Welcome to Oscillations!
Hello future Physics expert! This chapter on Oscillations might seem daunting, full of sines and cosines, but it’s actually about understanding predictable, rhythmic motion—like a swinging pendulum or a vibrating guitar string.
We’re studying Simple Harmonic Motion (SHM), the most fundamental type of oscillation. Mastering this topic is crucial not just for A Level Physics, but also for understanding waves, electricity (AC circuits), and even quantum mechanics later on. Ready to dive into the world of predictable wiggles? Let's go!
17.1 Simple Harmonic Motion (SHM) Basics
Defining the Key Terms
When an object is oscillating (moving back and forth), we use specific terms to describe its motion.
- Displacement (\(x\)): This is the distance of the oscillating object from its fixed central point, known as the equilibrium position (where the resultant force is zero). Remember: Displacement is a vector quantity.
- Amplitude (\(x_0\)): This is the maximum displacement from the equilibrium position. It measures 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 \(\text{s}^{-1}\).
- Angular Frequency (\(\omega\)): This relates the frequency to circular motion, measured in radians per second (\(\text{rad s}^{-1}\)).
Relating Period, Frequency, and Angular Frequency
These three quantities are closely linked.
Since period (\(T\)) is the reciprocal of frequency (\(f\)):
\[T = \frac{1}{f}\]
And angular frequency (\(\omega\)) is defined as \(2\pi\) multiplied by the frequency:
\[\omega = 2\pi f\]
Therefore, you can also write:
\[\omega = \frac{2\pi}{T}\]
Phase Difference (\(\phi\))
If you have two oscillating systems (or two parts of the same wave), phase difference tells you how far apart they are in their cycle.
- It is measured either in degrees (\(0^{\circ}\) to \(360^{\circ}\)) or radians (\(0\) to \(2\pi\)).
- If two objects are moving together, they are in phase (\(\phi = 0\)).
- If one object reaches maximum displacement when the other reaches maximum displacement in the opposite direction, they are antiphase (or 180° / \(\pi\) radians out of phase).
Quick Takeaway: The definitions of \(x_0\), \(T\), and \(f\) are essential starting points. Angular frequency \(\omega\) is just a convenient way to link linear frequency and period mathematically.
17.1 Simple Harmonic Motion: The Defining Condition
What is Simple Harmonic Motion (SHM)?
SHM is a very specific type of oscillation. It is defined by the rule governing the acceleration of the object.
The Defining Condition of SHM (Syllabus Requirement 17.1.2)
Simple harmonic motion occurs when the acceleration (\(a\)) of an object is:
- Proportional to its displacement (\(x\)) from a fixed equilibrium point.
- Always directed towards that fixed point (the equilibrium position).
Mathematically, this crucial relationship is written as:
\[a \propto -x\]
The negative sign is vital! It means that when the displacement is positive (moving right), the acceleration (and thus the resultant force) is negative (pulling left), and vice versa. This is the restoring force trying to bring the object back to the centre.
Analogy: Imagine a ball rolling back and forth in a smooth bowl. The further up the side (greater \(x\)) it is, the steeper the slope, and the greater the force (and acceleration) pulling it back down towards the centre.
The Fundamental Equation of SHM
The constant of proportionality that links acceleration and displacement is the square of the angular frequency, \(\omega^2\).
The equation of simple harmonic motion is:
\[a = -\omega^2 x\]
Why is this equation so important? It contains everything we need to know about the motion, including the angular frequency \(\omega\), which dictates the period and frequency of oscillation.
Equations for Displacement, Velocity, and Acceleration (Syllabus Requirements 17.1.3 & 17.1.4)
1. Displacement (\(x\))
For an oscillation starting at the equilibrium position (\(x=0\) at \(t=0\)), the displacement varies sinusoidally (following a sine wave).
\[x = x_0 \sin \omega t\]
Where:
- \(x_0\) is the amplitude (maximum displacement).
- \(\omega\) is the angular frequency.
- \(t\) is the time.
Note: If the oscillation starts at maximum displacement (\(x=x_0\) at \(t=0\)), the equation would be \(x = x_0 \cos \omega t\). Both are valid forms of SHM, but the syllabus specifically highlights the sine form as the solution to \(a = -\omega^2 x\).
