Physics 20: Comprehensive Guide to Simple Harmonic Motion and Mechanical Waves

Simple Harmonic Motion: Definitions and Fundamental Concepts

Oscillatory Motion: This occurs when an object repeats its motion with the same period. An example is the motion of pistons in a car engine rising and falling repeatedly to the same maximum and minimum heights.
Oscillation: One full cycle of motion, such as when a piston falls and returns to its original starting point.
Amplitude ( $A$ ): The magnitude of the maximum displacement of a system from its equilibrium position. It is measured in meters ( $m$ ).
Period ( $T$ ): The time required to complete one full oscillation. It is measured in seconds ( $s$ ).
Frequency ( $f$ ): The number of oscillations completed in one second. It is measured in hertz ( $Hz$ ), where $1.00\,Hz = 1.00\,s^{-1}$ . Frequency can also be expressed in rotations per minute ( $rpm$ ).
Mathematical Relationship Between Period and Frequency: - $T = \frac{1}{f}$ - $f = \frac{1}{T}$
Example Problem 1: Hummingbird Wing Frequency: - A hummingbird flaps its wings at $4680\,rpm$ . - To determine frequency in $Hz$ : $\frac{4680\,rotations}{60.0\,s} = 78\,Hz$ . - To determine Period in seconds ( $2$ sig digs): $T = \frac{1}{78\,Hz} \approx 0.013\,s$ .

Hooke’s Law and Restoring Forces

Hooke’s Law: Robert Hooke determined that the force generated in an elastic material ( $F_s$ , measured in Newtons) is proportional to the deformation of the material ( $x$ , measured in meters) and depends on the spring constant of the material ( $k$ , measured in $N/m$ ). - Formula: $F_s = -kx$ - The negative sign in the formula indicates that the spring force always opposes the direction of deformation.
Behavioral Sections of Elastic Materials: - Section 1: The material obeys Hooke’s Law. The slope of the line on a graph of applied force vs. displacement is equal to the spring constant ( $k$ ). - Section 2: The force applied is beyond the limits of the elastic material, leading to permanent deformation or breakage.
Restoring Force: When a system is disturbed from equilibrium, a net force exists to return it to that state. This net force is the restoring force ( $F_{net}$ ).
Definition of Simple Harmonic Motion (SHM): A system undergoes SHM when it experiences a restoring force that is proportional to the displacement of the system. SHM obeys Hooke’s Law.
Example Problem 2: Spring Deformation: - Given: Force ( $F_{app}$ ) = $50\,N$ , Spring constant ( $k$ ) = $80\,N/m$ . - a. How far will it stretch? $x = \frac{F}{k} = \frac{50\,N}{80\,N/m} = 0.63\,m$ . - b. How far will it compress? The distance remains the same: $0.63\,m$ .
Example Problem 3: Horizontal Mass-Spring System: - Given: Mass ( $m$ ) = $2.00\,kg$ , Spring constant ( $k$ ) = $200\,N/m$ , Displacement ( $x$ ) = $23.0\,cm = 0.230\,m$ . - The maximum restoring force is calculated at the amplitude: $F = -kx = -(200\,N/m)(0.230\,m) = -46.0\,N$ . - The amplitude is $0.230\,m$ .

Dynamics and Kinematics of Mass-Spring Systems

Acceleration in SHM: Using Newton’s Second Law ( $F_{net} = ma$ ) and Hooke's Law ( $F = -kx$ ): - $ma = -kx$ - $a = \frac{-kx}{m}$ - Acceleration and displacement are always opposite in direction. Maximum acceleration occurs at maximum displacement ( $x = A$ ); acceleration is zero at equilibrium ( $x = 0$ ).
Velocity in SHM: Velocity is constantly changing. It is zero at maximum displacement (amplitude) and reaches maximum magnitude at the equilibrium position.
Conservation of Energy in SHM: To find maximum velocity ( $v_{max}$ ), equate elastic potential energy at maximum displacement to kinetic energy at equilibrium: - $E_{pe} = E_k$ - $\frac{1}{2}kx_{max}^2 = \frac{1}{2}mv_{max}^2$ - $v_{max} = \sqrt{\frac{kA^2}{m}} = A\sqrt{\frac{k}{m}}$
Period of a Mass-Spring System: Derived by comparing SHM to circular motion. The maximum velocity in circular motion is $v_{max} = \frac{2\pi A}{T}$ . Setting this equal to the energy-derived $v_{max}$ , we get: - $T = 2\pi \sqrt{\frac{m}{k}}$ - The period of a mass-spring system does not depend on the amplitude.
System States for a Horizontal Spring: - At $+A$ : $F_{net} = -max$ , $a = -max$ , $v = 0$ . - At Equilibrium ( $0$ ): $F_{net} = 0$ , $a = 0$ , $v = +max$ or $-max$ . - At $-A$ : $F_{net} = +max$ , $a = +max$ , $v = 0$ .

