SHM Foundations Worksheet SHM Foundations
Lesson 1: Forces, Energy, and Periodicity
REf: PHY-WV-01
UNIT: WAVE MECHANICS
Student Name:
Date:
1. Analyzing Restoring Forces
Simple Harmonic Motion (SHM) occurs when the restoring force is directly proportional to the displacement from equilibrium. For a mass-spring system, this is defined by Hooke's Law: \( F_s = -kx \).
Problem A:
A 0.50 kg mass is attached to a vertical spring with a spring constant of 120 N/m. The mass is pulled down 15 cm from its equilibrium position and released.
Calculate the maximum restoring force.
Calculate the acceleration at the moment of release.
2. Period Equations
Mass-Spring System
\[ T = 2\pi\sqrt{\frac{m}{k}} \]
Simple Pendulum
\[ T = 2\pi\sqrt{\frac{L}{g}} \]
Task 1: The Martian Clock
A grandfather clock uses a 1.0 m pendulum. On Earth (\( g = 9.8 \text{ m/s}^2 \)), it keeps perfect time. If taken to Mars (\( g = 3.7 \text{ m/s}^2 \)), what would the new period be? Show all work.
Task 2: Spring Tuning
An engineer wants to double the period of a mass-spring system. They have two choices: change the mass or change the spring. Describe exactly how much the mass would need to change to achieve this.
3. Conservation of Energy
The total mechanical energy in an oscillating system is conserved (ignoring friction): \( E_{total} = K + U \).
Position Kinetic Energy (\( K \)) Potential Energy (\( U \)) Velocity Max Displacement (\( \pm A \)) Zero Maximum Zero Equilibrium (\( x=0 \)) At \( x = A/2 \)
Technical Diagram 1.4: Energy Balance in Oscillating Systems
Critical Synthesis
A bungee jumper oscillates at the end of a cord. If we model this as SHM, why does the jumper eventually stop? Relate your answer to the concepts of "restoring force" and "mechanical energy."
SHM Foundations Slides SHM Foundations
The Physics of Predictable Patterns
Unit: Wave Mechanics | Lesson 01
The Bungee Conundrum
A jumper drops. They bounce. Then they bounce again. And again.
"Can we predict exactly when they will stop moving based on physics alone?"
Visual: Bungee Motion Path
What variables matter? Height? Mass? Cord stiffness?
Conditions for SHM
1
Restoring Force
A force that always points back toward the equilibrium position.
2
Proportionality
The force magnitude is directly proportional to displacement (\( x \)).
Hooke's Law
\[ F_s = -kx \]
The Mathematical DNA of SHM
Spring Systems
The period depends solely on the inertia of the system (mass) and the stiffness of the restoring agent (spring constant).
\[ T = 2\pi\sqrt{\frac{m}{k}} \]
Derivation Logic
Start with \( F_{net} = ma \)
Substitute Hooke's Law: \( -kx = ma \)
Relate to angular frequency: \( \omega^2 = \frac{k}{m} \)
Use the identity \( T = \frac{2\pi}{\omega} \)
The Pendulum
Intriguingly, for small angles, the mass cancels out! Gravity serves as the restoring agent.
\[ T = 2\pi\sqrt{\frac{L}{g}} \]
Small Angle Approximation:
The Energy Seesaw
Max Displacement
\( U = \frac{1}{2}kA^2 \)
Kinetic Energy is Zero.
Equilibrium
\( K = \frac{1}{2}mv_{max}^2 \)
Potential Energy is Zero.
Total Energy
\( E = K + U \)
Always Constant.
Continuous Transformation
SHM Foundations Teacher Guide Teacher Guide: SHM Foundations
Lesson 1: Bridging Forces and Oscillations
Lesson Overview
This lesson introduces the fundamental mechanics of Simple Harmonic Motion (SHM). Students transition from high-level force analysis to deriving the periodic nature of springs and pendulums. The goal is to establish that SHM is not "magic," but a direct result of linear restoring forces and Newton's Second Law.
