DP IB Physics: SL
C. Wave behavior
C.1 Simple harmonic motion
DP IB Physics: SLC. Wave behaviorC.1 Simple harmonic motion | |
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| a) | What makes the harmonic oscillator model applicable to a wide range of physical phenomena? |
| b) | Why must the defining equation of simple harmonic motion take the form it does? |
| c) | How can the energy and motion of an oscillation be analysed both graphically and algebraically? |
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a) What makes the harmonic oscillator model applicable to a wide range of physical phenomena?
- Solution:
- For small oscillations, any system close to a stable equilibrium point can be roughly represented as a harmonic oscillator, making the harmonic oscillator model broadly applicable.
- The linear term, which represents a restoring force, predominates for small displacements in the Taylor expansion of the potential energy around the equilibrium position, which gives rise to this approximation.
- This makes the model applicable to a number of disciplines, such as quantum mechanics, electrical circuits, acoustics, and mechanics, to explain phenomena like oscillations, vibrations, and wave-like behaviour.
- Figure 1 Harmonic oscillator model
- Universality of Equilibrium:
- The universality of equilibrium refers to the tendency of many physical systems to fluctuate back towards their stable equilibrium states when they are marginally perturbed. The harmonic oscillator concept is frequently used to explain these oscillations.
- The Taylor Expansion
- The Taylor series expansion of a system’s potential energy around its equilibrium point gives rise to the harmonic oscillator concept. For minor oscillations, the linear term in this expansion—which denotes a restoring force proportionate to the displacement—dominates.
- ⇒ Wide Range of Applications:
- This model is used in many different fields:
- Quantum Mechanics:
- The quantum harmonic oscillator is a fundamental model for understanding the behaviour of quantum systems and is used to study phonons and lattice vibrations in solids.
- Mechanical Systems:
- Classic examples of harmonic oscillators include vibrating strings, pendulums, and mass-spring systems.
- Electrical Circuits:
- The harmonic oscillator model can be used to analyse AC circuits, filters, and oscillators.
- Molecular Vibrations:
- Atoms within molecules and crystal lattices vibrate, and this motion can be approximated by harmonic oscillators.
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b) Why must the defining equation of simple harmonic motion take the form it does?
- Solution:
- The restoring force exerted on an item in simple harmonic motion (SHM) results in the defining equation,
- [math]a = -\omega^2 x[/math]
- This restoring force is always directed towards the equilibrium point and is directly proportionate to the displacement from the equilibrium position.
- This relationship results in the defining equation, where the acceleration (a) is proportional to the negative of the displacement (x), with the constant of proportionality being the square of the angular frequency (ω²), as acceleration is proportional to force
- [math]F = ma[/math]
- Figure 2 Simple harmonic motion
- ⇒ Restoring force and displacement:
- In SHM, a restoring force serves to return an item to its equilibrium position after it has been moved.
- The displacement from the equilibrium position is precisely proportional to this restoring force.
- The restoring force works in the opposite direction of the displacement, as indicated by the negative sign.
- ⇒ Differential equation:
- Applying Newton’s second law
- [math]F = ma[/math]
- – By Hooke’s law
- [math]F = -kx[/math]
- – Combine together
- [math]ma = -kx \\
a = -\frac{k}{m}x \\
\frac{d^2 x}{dt^2} = -\frac{k}{m}x[/math] - – This is a second order linear homogeneous differential equation.
- ⇒ Sinusoidal solution:
- Sinusoidal functions (sine and cosine) are the usual solution to this differential equation.
- This is due to the fact that the equation is satisfied when the second derivative of a sine or cosine function is proportional to the original function with a negative sign.
- Consequently, the following equations may be used to explain an object’s displacement in SHM:
- [math]x(t) = A \cos(\omega t + \phi) \\
x(t) = A \sin(\omega t + \phi)[/math] - – Where A is the amplitude, is the angular frequency, and is the phase constant.
c) How can the energy and motion of an oscillation be analysed both graphically and algebraically?
- Solution:
- By looking at their displacement, velocity, acceleration, and energy, oscillations in a pendulum or a spring-mass system may be visually and algebraically analysed.
- Energy fluctuates between kinetic and potential forms on sinusoidal displacement versus time graphs, but the overall energy stays constant.
- Figure 3 Graphically analysis of energy and the motion of an oscillation
- ⇒ Graphical Analysis:
- Displacement vs. Time:
- In simple harmonic motion (SHM), a typical displacement graph is a sine or cosine wave, where the oscillation’s rate is determined by the angular frequency (ω) and the maximum displacement is represented by the amplitude (A).
- Figure 4 Displacement vs Time graph
- Time versus Velocity:
- Although it is 90 degrees (π/2 radians) displaced from the displacement graph, the velocity graph is likewise sinusoidal. At the equilibrium position, velocity is at its highest, and at the extreme locations, it is at zero.
- Figure 5 Time vs Velocity graph
- Time versus Acceleration:
- In addition to being sinusoidal, the acceleration graph is 180 degrees (π radians) away from the displacement graph. At the extreme locations, acceleration reaches its maximum, while at the equilibrium position, it is zero.
- Figure 6 Time versus Acceleration graph
- Energy vs Time:
- Kinetic energy (KE) and potential energy (PE) graphs are sinusoidal but out of phase with one another when plotted against time. KE is 0 at the extreme locations (where velocity is zero) and highest at the equilibrium position (where velocity is maximum). PE is 0 in the equilibrium position and reaches its highest at the extreme locations. The sum of KE and PE equals the total energy (TE), which stays constant.
- Figure 7 Energy vs time graph
- Diagrams of phase spaces:
- Velocity and displacement are shown in a phase space diagram. This creates an ellipse for SHM, which graphically illustrates energy conservation and the relationship between displacement and velocity.