Pendulum Systems and Small Angle Approximation

Restoring Force of a Pendulum: The restoring force for a pendulum is proportional to $\sin(\theta)$ . Strictly speaking, this does not follow Hooke’s Law.
Small Angle Approximation: For angles less than $15^\circ$ , $\sin(\theta) \approx \theta$ . In this regime, the restoring force becomes proportional to displacement, and the pendulum undergoes Simple Harmonic Motion.
Equivalent Spring Constant for Pendulums: By comparing the pendulum restoring force to Hooke's Law: - $kx = mg\left(\frac{x}{l}\right)$ - $k = \frac{mg}{l}$
Period of a Pendulum: Substituting the equivalent $k$ into the mass-spring period formula: - $T = 2\pi \sqrt{\frac{l}{g}}$ - The period depends only on the length of the string ( $l$ ) and the gravitational field strength ( $g$ ). It does not depend on the mass of the bob or the amplitude (within small angles).
Determining Gravitational Field Strength: Astronauts can use a pendulum to find $g$ by rearranging the period formula: - $g = \frac{4\pi^2 l}{T^2}$

Damping and Resonance

Damping: In real-world systems, friction, heat, and sound energy are lost, causing the amplitude of oscillation to decrease over time. This reduction in amplitude is known as damping.
Resonant Frequency: The natural frequency of vibration for an object.
Forced Frequency: The frequency with which an external force is applied to an object.
Mechanical Resonance: An increase in the amplitude of a system that occurs when the forced frequency matches the system's resonant frequency. - Example: Pushing a person on a swing is most effective (highest amplitude) when the push occurs at the peak of the backswing. - Practical Implications: Engineers must account for resonance to prevent disasters (e.g., the Tacoma Narrows Bridge collapse) or use dampeners in appliances (e.g., clothes dryers) to avoid resonant vibrations.

Fundamentals of Mechanical Waves

Wave Definition: A disturbance that moves outward from its point of origin, transmitting energy without transmitting mass.
Mechanical Waves: Waves that require a medium (social, liquid, or gas) to travel.
Transverse Waves: Particles of the medium vibrate perpendicularly to the direction of wave travel. These occur in solids and on the surface of fluids (where gravity acts as a restoring force).
Longitudinal Waves: Particles of the medium vibrate parallel to the direction of wave travel. These occur in solids, fluids, and gases. Sound waves are a primary example.
Wave Properties: - Equilibrium Position: The rest or undisturbed position of the medium. - Crest/Trough: The highest and lowest points of a transverse wave. - Compression/Rarefaction: High-density and low-density regions of a longitudinal wave. - Wavelength ($\lambda$): The distance between two points in phase (e.g., crest to crest or trough to trough). - In Phase: Points at the same location moving in the same direction.
Wave Sources: - Point Source: A single point disturbance generating circular waves. - Wavefront: An imaginary line joining all points reached by the wave concurrently. - Wave Train: A series of wavefronts produced by a continuous disturbance.
The Universal Wave Equation: - $v = \frac{\Delta d}{\Delta t} = \frac{\lambda}{T} = f\lambda$

