👋 Welcome to the World of Oscillations!
Hello future Physicist! This chapter, Oscillations, is all about things that wiggle, wave, and repeat their motion. Don't worry if the maths seems a bit intimidating at first; we will break down the crucial concepts—like Simple Harmonic Motion—into easy, digestible steps.
Understanding oscillations is fundamental. It helps us design earthquake-proof buildings, tune radio receivers, and understand how sound waves travel. Let’s dive into how the universe repeats itself!
1. Defining Periodic Motion and Key Terms
Any motion that repeats itself regularly over a fixed period of time is called Periodic Motion. Oscillations are a specific type of periodic motion where an object moves back and forth around a fixed central point (the equilibrium position).
Key Definitions in Periodic Motion
- Displacement (\(x\)): The distance (and direction) of the oscillating object from the equilibrium position. It is maximum at the ends of the swing.
- Amplitude (\(A\)): The maximum displacement from the equilibrium position. It represents the size of the oscillation.
- Period (\(T\)): The time taken for one complete oscillation or cycle. Measured in seconds (s).
- Frequency (\(f\)): The number of complete oscillations per unit time. Measured in Hertz (Hz) or \(s^{-1}\).
The relationship between Period and Frequency is straightforward:
$$f = \frac{1}{T}$$
Angular Frequency (\(\omega\))
In physics, we often find it easier to work with Angular Frequency (\(\omega\)), especially when relating oscillatory motion to circular motion. Think of an oscillation as the projection of uniform circular motion onto a straight line.
Angular frequency is the rate of change of the angle (or phase) of the object, measured in radians per second (\(rad \ s^{-1}\)).
$$\omega = 2\pi f = \frac{2\pi}{T}$$
Analogy: Imagine a point moving in a circle. Frequency (\(f\)) tells you how many laps per second. Angular frequency (\(\omega\)) tells you how many radians per second the point rotates. Since one lap is \(2\pi\) radians, the relationship makes perfect sense!
Quick Review: \(A\) is max displacement. \(T\) is time per cycle. \(f\) is cycles per time. \(\omega\) converts cycles to radians.
2. Simple Harmonic Motion (SHM)
Not all oscillations are created equal! Simple Harmonic Motion (SHM) is a special, fundamental type of oscillation that follows a specific rule.
The Defining Condition of SHM
An object executes SHM if and only if its acceleration is:
- Directly proportional to its displacement (\(x\)) from the equilibrium position.
- Always directed towards the equilibrium position (meaning it is always in the opposite direction to the displacement).
This definition is written mathematically as:
$$a \propto -x$$
Introducing the constant of proportionality, which is the square of the angular frequency (\(\omega^2\)), we get the master equation for SHM:
$$a = -\omega^2 x$$
The negative sign is absolutely critical. It tells you that when the displacement \(x\) is positive (e.g., mass moving right), the acceleration \(a\) is negative (acting left, back towards equilibrium). This is the restoring force in action.
Common Mistake to Avoid: Forgetting the negative sign! If you forget it, your calculation will imply that the acceleration pushes the mass away from equilibrium, which would lead to runaway motion, not oscillation.
3. Describing SHM Mathematically (The Equations)
Since acceleration depends on displacement, the motion is constantly changing. We use trigonometric functions (sine and cosine) to model this repeating motion.
3.1 Displacement Equation (\(x\))
The displacement of an object performing SHM varies sinusoidally (like a sine or cosine wave) with time. If the object starts at maximum positive displacement (\(x = A\)) when \(t=0\), we use cosine:
$$x = A \cos(\omega t)$$
If the object starts at the equilibrium position (\(x=0\)) when \(t=0\), we use sine:
$$x = A \sin(\omega t)$$
Remember, \(A\) is the amplitude, and \(\omega\) is the angular frequency.
3.2 Velocity Equation (\(v\))
Velocity is the rate of change of displacement. By differentiating the displacement equation, we find the velocity:
$$v = \pm \omega \sqrt{A^2 - x^2}$$
What this equation tells us:
- Velocity is maximum when displacement \(x\) is zero (at the equilibrium position).
- Maximum Velocity: \(v_{max} = \omega A\)
- Velocity is zero when displacement \(x\) is \(\pm A\) (at the extremes of the oscillation).
3.3 Acceleration Equation (\(a\))
We already know this from the definition of SHM, but using \(\omega\):
$$a = -\omega^2 x$$
What this equation tells us:
- Acceleration is zero when displacement \(x\) is zero (at the equilibrium position).
- Acceleration is maximum when displacement \(x\) is \(\pm A\) (at the extremes).
- Maximum Acceleration: \(a_{max} = \omega^2 A\)
Step-by-Step Thinking: When the mass is far out (large \(x\)), the restoring force is high, so acceleration is maximum, but velocity is zero. When the mass passes through the centre (\(x=0\)), the restoring force is zero (minimum acceleration), but it's moving fastest (maximum velocity).
3.4 Phase Difference (\(\phi\))
When two oscillators are moving, their motion can be compared using Phase Difference. Phase difference (\(\phi\)) measures how far "out of step" one oscillation is compared to another, measured in radians or degrees.
If one oscillation starts before the other, we say it leads. If it starts later, it lags.
Important relationship for SHM:
- Displacement and Velocity are \(\frac{\pi}{2}\) radians (90°) out of phase.
- Displacement and Acceleration are \(\pi\) radians (180°) out of phase. (This confirms the negative sign!).
