DP IB Physics: SL

C. Wave Behaviour

C.1 Simple Harmonic motion

DP IB Physics: SL C. Wave Behaviour C.1 Simple Harmonic motion Understandings Students should understand:
a)	Conditions that lead to simple harmonic motion
b)	The defining equation of simple harmonic motion as given by [math]a = -\omega^2 x[/math]
c)	A particle undergoing simple harmonic motion can be described using time period T, frequency ƒ, angular frequency ω, amplitude, equilibrium position, and displacement
d)	The time period in terms of frequency of oscillation and angular frequency as given by [math]T = \frac{1}{f} = \frac{2\pi}{\omega}[/math]
e)	The time period of a mass–spring system as given by [math]T = 2\pi \sqrt{\frac{m}{k}}[/math]
f)	The time period of a simple pendulum as given by [math]T = 2\pi \sqrt{\frac{l}{g}}[/math]
g)	A qualitative approach to energy changes during one cycle of an oscillation.

1 Simple Harmonic motion
a) Conditions that Lead to Simple Harmonic Motion (SHM)
Simple harmonic motion (SHM) is a type of periodic motion where a particle oscillates about an equilibrium position under a restoring force that is directly proportional to its displacement and directed towards the equilibrium position.
A system exhibits SHM if it satisfies the following conditions:
1. A Restoring Force Acting Towards the Equilibrium Position:
– There must be a force pulling or pushing the particle back toward its mean (equilibrium) position.
– This force should be proportional to the displacement from equilibrium.
2. Linear Relationship Between Force and Displacement:
– The force should follow Hooke’s Law:
[math]F = -kx[/math]
– where k is a proportionality constant and x is the displacement.
3. Negligible Damping:
– The system should experience minimal resistive forces (like friction or air resistance), which would otherwise reduce oscillations over time.
4. No External Driving Forces:
– External forces (like a continuous push) should not interfere, as SHM is a free
Examples of SHM Systems:
– A mass-spring system oscillating back and forth.
– A simple pendulum (for small angles).
– Vibrations of a tuning fork.
b) Defining Equation of Simple Harmonic Motion
Newton’s Second Law states that force is the product of mass and acceleration:
[math]F = ma[/math]
Since SHM has a restoring force [math]F = -kx[/math], we can write:
[math]ma = -kx[/math]
Dividing by m(mass):
[math]a = – \frac{k}{m}x[/math]
Since [math]\omega^2 = \frac{k}{m}[/math] (where ω is the angular frequency), we get the defining equation of SHM:
[math]a = -\omega^2 x[/math]
Figure 1 Simple Harmonic motion
⇒ Explanation:
– a is the acceleration of the particle.
– x is the displacement from the equilibrium position.
– [math]\omega^2[/math] is a constant that depends on the system.
– The negative sign indicates that acceleration is always directed opposite to displacement (toward equilibrium).
This equation shows that acceleration in SHM is directly proportional to displacement and always opposite in direction.
c) Description of a Particle Undergoing SHM
A particle in SHM can be characterized by the following parameters:
1. Equilibrium Position
– The central position where no net force acts on the particle.
– When the particle passes through equilibrium, velocity is maximum, and acceleration is zero.
2. Amplitude (A)
– The maximum displacement from the equilibrium position.
– Determines the energy of oscillation.
3. Displacement (x)
– The instantaneous position of the particle at any time
– It varies sinusoidally and is given by:
[math]x = A \cos(\omega t + \phi)[/math]
where:
– A = Amplitude
– ω = Angular frequency
– ϕ = Phase constant
4. Angular Frequency (ω)
– The rate at which the particle moves through its oscillatory cycle.
– Measured in radians per second (rad/s).
Given by:
[math]\omega = 2\pi f[/math]
– Where f is the frequency.
5. Frequency (f)
– The number of oscillations per second.
– Measured in Hertz (Hz).
Given by:
[math]f = \frac{1}{T}[/math]
Where T is the time period.
6. Time Period (T)
– The time taken for one complete oscillation.
– Related to frequency and angular frequency as:
[math]T = \frac{1}{f} = \frac{2\pi}{\omega}[/math]
d) Time Period in Terms of Frequency and Angular Frequency
The time period of oscillation T is related to frequency and angular frequency as:
[math]T = \frac{1}{f} = \frac{2\pi}{\omega}[/math]
Where:
– T = Time period (seconds)
– f = Frequency (Hz)
– ω = Angular frequency (rad/s)