2. Velocity (\(v\))
The velocity changes constantly throughout SHM. There are two key velocity equations you need to use:
a) Velocity as a function of time:
Since velocity is the rate of change of displacement, this equation is typically:
\[v = v_0 \cos \omega t\]
Where \(v_0\) is the maximum speed, found when the object passes through the equilibrium position.
b) Velocity as a function of displacement:
This is often the most useful equation for calculations, as it lets you find the speed at any position \(x\):
\[v = \pm \omega \sqrt{(x_0^2 - x^2)}\]
Maximum Speed (\(v_0\)): This occurs when the object is at the equilibrium position, so \(x=0\).
Setting \(x=0\): \(v_0 = \omega \sqrt{(x_0^2 - 0)} = \omega x_0\).
3. Acceleration (\(a\))
We already have the fundamental defining equation:
\[a = -\omega^2 x\]
Since \(x\) is maximum at \(x_0\), the maximum acceleration (\(a_0\)) is:
\[a_0 = -\omega^2 x_0\]
Maximum acceleration occurs at the points of maximum displacement (the amplitude).
Common Mistake Alert! Always use the angular frequency \(\omega\) in SHM formulas, not the linear frequency \(f\). If you are given \(f\), first calculate \(\omega = 2\pi f\).
Quick Review: SHM Conditions
\(\bullet\) \(a\) is max when \(x\) is max (\(x = \pm x_0\)). \(v\) is zero.
\(\bullet\) \(v\) is max when \(x\) is zero (equilibrium). \(a\) is zero.
\(\bullet\) \(a\) and \(x\) are always in opposite directions (the negative sign).
17.1 Graphical Analysis of SHM
Understanding the relationships between displacement, velocity, and acceleration is easiest when looking at their graphs over time (Syllabus Requirement 17.1.5).
Key Phase Relationships
If displacement is described by a sine function, notice how the other quantities relate in terms of phase difference (which can be described as a time lag or lead).
Displacement (\(x\)) vs Time (\(t\))
(Starts at zero, follows a sine curve)
\(x = x_0 \sin \omega t\)
Maximums occur at \(t = T/4, 5T/4, \dots\) and minimums at \(t = 3T/4, 7T/4, \dots\)
Velocity (\(v\)) vs Time (\(t\))
Velocity is maximum when the displacement is zero (passing through equilibrium).
(Starts at maximum positive value, follows a cosine curve)
\(v = v_0 \cos \omega t\)
The velocity graph is \(\mathbf{90^{\circ}}\) (or \(\pi/2\) rad) ahead of the displacement graph. When \(x\) is zero, \(v\) is maximum.
Acceleration (\(a\)) vs Time (\(t\))
Acceleration is always opposite to displacement (\(a = -\omega^2 x\)).
(Starts at zero, follows a negative sine curve, inverted relative to displacement)
\(a = -\omega^2 x_0 \sin \omega t\)
The acceleration graph is \(\mathbf{180^{\circ}}\) (or \(\pi\) rad) ahead of the displacement graph. When \(x\) is max positive, \(a\) is max negative.
Memory Aid: Remember the phase differences in quarter cycles (90°):
Acceleration leads Velocity leads X displacement.
A is 90° ahead of V. V is 90° ahead of X. A is 180° ahead of X.
Quick Takeaway: When drawing the graphs, ensure that when displacement is maximum, velocity is zero, and acceleration is maximum negative (and vice versa).
17.2 Energy in Simple Harmonic Motion
When an ideal system undergoes SHM, the total mechanical energy remains constant, but energy continually swaps between Kinetic Energy (KE) and Potential Energy (PE) (Syllabus Requirement 17.2.1).
The Energy Interchange
Think about a mass oscillating on a spring:
-
At the Equilibrium Position (\(x=0\)):
- The displacement is zero, so the extension/compression is zero. Potential Energy (\(E_p\)) is zero (or minimum).
- The speed (\(v\)) is maximum. Kinetic Energy (\(E_k\)) is maximum.
