Simple Harmonic Motion, Elasticity, and Mechanical Waves Study Notes

Elasticity

• Elasticity is categorized into three types: Young's modules, shear modules, and Bulle modules.

• Stress is defined as force per unit area: $\sigma = \frac{F}{A}$. Its units are $N/m^2$ or Pascal ($Pa$).

• Strain ($\epsilon$) is the change in length relative to the original length: $\epsilon = \frac{\Delta L}{L}$. It is a dimensionless number.

• Modulus is defined generally as the ratio of stress to strain.

• Young's modules ($Y$ or $E$) is measured in Pascal ($Pa$).

• The elastic limit is the maximum stress an object can withstand before permanent deformation occurs.

• Bulle modulus ($B$) relates a change in pressure ($\Delta p$) to a fractional change in volume: $B = -\frac{\Delta p}{\Delta V / V}$. If $\Delta p$ is positive, the volume decrease ($\Delta V$) will be negative.

Simple Harmonic Motion (SHM)

• SHM is motion that can be described by sinusoidal functions ($\sin$ or $\cos$).

• Amplitude ($A$) is the maximum displacement from equilibrium.

• Period ($T$) is the shortest time for the motion to repeat itself: $T = \frac{2\pi}{\omega} = \frac{1}{f}$.

• Frequency ($f$) and angular frequency ($\omega$) are related by $\omega = 2\pi f$. Units for $\omega$ are $rad/s$.

• Standard wave functions for displacement include:

  • $x(t) = A\sin(\omega t + \phi)$

  • $x(t) = A\cos(\omega t + \phi)$

• Velocity ($v(t)$) and acceleration ($a(t)$) formulas:

  • $v(t) = \omega A\cos(\omega t + \phi)$

  • $a(t) = -\omega^2 A\sin(\omega t + \phi) = -\omega^2 x(t)$

• For a simple pendulum, the acceleration is $a_x = -\frac{g}{L}x$, resulting in $\omega = \sqrt{\frac{g}{L}}$ and $T = 2\pi \sqrt{\frac{L}{g}}$.

• For a mass-spring system, the angular frequency is determined by $\omega = \sqrt{\frac{k}{m}}$.

Mechanical Waves

• A wave is a propagation of disturbance away from a source without the bulk movement of the medium.

• Mechanical waves (e.g., sound, waves on a string) require a medium, whereas non-mechanical waves (e.g., electromagnetic waves) do not.

• Transverse waves: Disturbance is perpendicular to the direction of propagation.

• Longitudinal waves: Disturbance is parallel to the direction of propagation.

• The speed of a transverse wave on a string is given by $v = \sqrt{\frac{F}{\mu}}$, where $F$ is tension and $\mu = \frac{M}{L}$ is the linear mass density.

• Intensity ($I$) is energy per unit time per unit area: $I = \frac{P}{A}$. For an isotropic point source, $I = \frac{P}{4\pi r^2}$.

• Threshold of hearing ($I_0$) is $10^{-12}\,W/m^2$.

Mathematical Description of a Wave

• The wave function $y(x, t)$ describes the displacement as a function of position and time.

• For a harmonic wave traveling in the $+x$ direction: $y(x, t) = A\sin(\omega t - kx + \phi)$.

• For a harmonic wave traveling in the $-x$ direction: $y(x, t) = A\sin(\omega t + kx + \phi)$.

• Wave number ($k$) is defined as $k = \frac{2\pi}{\lambda}$.

• The relationship between speed, frequency, and wavelength is $v = f\lambda = \frac{\omega}{k}$.

Interference and Standing Waves

• Principle of Superposition: The total displacement is the sum of individual displacements: $y = y_1 + y_2$.

• Constructive interference occurs when the path difference between two coherent sources ($|S_1P - S_2P|$) is an integer multiple of the wavelength ($n\lambda$).

• Destructive interference occurs when the path difference is an odd integer multiple of half wavelengths ($(n + 1/2)\lambda$).

• Standing waves are formed by the interference of traveling waves and contain nodes (zero amplitude) and antinodes (maximum amplitude).

• Normal modes on a string have frequencies $f_n = n f_1 = n\frac{v}{2L}$, where $f_1$ is the fundamental frequency.

Physical Optics

• Light is a transverse electromagnetic wave consisting of electric ($E$) and magnetic ($B$) fields.

• In Young's Double Slit experiment, the positions of bright fringes (maxima) are determined by $d\sin(\theta) = n\lambda$, where $d$ is the slit separation.

• For small angles, $\sin(\theta) \approx \tan(\theta) = \frac{y}{D}$, where $y$ is the distance from the central maximum and $D$ is the distance to the screen.

• Refractive index ($n$) is the ratio of the speed of light in a vacuum to the speed in a medium: $n = \frac{c}{v}$.

• Snell's Law of refraction: $n_1\sin(\theta_1) = n_2\sin(\theta_2)$.