Key Misconceptions
Students often believe the period of a pendulum depends on the mass. (Emphasize the derivation where mass cancels).
Confusing "Amplitude" with "Period." (Remind them that for ideal SHM, amplitude does not affect frequency).
Misidentifying restoring forces. (Forces always point toward equilibrium, not the direction of motion).
Pacing Guide
00-10m Bungee Hook & Discussion
10-25m Derivations (Slides 3-5)
25-45m Workshop: Worksheet
45-60m Energy Debrief
Worksheet Answer Key
Part 1: Problem A (Mass-Spring)
Max Force: \( F = kx = (120 \text{ N/m})(0.15 \text{ m}) = 18 \text{ N} \).Acceleration: \( a = F/m = 18 \text{ N} / 0.50 \text{ kg} = 36 \text{ m/s}^2 \).
Part 2: Task 1 (Martian Clock)
Earth Period: \( T_e = 2\pi\sqrt{1.0/9.8} \approx 2.0 \text{ s} \).Insight: The clock would run significantly slower on Mars.
Part 2: Task 2 (Spring Tuning)
Since \( T \propto \sqrt{m} \), to double the period (\( 2T \)), the mass must increase by a factor of 4 (\( \sqrt{4} = 2 \)).
Part 3: Energy Table
Equilibrium: Kinetic = Maximum; Potential = Zero; Velocity = Maximum.At \( x = A/2 \): Potential = \( \frac{1}{4} \) Max; Kinetic = \( \frac{3}{4} \) Max. (Since \( U \propto x^2 \)).
Facilitating the Hook
When discussing the bungee jumper, guide students toward the idea of damping . In an ideal world (no friction), they would oscillate forever. In reality, energy is lost to air resistance and internal heating of the cord. This sets the stage for "conservation of energy" and its real-world limitations.
Graphing Periodic Motion Worksheet The Motion Blueprint
Lesson 2: Graphing Sinusoidal Motion
REF: PHY-WV-02
UNIT: WAVE MECHANICS
Student Name:
Date:
1. Position and Time
The position of an object in SHM follows a cosine or sine function: \( x(t) = A \cos(\omega t + \phi) \).
Task: Plotting the Oscillator
A mass oscillates with an amplitude of 10 cm and a period of 4.0 seconds. It starts at maximum positive displacement at \( t = 0 \). Sketch the displacement-time graph for two full cycles.
0 cm
+10
-10
Time (s)
0s
2s
4s
6s
8s
2. Derivative Relationships
Conceptual Logic
When the displacement is maximum, velocity is _______ and acceleration is _______.
Acceleration is always 180° out of phase with _______.
Phase Equations
\[ x = A \cos(\omega t) \]
\[ v = -A\omega \sin(\omega t) \]
\[ a = -A\omega^2 \cos(\omega t) \]
3. The Reference Circle
SHM can be viewed as the one-dimensional projection of uniform circular motion. The radius of the circle corresponds to the amplitude.
Reflective Task:
A light shines from above onto a rotating horizontal peg. If the peg rotates at 2.0 rad/s and has a radius of 0.5 m, calculate the period of the shadow's motion on the floor.
Graphing Periodic Motion Slides Graphing Motion
Visualizing the Sinusoidal Blueprint
From Circles to Lines
In a car engine, the rotational motion of the crankshaft is translated into the linear motion of the piston.
How do we describe the position of that piston at any given millisecond?
Linear projection of circular path
The Displacement Function
\[ x(t) = A \cos(\omega t) \]
A Amplitude (Max displacement)
ω Angular Frequency (\( 2\pi f \))
t Time
Visual: Cosine Graph Animation
The Phase Trio
Position (\( x \))
\( \cos(\omega t) \)
Start at Max
Velocity (\( v \))
\( -\sin(\omega t) \)
90° Phase Shift
Acceleration (\( a \))
\( -\cos(\omega t) \)
180° Out of Phase
KEY: When \( x \) is at its POSITIVE peak, \( a \) is at its NEGATIVE peak!
x = A cos θ
Unified Mechanics
The linear oscillator is just a shadow of circular motion.