-
At the Amplitude Position (\(x=\pm x_0\)):
- The displacement is maximum. The spring is fully extended or compressed. Potential Energy (\(E_p\)) is maximum.
- The speed (\(v\)) is momentarily zero (as the mass changes direction). Kinetic Energy (\(E_k\)) is zero.
The Total Energy (E) of the system is the sum of KE and PE at any point:
\[E = E_k + E_p\]
Since the total energy is conserved (in ideal SHM), we can calculate it using the point where KE is maximum (and PE is zero). Maximum KE occurs when the speed is \(v_0 = \omega x_0\).
Equation for Total Energy (Syllabus Requirement 17.2.2)
The total energy (\(E\)) is equal to the maximum kinetic energy, which is:
\[E = \frac{1}{2} m v_0^2\]
Substituting \(v_0 = \omega x_0\) into the equation gives the required formula:
\[E = \frac{1}{2} m (\omega x_0)^2\]
Therefore, the total energy of a system undergoing SHM is:
\[E = \frac{1}{2} m \omega^2 x_0^2\]
This equation shows that the total energy is proportional to the square of the amplitude (\(x_0^2\)) and the square of the angular frequency (\(\omega^2\)).
Did you know? This quadratic relationship (\(E \propto x_0^2\)) is why turning up the volume (increasing the amplitude) on a speaker requires a lot more power (energy) than a small change in volume might suggest!
Quick Takeaway: Total energy in SHM is proportional to \((\text{amplitude})^2\). Energy constantly transforms between KE (max at centre) and PE (max at ends).
17.3 Damped and Forced Oscillations, Resonance
Damping (Syllabus Requirements 17.3.1 & 17.3.2)
So far, we have looked at ideal SHM where total energy is conserved. In reality, all oscillations lose energy to their surroundings, usually due to resistive forces like air resistance or friction. This loss of energy is called damping.
Damping causes the amplitude (\(x_0\)) of the oscillation to decrease over time. The energy is converted into thermal energy (heat).
Types of Damping and their Graphs
We categorize damping based on how quickly the amplitude decreases:
1. Light Damping (Under-damped)
- Description: The resistive forces are small.
- Motion: The object oscillates with a progressively decreasing amplitude. The period of oscillation remains nearly constant, but slightly longer than the natural period.
- Example: A pendulum swinging in the air.
2. Heavy Damping (Over-damped)
- Description: The resistive forces are large.
- Motion: The object moves slowly back towards the equilibrium position without oscillating at all.
- Example: A mass oscillating submerged in thick oil or honey.
3. Critical Damping
- Description: This is the ideal damping level.
- Motion: The object returns to the equilibrium position in the shortest possible time without oscillating or overshooting.
- Example: Car shock absorbers are designed to be critically damped so your suspension doesn't bounce excessively after hitting a bump.
Forced Oscillations and Resonance (Syllabus Requirement 17.3.3)
When an object oscillates naturally without any external influence, it oscillates at its natural frequency (\(f_0\)).
A forced oscillation occurs when an external periodic force is applied to the system, causing it to oscillate at the frequency of the external force (the driving frequency, \(f\)).
Resonance
Resonance occurs when the driving frequency (\(f\)) becomes equal to the natural frequency (\(f_0\)) of the oscillating system.
When resonance occurs:
- The system absorbs energy very efficiently from the driving force.
- The amplitude of the resulting oscillations becomes maximum.
Analogy: Pushing a child on a swing. If you push at the exact natural rhythm (\(f = f_0\)), the amplitude builds up quickly and dramatically. If you push at a different rate, the motion is awkward and the amplitude stays small.
The Role of Damping in Resonance
Damping dramatically affects the maximum amplitude at resonance:
- Light Damping: Results in a very high, sharp resonance peak (very large maximum amplitude).
- Heavy Damping: Results in a very low, broad resonance peak (small maximum amplitude, occurring at a slightly lower frequency than \(f_0\)).
Resonance is a double-edged sword: it is useful in things like tuning radio circuits, but destructive in structures like bridges if the wind makes them vibrate at their natural frequency.
Quick Takeaway: Damping reduces amplitude over time. Resonance happens when the external driving frequency matches the natural frequency, leading to maximum (and potentially dangerous) amplitude.