Elasticity Questions:

1. What is the definition of stress and its formula?

2. How is strain calculated, and what does it represent?

3. Explain Young's modulus and its importance in material science.

4. What is the elastic limit, and why is it significant?

5. Describe Bulle modulus and its application in understanding volume changes.

Formulas:

• Stress: $au = rac{F}{A}$

• Strain: $ext{Strain} = rac{L}{L}$

• Young's Modulus: $Y = rac{ au}{ ext{Strain}}$

• Bulle Modulus: $B = - rac{p}{V/V}$

Simple Harmonic Motion (SHM) Questions:

1. How can SHM be represented using sinusoidal functions?

2. What are the definitions of amplitude and period in SHM?

3. Explain the relationship between angular frequency and frequency.

4. What are the equations for velocity and acceleration in SHM?

5. How does the simple pendulum relate to SHM?

Formulas:

• Period: $T = rac{2 ext{ }}{ ext{ and }}$

• Frequency: $f = rac{1}{T}$

• Angular frequency: $= 2 ext{pi}f$

• Displacement: $x(t) = A ext{sin} (t + )$

Mechanical Waves Questions:

1. What distinguishes mechanical waves from non-mechanical waves?

2. Explain the difference between transverse and longitudinal waves.

3. How is the speed of a transverse wave on a string calculated?

4. What is intensity, and how is it related to power and area?

5. What is the threshold of hearing?

Formulas:

• Wave speed: $v = ext{sqrt} rac{F}{ ext{u}}$

• Intensity: $I = rac{P}{A}$

• Intensity for isotropic point source: $I = rac{P}{4 ext{pi}r^2}$

• Threshold of hearing: $I_0 = 10^{-12} ext{W/m}^2$

Interference and Standing Waves Questions:

1. What is the principle of superposition?

2. Explain constructive and destructive interference in waves.

3. What are standing waves, and how are they formed?

4. Define nodes and antinodes in the context of standing waves.

5. How do normal modes differ in frequency for a vibrating string?

Formulas:

• Constructive interference: $|S<em>1P - S</em>2P| = n$

• Destructive interference: $(n + 1/2)$

• Normal mode frequencies: $f<em>n = n f</em>1 = n rac{v}{2L}$

Physical Optics Questions:

1. What is the nature of light in terms of its wave properties?

2. Describe Young's Double Slit experiment and its outcome.

3. How is the refractive index defined?

4. What is Snell's Law, and how does it describe refraction?

5. How does the slit separation affect the position of bright fringes?

Formulas:

• Bright fringe condition: $d ext{ sin}( heta) = n ext{lambda}$

• Snell's Law: $n<em>1 ext{ sin}( heta</em>1) = n<em>2 ext{ sin}( heta</em>2)$

• Refractive index: $n = rac{c}{v}$

Elasticity Problems:

1. Problem: A force of 1000 N is applied to a bar with a cross-sectional area of 0.01 m². What is the stress in the bar?
   Solution: $ext{Stress} = rac{F}{A} = rac{1000 ext{ N}}{0.01 ext{ m}^2} = 100,000 ext{ Pa}$

2. Problem: A steel rod originally 2 m long is stretched to 2.002 m when a force is applied. Calculate the strain.
   Solution: $ext{Strain} = rac{ ext{Change in Length}}{ ext{Original Length}} = rac{0.002 ext{ m}}{2 ext{ m}} = 0.001$

3. Problem: If the Young's modulus of a material is 200 GPa and the stress is 50 MPa, what is the strain?
   Solution: $ext{Strain} = rac{ ext{Stress}}{ ext{Young's Modulus}} = rac{50 ext{ MPa}}{200 ext{ GPa}} = rac{50 imes 10^6}{200 imes 10^9} = 0.00025$

Simple Harmonic Motion Problems:

1. Problem: A pendulum has a period of 2 seconds. What is its frequency?
   Solution: $f = rac{1}{T} = rac{1}{2 ext{ s}} = 0.5 ext{ Hz}$

2. Problem: If the amplitude of a wave is 0.5 m and its angular frequency is 4 rad/s, what is the maximum velocity?
   Solution: $v_{ ext{max}} = ext{Amplitude} imes ext{Angular Frequency} = 0.5 ext{ m} imes 4 ext{ rad/s} = 2 ext{ m/s}$

3. Problem: Calculate the period of a mass-spring system with spring constant 100 N/m and mass 0.5 kg.
   Solution: $T = 2 ext{π} imes rac{ ext{m}}{ ext{k}} = 2 ext{π} imes rac{0.5 ext{ kg}}{100 ext{ N/m}} = 0.44 ext{ s}$

Mechanical Waves Problems:

1. Problem: A wave on a string has a tension of 200 N and a linear density of 0.5 kg/m. What is the speed of the wave?
   Solution: $v = ext{sqrt} rac{F}{ ext{u}} = ext{sqrt} rac{200 ext{ N}}{0.5 ext{ kg/m}} = 20 ext{ m/s}$

2. Problem: If a wave has a power of 50 W and an area of 2 m², what is the intensity?
   Solution: $I = rac{P}{A} = rac{50 ext{ W}}{2 ext{ m}^2} = 25 ext{ W/m}^2$

Interference and Standing Waves Problems:

1. Problem: Two waves interfere. If the path difference is 0.5 m and the wavelength is 1 m, is the interference constructive or destructive?
   Solution: Since $rac{0.5}{1} = 0.5$ is not an integer multiple, it is destructive.

2. Problem: Determine the frequency of the first normal mode of a string of length 1 m vibrating at a wave speed of 30 m/s.
   Solution: $f_1 = rac{v}{2L} = rac{30 ext{ m/s}}{2 imes 1 ext{ m}} = 15 ext{ Hz}$

Physical Optics Problems:

1. Problem: Light with a wavelength of 500 nm passes through a double slit with a separation of 0.01 m. Find the angle for the first bright fringe.
   Solution: $d ext{ sin}( heta) = n ext{ lambda} <br>ightarrow ext{for } n=1; ext{ sin}( heta) = rac{1 imes 500 imes 10^{-9}}{0.01} = 0.05 <br>ightarrow heta = ext{sin}^{-1}(0.05) ext{ rad}$

2. Problem: If the speed of light in a medium is 2.0 x 10^8 m/s, what is its refractive index?
   Solution: $n = rac{c}{v} = rac{3.0 imes 10^8 ext{ m/s}}{2.0 imes 10^8 ext{ m/s}} = 1.5$