Radius = Amplitude (\( A \))
Angular Velocity (\( \omega \)) = Angular Frequency
Graphing Periodic Motion Teacher Guide Teacher Guide: Graphing Periodic Motion
Lesson 2: The Calculus and Geometry of SHM
Lesson Overview
This lesson shifts from forces to kinematics. Students must master the phase relationships between position, velocity, and acceleration. This is a critical building block for understanding wave interference later in the unit. The reference circle is used to provide a geometric intuition for the trigonometric functions used in SHM.
Pedagogical Note: Calculus
For students in AP Physics C or calculus-heavy tracks, this is a great time to introduce the derivative relationship (\( v = dx/dt \)). For general physics, focus on the "slope" relationship: at the peak of the position graph, the slope (velocity) is zero.
Learning Targets
Construct x-t graphs from given physical parameters (\( A, T \)).
Identify phase differences between \( x, v, \) and \( a \).
Calculate period from angular velocity in circular systems.
Worksheet Answer Key
Part 1: The Oscillator Graph
The graph should be a cosine curve starting at +10 cm.Key points: Peak at 0s and 4s. Trough (-10 cm) at 2s and 6s. Equilibrium crossings (0 cm) at 1s, 3s, 5s, 7s.
Part 2: Conceptual Logic
When \( x = max \), \( v = \) Zero .
When \( x = max \), \( a = \) Max Negative .
Acceleration is 180° out of phase with Displacement .
Part 3: Reference Circle
Given \( \omega = 2.0 \text{ rad/s} \).
Closing Discussion
"If the piston in your car engine is moving in SHM, why does your car vibrate more at high RPMs?"
Guide students to the acceleration equation: \( a = -A\omega^2 \cos(\omega t) \). As RPM (\( \omega \)) increases, the acceleration (and thus the force \( F=ma \)) increases by the square of the frequency. This is why high-performance engines require stronger materials!
Wave Equation Worksheet Wave Anatomy and Speed
Lesson 3: The Universal Wave Equation
REF: PHY-WV-03
UNIT: WAVE MECHANICS
Student Name:
Date:
1. Spatial vs. Temporal Analysis
Unlike a simple oscillator, a wave exists in both time and space . We must distinguish between displacement-time graphs and displacement-position graphs.
Wavelength (\(\lambda\))
Crest
Trough
Transverse Waves:
Particles move perpendicular to wave motion.
Sketch Particle Motion vs Wave Direction
Longitudinal Waves:
Particles move parallel to wave motion.
Sketch Compressions and Rarefactions
2. The Universal Wave Equation
Speed
v
=
Frequency
f
×
Wavelength
\(\lambda\)
Problem Set 3.1: Medium Shifts
A sound wave with a frequency of 440 Hz travels from air (\( v = 340 \text{ m/s} \)) into water (\( v = 1500 \text{ m/s} \)).
Calculate the wavelength of the sound in air.
The frequency does not change when crossing media. Calculate the new wavelength in water.
3. Synthesis Question
If a tsunami travels at 800 km/h (the speed of a jet), but its period is 20 minutes, calculate its wavelength. Explain why such a wave is barely noticeable in the deep ocean but catastrophic at the shore.
Wave Equation Slides The Wave Equation
Propagating Energy Through Space
The Tsunami Paradox
A tsunami travels at 800 km/h in deep water. In the middle of the ocean, ships barely feel it pass.
"Why does it only become destructive when it hits the shore?"
Visual: Wave shoaling diagram
Hint: It's all about speed and depth.
Wave Anatomy
Spatial Wave Model
Amplitude
Spatial Measures
• Wavelength (\(\lambda\)): Crest to Crest
• Amplitude (\(A\)): Center to Crest
Temporal Measures
• Period (\(T\)): Time for one cycle
• Frequency (\(f\)): Cycles per second
The Core Relationship
\[ v = f \lambda \]
Frequency (\( f \))
Determined ONLY by the Source . It does not change when the wave enters a new medium.