Key Takeaway: Maximum velocity is at \(x=0\). Maximum acceleration is at \(x=A\). Acceleration and displacement are always pointing opposite directions.
4. Energy in Simple Harmonic Motion
Oscillations involve a continuous interchange between Kinetic Energy (KE) and Potential Energy (PE). Energy is conserved if there is no damping.
4.1 Potential Energy (PE)
For a mass on a spring, the potential energy is stored as Elastic Potential Energy (EPE). For a pendulum, it is Gravitational Potential Energy (GPE).
PE is maximum at the points of maximum displacement (\(x = \pm A\)) because the object is momentarily stationary, and all its energy is stored.
$$PE = \frac{1}{2} m \omega^2 x^2$$
4.2 Kinetic Energy (KE)
KE depends on the speed of the object.
KE is maximum at the equilibrium position (\(x=0\)) because the object is moving at its maximum speed (\(v_{max} = \omega A\)).
$$KE = \frac{1}{2} m v^2$$
4.3 Total Energy (E)
The Total Energy (E) of the oscillator is the sum of KE and PE at any point, and in SHM (without damping), this total energy is constant.
$$E = KE + PE$$
We can find the total energy by considering the point of maximum displacement, where \(KE=0\) and \(PE\) is maximum (using \(x=A\)):
$$E = \frac{1}{2} m \omega^2 A^2$$
Observation: Total Energy is proportional to the square of the amplitude (\(E \propto A^2\)). If you double the amplitude, you quadruple the energy!
5. Specific Oscillating Systems
Although the general equations for SHM are the same, the angular frequency (\(\omega\)) and the period (\(T\)) depend on the physical properties of the specific system.
5.1 Mass-Spring System (Horizontal or Vertical)
The restoring force is provided by the spring's stiffness (Hooke's Law: \(F = -kx\)).
Since \(a = -\omega^2 x\) and we know \(a = F/m = -kx/m\), we can equate the terms:
$$\omega^2 = \frac{k}{m}$$
Therefore, the Period (\(T\)) is:
$$T = 2\pi \sqrt{\frac{m}{k}}$$
Where: \(m\) is the mass (kg), and \(k\) is the spring constant (\(N \ m^{-1}\)).
5.2 Simple Pendulum
A simple pendulum consists of a small mass (bob) on a light string of length \(L\). SHM only occurs if the angle of swing is small (typically less than 10°).
The period \(T\) is determined by the length of the string and gravity:
$$T = 2\pi \sqrt{\frac{L}{g}}$$
Where: \(L\) is the length of the pendulum (m), and \(g\) is the acceleration due to gravity (\(m \ s^{-2}\)).
Did You Know? Since the period of a pendulum does not depend on the mass of the bob, Galileo reputedly used his pulse to time the swings of lamps in church, noting the period remained constant regardless of the amplitude (for small angles).
6. Damping: Taking Energy Away
In the real world, oscillations don't continue forever. Energy is always lost, usually due to resistive forces like air resistance or friction. This progressive reduction in amplitude is called Damping.
When an oscillator is damped, the total energy decreases over time, and consequently, the amplitude decreases exponentially.
Types of Damping
Light Damping (Underdamped)
The damping force is small. The oscillator completes many cycles before coming to rest. The period remains almost constant, but the amplitude slowly decreases.
Example: A swing slowly coming to a stop.
Critical Damping
This is the optimum level of damping. The object returns to the equilibrium position in the shortest possible time without oscillating.
Example: Car shock absorbers (dampers) are designed to be critically damped to ensure a comfortable and safe ride.
Heavy Damping (Overdamped)
The damping force is very large. The object returns slowly to the equilibrium position but takes a longer time than critical damping and does not oscillate.
Example: A heavy door closing slowly because it is pushing through thick hydraulic fluid.
Key Takeaway: Damping reduces amplitude and energy. Critical damping is the quickest way to return to equilibrium without overshoot.
7. Forced Oscillations and Resonance
So far, we have looked at Free Oscillations, where the system oscillates at its own unique frequency, called the Natural Frequency (\(f_0\)).
However, if we apply an external, periodic force to the system, it is performing Forced Oscillations. The frequency of the applied force is called the Driving Frequency (\(f\)).
Resonance
Resonance occurs when the driving frequency (\(f\)) is equal (or very close) to the natural frequency (\(f_0\)) of the system.
At resonance:
- The system absorbs maximum energy from the driving force.
- The amplitude of oscillation becomes very large, potentially destructive.
Analogy: Pushing a child on a swing. The natural frequency (\(f_0\)) is how fast the swing wants to go. If you push at exactly that frequency, the amplitude (height) builds up dramatically. If you push too quickly or too slowly, the swing barely moves.
The Effect of Damping on Resonance
Damping limits the amplitude at resonance:
- Low Damping: Produces a very sharp, high resonance peak. A small change in the driving frequency causes a large drop in amplitude.
- High Damping: Produces a broad, low resonance peak. The maximum amplitude reached is much smaller.
Real-World Example (Engineering): Engineers try to design structures (like bridges or buildings) so that their natural frequencies (\(f_0\)) are far away from potential driving frequencies (like wind, footsteps, or traffic) to avoid destructive resonance. The famous 1940 Tacoma Narrows Bridge disaster was caused by wind generating a driving frequency that matched the bridge's natural torsional frequency.
Congratulations! You have mastered the mechanics of repeating motion. Keep practicing the relationship between \(a = -\omega^2 x\) and the energy transformations, and you'll ace this chapter!