the Total Energy remains a straight horizontal line, showing that the total energy is conserved (in the ideal case).
4. The Effects of Damping
In the real world, systems are rarely ideal. They experience resistive forces like air resistance or friction. This process is called damping.
What is Damping?
Damping is the process where the energy of an oscillating system is gradually removed due to resistive forces, converting the mechanical energy into thermal (heat) energy or sound.
Effects of Damping
- The amplitude of the oscillation gradually decreases over time.
- The total energy of the system decreases over time.
Did you know? Damping is often intentional! Car suspensions use hydraulic shock absorbers to introduce damping, preventing the car from bouncing endlessly after hitting a bump.
Types of Damping (Qualitative)
The syllabus requires a qualitative understanding of the effects:
- Light Damping:
This occurs when resistive forces are small. The amplitude slowly decreases, and the oscillator completes many cycles before stopping.
Example: A tuning fork vibrating in air. - Heavy Damping (or Overdamping):
This occurs when resistive forces are large. The system returns to the equilibrium position very slowly and takes a long time, but does not oscillate past the equilibrium point.
Example: A mass oscillating in thick oil or treacle. - Critical Damping:
This is the ideal case where the system returns to the equilibrium position in the shortest possible time without oscillating at all. This is used in measurement instruments and vehicle shock absorbers for quick stability.
Example: The door closure mechanism on a fire door.
Key Takeaway from Oscillating Systems
You must be confident in identifying the factors affecting the period of a mass-spring system (\(m\) and \(k\)) and a simple pendulum (\(l\) and \(g\)). Furthermore, remember that energy constantly switches between kinetic and potential, peaking when displacement is zero (\(E_k\)) and maximum (\(E_p\)), and that damping always reduces amplitude and total energy over time.