Wave Speed (\( v \))
Determined ONLY by the Medium (tension, density, depth, temperature).
Transverse
Particle motion is perpendicular to wave energy.
Ex: Light, Water Waves
Longitudinal
Particle motion is parallel to wave energy.
Ex: Sound, P-Waves (Earthquake)
Quick Quiz: Is a Slinky transverse or longitudinal?
Answer: Both! It depends on how you pulse it.
Wave Equation Teacher Guide Teacher Guide: The Wave Equation
Lesson 3: From Point Oscillations to Moving Energy
Lesson Overview
In this lesson, we introduce the spatial dimension of wave mechanics. Students must understand that while a point in the medium oscillates in time, the wave pattern itself moves in space. The wave equation \( v = f\lambda \) is the mathematical bridge between these two perspectives.
Instructional Priority: Media and Source
Students often struggle with which variable is fixed. Stress this:Frequency is fixed by the source (e.g., the vocal cord).Speed is fixed by the medium (e.g., the water).Wavelength is the result of their interaction.
Key Benchmarks
Distinguish between transverse and longitudinal motion.
Calculate wavelength changes when waves cross boundaries.
Model spatial anatomy (crests, troughs, rarefactions).
Worksheet Answer Key
Part 2: Problem 3.1 (Medium Shifts)
1. In Air: \( \lambda = v/f = 340 \text{ m/s} / 440 \text{ Hz} \approx 0.77 \text{ m} \).2. In Water: \( \lambda = v/f = 1500 \text{ m/s} / 440 \text{ Hz} \approx 3.41 \text{ m} \).Discussion: Sound waves stretch out as they speed up in more rigid media.
Part 3: Tsunami Synthesis
Calculation:
The Paradox: Because the wavelength is hundreds of kilometers long, the "slope" of the wave is incredibly shallow in deep water. Ships just lift and lower slightly over 20 minutes. It becomes catastrophic at the shore because as the water shallows, speed (\(v\)) drops, causing the wave to "pile up" (wavelength compresses, amplitude increases) to conserve energy.
Pacing Guide
00-10: Tsunami Hook discussion10-25: Wave Anatomy & Equation lecture25-45: Collaborative Worksheet practice45-60: Media Boundary discussion & Exit Ticket
Demonstration Tip
Use a "Long Slinky" to demonstrate both transverse (side-to-side) and longitudinal (push-pull) waves. Note that the slinky doesn't move across the room—only the energy pulse does!
Superposition Worksheet Wave Superposition
Lesson 4: Calculating Resultant Amplitudes
REF: PHY-WV-04
UNIT: WAVE MECHANICS
Student Name:
Date:
1. Linear Addition
The Principle of Superposition states that when two or more waves meet, the resultant displacement at any point is the algebraic sum of the displacements of the individual waves.
\[ y_{total} = y_1 + y_2 + ... + y_n \]
Constructive Interference
Waves are "in phase" (crests meet crests).
Sketch Wave A + B
Destructive Interference
Waves are "out of phase" (crests meet troughs).
Sketch Wave A + B
2. Point-by-Point Analysis
Two square pulses travel toward each other on a string. Pulse A has an amplitude of +2.0 cm, and Pulse B has an amplitude of -1.5 cm. At the moment they perfectly overlap, calculate and sketch the resultant string shape.
+2 cm
-2 cm
0 cm
Calculation:
Resultant Displacement = (+2.0 cm) + (-1.5 cm) = ________ cm
3. Real-World Engineering
Noise-canceling headphones use a microphone to record incoming sound waves (noise). The headphone's computer then generates a secondary wave.
Technical Task:
If the incoming noise wave is \( y_{noise} = 0.5 \sin(500t) \), what equation must the headphone's computer generate to create perfect silence?
y_generated = ______________________
Note: Consider phase shifts or amplitude negation.