Figure 2 Parameters of wave
⇒ Explanation:
– The higher the frequency, the shorter the time period (faster oscillations).
– The larger the angular frequency, the shorter the time period.
e) Time Period of a Mass-Spring System
A mass-spring system undergoes simple harmonic motion (SHM) when a mass mmm is attached to a spring with a force constant (spring stiffness) k.
⇒ Derivation of the Time Period Formula
From Hooke’s Law, the restoring force exerted by the spring is:
[math]F = -kx[/math]
Using Newton’s Second Law:
[math]ma = -kx[/math]
Since acceleration a in SHM is given by:
[math]a = -\omega^2 x[/math]
Comparing with[math]ma = -kx[/math] we get:
[math]m \omega^2 = k[/math]
Solving for ω:
[math]\omega = \sqrt{\frac{k}{m}}[/math]
Since time period T is related to angular frequency by:
[math]T =\frac{ 2\pi }{\omega}[/math]
Substituting [math]\omega = \sqrt{\frac{k}{m}}[/math]:
[math]T = 2\pi \sqrt{\frac{k}{m}}[/math]
Figure 3 Simple Harmonic motion (mass attach to spring)
Interpretation
– The time period depends on mass mmm and spring constant
– A larger mass increases the time period (slower oscillations).
– A stiffer spring (higher k) decreases the time period (faster oscillations).
– This equation applies only to ideal springs with negligible damping.
f) Time Period of a Simple Pendulum
A simple pendulum consists of a small mass (bob) attached to a string of length l, oscillating under gravity.
Figure 4 Simple pendulum
⇒ Derivation of the Time Period Formula
For small angular displacements (θ), the restoring force is:
[math]F = -mg \sin\theta[/math]
For small angles (sine e in radians), the equation simplifies to:
[math]F \approx -mg \frac{x}{\ell}[/math]
Applying Newton’s Second Law:
[math]ma = -mg \frac{x}{\ell}[/math]
Since[math]a = -\omega^2 x[/math] comparing terms gives:
[math]m\omega^2 = \frac{mg}{\ell}[/math]
Solving for [math]\omega[/math]:
[math]\omega = \sqrt{\frac{g}{\ell}}[/math]
Since T =[math]\frac{2\pi}{\omega}[/math]
[math]T = 2\pi \sqrt{\frac{\ell}{g}}[/math]
Interpretation
– The time period depends only on length l and gravitational acceleration g.
– A longer pendulum has a longer time period (slower oscillations).
– A stronger gravitational field (higher g) decreases the time (fister oscillations)-
– The formula holds small oscillations; at larger angles, nonlinear arise.
g) Energy Changes in Simple Harmonic Motion (SHM)
An oscillating system continuously converts kinetic energy (KE) and potential energy (PE) during each cycle. The total energy remains constant, assuming no energy loss (no damping).
Energy Types in SHM
1. Kinetic Energy (KE) – Energy due to motion
2. Potential Energy (PE) – Energy stored due to position
3. Total Mechanical Energy (E) – Sum of kinetic and potential energy, which remains constant
At any displacement x, the total energy is:
[math]E=KE+PE[/math]
⇒ Kinetic Energy in SHM
Kinetic energy is given by:
[math]KE = \frac{1}{2}mv^2[/math]
Since velocity in SHM is:
[math]v = \omega \sqrt{A^2 – x^2}[/math]
Substituting v in KE:
[math]KE = \frac{1}{2} m\omega^2 \left(A^2 – x^2\right)[/math]
At equilibrium (x=0):
KE is maximum:
[math]KE_{\text{max}} = \frac{1}{2} m\omega^2 A^2[/math]
– The object moves at maximum velocity [math]v_{\text{max}} = \omega A[/math]
At maximum displacement (x=A):
– KE=0 (momentarily stops).
⇒ Potential Energy in SHM
Potential energy is stored as elastic energy (in a spring) or gravitational energy (in a pendulum).
For a mass-spring system:
[math]PE = \frac{1}{2} kx^2[/math]
Using [math]k = m\omega^2[/math]:
[math]PE = \frac{1}{2} m\omega^2 x^2[/math]
At maximum displacement (x=A):
– PE is maximum:
[math]PE_{\text{max}} = \frac{1}{2} m\omega^2 A^2[/math]
– The object momentarily stops moving.
At equilibrium (x=0x = 0x=0):
– PE=0 (all energy is kinetic).
Total Energy in SHM
Total energy remains constant:
[math]E = KE + PE \\
E = \frac{1}{2} m\omega^2 A^2[/math]
Figure 5 Energy changes in simple harmonic motion
At different positions:
– At equilibrium (x=0x = 0x=0):
– KE=E, PE=0
Maximum speed.
– At maximum displacement (x=A):
– PE=E, KE=0
Object momentarily stops.
– At an intermediate position ([math]0 < x < A[/math] ):
– KE and PE share the total energy.
Graphical Representation of Energy in SHM
– KE vs. x → Parabolic (KE is max at equilibrium, zero at extremes).
– PE vs. x → Parabolic (PE is zero at equilibrium, max at extremes).
– Total energy vs. x → Constant horizontal line (energy is conserved).