Wave Interactions: Reflection, Refraction, and Interference

Reflection: - Fixed End: A wave pulse encountering a fixed end reflects back inverted. According to Newton's Third Law, the fixed support pulls back in the opposite direction. - Free End: A wave pulse reflects back upright. - Law of Reflection: The angle of incidence equals the angle of reflection ( $\theta_i = \theta_R$ ) relative to the normal line.
Changing Mediums: - When a wave changes medium, its frequency remains constant ( $f_1 = f_2$ ). - The relationship between velocity and wavelength is: $\frac{v_1}{v_2} = \frac{\lambda_1}{\lambda_2}$ . - Low Density to High Density: Part is reflected inverted (same $v, \lambda$ ) and part is transmitted upright with reduced speed and wavelength. - High Density to Low Density: Part is reflected upright (same $v, \lambda$ ) and part is transmitted upright with increased speed and wavelength.
Principle of Superposition: The displacement of a medium at a point where two or more waves interfere is the sum of the displacements of the individual waves.
Interference: - Constructive Interference: Overlapping pulses create an increased amplitude. The maximum displacement is an antinode. - Destructive Interference: Overlapping pulses create a decreased or zero amplitude. A point of zero amplitude is a node.
Two-Point Source Interference: - Produces a pattern of nodal and antinodal lines. - Antinodal lines ( $A_0, A_1, A_2…$ ): Occur where path difference is a whole number of wavelengths ( $0, 1\lambda, 2\lambda…$ ). - Nodal lines ( $n_1, n_2…$ ): Occur where path difference is a half-integer number of wavelengths ( $0.5\lambda, 1.5\lambda…$ ).

Standing Waves and Harmonics

Standing Wave: A wave that oscillates in place without moving through space, caused by the interference of incident and reflected waves at specific resonant frequencies.
Harmonics for a Fixed String: - Fundamental Frequency (1st Harmonic): $L = \frac{1}{2}\lambda$ , so $\lambda = 2L$ . - 2nd Harmonic (1st Overtone): $L = \lambda$ , so $\lambda = L$ . - 3rd Harmonic (2nd Overtone): $L = \frac{3}{2}\lambda$ , so $\lambda = \frac{2}{3}L$ . - General Formula: $L = \frac{n}{2}\lambda$ and $f_n = nf_1$ .
Resonance in Air Columns: - Open Pipes (Antinodes at both ends): Follow the same pattern as fixed strings: $L = \frac{n}{2}\lambda$ and $f_n = nf_1$ . - Closed Pipes (Node at one end, antinode at the other): Only odd harmonics are possible ( $n = 1, 3, 5…$ ). - 1st Harmonic: $L = \frac{1}{4}\lambda$ , $\lambda = 4L$ . - 3rd Harmonic (1st Overtone): $L = \frac{3}{4}\lambda$ , $\lambda = \frac{4}{3}L$ . - 5th Harmonic (2nd Overtone): $L = \frac{5}{4}\lambda$ , $\lambda = \frac{4}{5}L$ . - General Formula: $L = \frac{n}{4}\lambda$ .

The Doppler Effect

Definition: The apparent change in wave frequency due to relative motion between the source and the observer.
Physical Cause: Moving toward an observer compresses wavefronts (higher frequency); moving away spreads them out (lower frequency).
Doppler Formula: - $f = \left(\frac{v_w}{v_w \pm v_s}\right)f_s$ - $v_w$ : speed of the wave; $v_s$ : speed of the source; $f_s$ : actual frequency of the source. - Approaching: Subtract $v_s$ (results in higher perceived frequency). - Receding: Add $v_s$ (results in lower perceived frequency).
Sonic Booms: Occur when a source travels faster than the wave speed (v_s > v_{sound}). Sound waves pile up behind the source, creating a high-pressure shockwave.

Questions & Discussion

Q: Determine the frequency of a guitar string with a period of 0.00400 s. - A: $f = \frac{1}{0.00400} = 2.50 \times 10^2\,Hz$ .
Q: A mass of 2.0 kg is on a horizontal frictionless surface with k = 40.0 N/m. Find the restoring force at a displacement of -0.15 m. - A: $F = -kx = -(40.0\,N/m)(-0.15\,m) = 6.0\,N$ opposite to displacement.
Q: What is the period of a 50.0 cm pendulum on the Moon ( $g = 1.62\,N/kg$ )? - A: $T = 2\pi \sqrt{\frac{0.500\,m}{1.62\,m/s^2}} \approx 3.49\,s$ .
Q: One pulse peak is 15 cm above equilibrium, another is 10 cm below. Where is the interference peak? - A: Using superposition: $15\,cm + (-10\,cm) = 5\,cm$ above equilibrium.
Q: A train moving at 100 km/h sounds its 400 Hz whistle ( $v_{sound} = 344\,m/s$ ). What frequency is heard approaching? - A: Convert speed: $100\,km/h \approx 27.8\,m/s$ . $f = \left(\frac{344}{344 - 27.8}\right)400 \approx 435\,Hz$ .
Q: How could a person walking across a rope bridge prevent resonant vibration? - A: By breaking their stride rhythm so their walking frequency does not match the natural resonant frequency of the bridge.