Superposition Slides Superposition
The Geometry of Interference
Engineering Silence
Noise-canceling headphones don't "block" sound. They create sound to fight sound.
"Silence is not the absence of energy; it's the perfect balance of opposing energies."
Calculated in real-time, thousands of times per second.
Active Noise Control (ANC)
The Principle of Superposition
When waves cross, their displacements simply add together. They don't bounce off each other; they pass through each other.
\[ y_{res} = y_1 + y_2 \]
Constructive
Reinforcement. High + High = Double High.
Destructive
Cancellation. High + Low = Zero.
Point-by-Point Analysis
The Logic
Identify the displacement of Wave 1 at point \( x \).
Identify the displacement of Wave 2 at the SAME point \( x \).
Calculate the sum.
Repeat for every point on the grid.
Phasing visualization: Waves meeting in transit
Interference Conditions
In Phase
\( \Delta \phi = 0, 2\pi, 4\pi \)
Waves match perfectly. Maximum noise/light.
Out of Phase
\( \Delta \phi = \pi, 3\pi, 5\pi \)
Waves are inverted. Silence/Darkness.
Question: What happens when the waves have different frequencies?
Stay tuned for Beats and Synthesis!
Superposition Teacher Guide Teacher Guide: Superposition
Lesson 4: Addition of Energy in Systems
Lesson Overview
This lesson covers the "Principle of Superposition." The core challenge for students is moving from seeing waves as objects that collide to seeing them as disturbances that overlap. The mathematical operation (simple addition) is easy, but the visual execution (adding positive and negative values on a grid) requires precision.
Misconception Alert: Reflection
Students often think waves "bounce" off each other like pool balls. Clarify that waves travel through each other. Use a slinky or ripple tank to show pulses passing through and continuing their original path unaffected after the overlap.
Pacing Guide
00-12m ANC/Headphone Discussion
12-25m Superposition Math Lecture
25-50m Workshop: Grid Graphing
50-60m Synthesizing Phase shifts
Worksheet Answer Key
Part 2: Grid Analysis
Calculation: \( y_{res} = (+2.0 \text{ cm}) + (-1.5 \text{ cm}) = +0.5 \text{ cm} \).Sketching: The resultant shape should be a single square pulse with a height of +0.5 cm, centered on the grid where the two waves overlap.
Part 3: Noise Cancellation Math
Given noise: \( y_{noise} = 0.5 \sin(500t) \).negative of the noise.Option 1: \( y_{generated} = -0.5 \sin(500t) \).Option 2: \( y_{generated} = 0.5 \sin(500t + \pi) \).(Both are mathematically equivalent and represent a 180° phase shift).
Demonstration Tip
"Destructive interference doesn't mean the energy is gone! It just means it's stored elsewhere (often as potential energy or transferred via different modes). In noise-canceling headphones, the electrical energy used to create the anti-noise wave is eventually dissipated as heat."
Wave Mastery Problem Set Wave Mastery Analysis
Lesson 5: Synthesis and Error Correction
REF: PHY-WV-05
UNIT: WAVE MECHANICS
Student Name:
Date:
1. Medium Synthesis
"When you inhale helium, your voice sounds higher. Most people think your vocal cords are vibrating faster, but this is a misconception."
The Physics:
Speed of sound in Air ≈ 340 m/s
Speed of sound in Helium ≈ 970 m/s
The geometry of your throat (vocal tract) sets a fixed wavelength (\(\lambda\)) for the resonant sound.
Calculate the Frequency Shift
If \(\lambda = 0.5 \text{ m}\), calculate the frequency in Air vs. Helium.
2. Debugging the Scientist
Read the student claim below. Identify the two mathematical or conceptual errors made.
"A pendulum with a mass of 2 kg and a length of 1 m oscillates on Earth. If I double the mass to 4 kg, the period will also double because of Newton's Second Law. Furthermore, as the pendulum swings, its kinetic energy is at a maximum when it reaches the highest point of its arc because that's where its displacement is greatest."
Error 1:
Error 2:
3. The Master Problem
Mastery Level
The Oscillating String Generator:
A 0.10 kg mass is attached to a vertical spring (\(k = 40 \text{ N/m}\)). This mass is used as the source for a transverse wave on a long string with a tension that allows waves to travel at 12 m/s.
Calculate the period of the mass-spring oscillation.
Identify the frequency of the waves produced on the string.
Calculate the wavelength of the waves on the string.
Wave Mechanics Mastery Slides Wave Mastery
Synthesizing the Mechanics of Motion
The Helium Myth
Does helium make your vocal cords vibrate faster?
"No. It changes the medium, not the source."
Air
340 m/s
Helium
970 m/s
If \(\lambda\) is fixed by your throat size, then \(f\) must increase as \(v\) increases!
The "Common Traps"
Trap 1: Mass
Thinking mass affects the period of a pendulum.
Real-Talk: \( T = 2\pi\sqrt{L/g} \). No \( m \) in sight!
Trap 2: Speed
Thinking frequency affects wave speed.
Real-Talk: Wave speed is a property of the medium only.
The Master Path
Step 1: The Source
Find \( f_{shm} \)
Oscillating mass defines the wave frequency.
Step 2: The Medium
Identify \( v \)
Medium properties (tension/density) fix the speed.
Step 3: Synthesis
Solve \( \lambda = v/f \)
Combine source and medium data.
What if?
If the tension in the string is quadrupled, while the mass-spring source remains the same...
What exactly happens to the wavelength ?
Doubles?
Halves?
Stays Same?
Hint: \( v \propto \sqrt{\text{Tension}} \)
Wave Mastery Teacher Guide Teacher Guide: Wave Mastery
Lesson 5: Synthesis and Quantitative Rigor
Lesson Overview
This final session acts as a capstone. Students are no longer working with isolated variables; they must now connect the "source" (SHM) to the "medium" (propagation) and the "interaction" (superposition). The Helium Voice hook is used to solidify the distinction between source-dependent and medium-dependent variables.
Instructional Strategy: Error Analysis
By Lesson 5, students can often solve problems by "formula hunting." The Error Analysis section (Section 2 of the worksheet) is designed to break this habit by forcing them to identify conceptual flaws in plain English claims.
Mastery Targets
Relate angular frequency (\(\omega\)) to wave period.
Differentiate between mass effects in springs vs. pendulums.
Synthesize multi-step wave propagation problems.
Mastery Set Answer Key
Section 1: Helium Frequency Shift
Air: \( f = v/\lambda = 340 / 0.5 = 680 \text{ Hz} \).Helium: \( f = v/\lambda = 970 / 0.5 = 1940 \text{ Hz} \).Insight: The higher speed in helium results in a significantly higher frequency for the same resonant wavelength.
Section 2: Error Analysis
Error 1: Claiming mass doubles the period of a pendulum. (Mass cancels out in the pendulum derivation; period only depends on \(L\) and \(g\)).Error 2: Claiming kinetic energy is max at the highest point. (Kinetic energy is max at equilibrium; potential energy is max at the highest point).
Section 3: The Master Problem
Period: \( T = 2\pi\sqrt{m/k} = 2\pi\sqrt{0.1/40} \approx 0.314 \text{ s} \).Frequency: \( f = 1/T \approx 3.18 \text{ Hz} \). (The wave frequency is forced by the source).Wavelength: \( \lambda = v/f = 12 \text{ m/s} / 3.18 \text{ Hz} \approx 3.77 \text{ m} \).
Synthesis Debrief Guide
To wrap up the unit, ask: "If we changed the string to a heavier rope (making sound travel slower), but kept the same spring-mass oscillator, which variable would change: frequency or wavelength?"
Guide them to understand that wavelength would decrease (compress) because the source's frequency is "boss" and the medium's slower speed forces the wavelength to adapt.
