🔄 What is Simple Harmonic Motion?

Part 1 of 7 — Restoring Force Proportional to Displacement

Simple Harmonic Motion (SHM) is one of the most fundamental types of motion in physics. It occurs whenever a restoring force is proportional to the displacement from equilibrium.

Defining SHM

A system exhibits Simple Harmonic Motion when the net force is:

$F = -kx$

Where:

• $F$ = restoring force
• $k$ = a positive constant (spring constant for a mass-spring system)
• $x$ = displacement from equilibrium
• The negative sign means the force always points back toward equilibrium

Why "Simple"?

The force is directly proportional to displacement (linear), making the math relatively straightforward. Real systems often approximate SHM for small displacements.

Why "Harmonic"?

The resulting motion is sinusoidal (sine/cosine waves), which are the fundamental building blocks of harmonic analysis.

Examples of SHM

• Mass on a spring
• Simple pendulum (small angles)
• Vibrating guitar string
• Atoms oscillating in a crystal lattice
• LC electrical circuits

The Restoring Force

The key feature of SHM is the restoring force — it always acts to pull the object back to its equilibrium position.

At Different Positions

Acceleration

Using Newton's Second Law:

$a = \frac{F}{m} = -\frac{k}{m}x = -\omega^2 x$

where $\omega = \sqrt{k/m}$ is the angular frequency.

The acceleration is proportional to displacement and always directed toward equilibrium. This is the hallmark of SHM.

SHM Concept Check 🎯

SHM Basics Drill 🧮

1. A spring with $k = 200$ N/m is stretched 0.1 m from equilibrium. What is the restoring force magnitude? (in N)

2. A 2 kg mass on a spring ($k = 50$ N/m) is displaced 0.3 m. What is the magnitude of its acceleration? (in m/s²)

3. For a mass-spring system, $\omega = \sqrt{k/m}$. If $k = 80$ N/m and $m = 5$ kg, what is $\omega$? (in rad/s)

Round all answers to 3 significant figures.

SHM Foundations 🔍

Exit Quiz — Intro to SHM ✅

🔩 Mass-Spring System

Part 2 of 7 — $T = 2\pi\sqrt{m/k}$

The mass-spring system is the most fundamental example of SHM. A mass $m$ attached to a spring with spring constant $k$ oscillates with a period that depends only on $m$ and $k$.

Period of a Mass-Spring System

$T = 2\pi\sqrt{\frac{m}{k}}$

Frequency and Angular Frequency

$f = \frac{1}{T} = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$

$\omega = 2\pi f = \sqrt{\frac{k}{m}}$

Key Dependencies

The Amazing Independence of Amplitude

The period of SHM does not depend on the amplitude. Whether you stretch the spring a little or a lot, the period is the same. This is called isochronism.

Horizontal vs. Vertical Springs

Horizontal Spring

A mass slides on a frictionless surface. The equilibrium position is where the spring is at its natural length.

Vertical Spring

A mass hangs from a spring. The equilibrium position is where the spring force balances gravity:

$kx_0 = mg \quad \Rightarrow \quad x_0 = \frac{mg}{k}$

The period is the same as the horizontal case: $T = 2\pi\sqrt{m/k}$

The equilibrium point just shifts downward by $x_0$. The oscillation about this new equilibrium is still SHM with the same $T$.

Springs in Combination

Mass-Spring Period Quiz 🎯

Mass-Spring Calculations 🧮

1. A 0.5 kg mass on a spring ($k = 200$ N/m) oscillates. What is the period? (in seconds, round to 3 significant figures)

2. A mass-spring system has $T = 0.8$ s and $k = 50$ N/m. What is the mass? (in kg, round to 3 significant figures)

3. Two springs ($k_1 = 100$ N/m, $k_2 = 300$ N/m) are connected in parallel to a 2 kg mass. What is the period? (in seconds, round to 3 significant figures)

Spring Period Review 🔍

Exit Quiz — Mass-Spring System ✅

🕰️ The Simple Pendulum

Part 3 of 7 — $T = 2\pi\sqrt{L/g}$

A simple pendulum — a mass on a string swinging back and forth — is another classic SHM system (for small angles). Its period depends on the string length and gravitational acceleration.

Period of a Simple Pendulum

For small angular displacements ($\theta < 15°$):

$T = 2\pi\sqrt{\frac{L}{g}}$

Where:

• $L$ = length of the pendulum (from pivot to center of mass)
• $g$ = gravitational acceleration ($9.8$ m/s² on Earth)

Key Dependencies

Why No Mass Dependence?

The restoring force is the tangential component of gravity: $F = -mg\sin\theta$. For small angles, $\sin\theta \approx \theta$:

$F \approx -mg\theta = -mg\frac{x}{L} = -\frac{mg}{L}x$

The effective spring constant is $k_{\text{eff}} = mg/L$. Period: $T = 2\pi\sqrt{m/k_{\text{eff}}} = 2\pi\sqrt{m/(mg/L)} = 2\pi\sqrt{L/g}$. Mass cancels!

Comparing Spring and Pendulum

The Pendulum as a Clock

The independence of period from mass and (small) amplitude is what makes pendulums excellent timekeeping devices. Galileo first noticed this property!

Physical Pendulum

For an extended object (not a point mass on a string):

$T = 2\pi\sqrt{\frac{I}{mgd}}$

where $I$ is the rotational inertia about the pivot and $d$ is the distance from the pivot to the center of mass.

Pendulum Quiz 🎯

Pendulum Calculations 🧮

1. A pendulum is 1.0 m long. What is its period on Earth? (in seconds, round to 3 significant figures, use $g = 9.8$ m/s²)

2. A pendulum has a period of 3.0 s on Earth. What is its length? (in m, round to 3 significant figures)

3. A pendulum has period 2.0 s on Earth. What would its period be on a planet where $g = 2.45$ m/s²? (in s)

Pendulum Review 🔍

Exit Quiz — Simple Pendulum ✅

📊 Position, Velocity, and Acceleration in SHM

Part 4 of 7 — Sinusoidal Relationships

The position, velocity, and acceleration of an object in SHM all vary sinusoidally with time, but they are out of phase with each other.

The SHM Equations

Starting from maximum displacement at $t = 0$:

$x(t) = A\cos(\omega t)$

$v(t) = -A\omega\sin(\omega t)$

$a(t) = -A\omega^2\cos(\omega t)$

Where $\omega = 2\pi/T = 2\pi f$ is the angular frequency.

Phase Relationships

Key insight: Acceleration is always opposite to position!

$a = -\omega^2 x$

This is the defining equation of SHM.

Reading the Graphs

At Maximum Displacement ($x = +A$)

• Position: maximum positive
• Velocity: zero (turning point)
• Acceleration: maximum negative (pointing back to equilibrium)

At Equilibrium ($x = 0$)

• Position: zero
• Velocity: maximum ($\pm A\omega$)
• Acceleration: zero (no net force)

Summary Table

Notice: when $|x|$ is maximum, $|a|$ is maximum and $v = 0$. When $x = 0$, $|v|$ is maximum and $a = 0$.

Phase Relationship Quiz 🎯

SHM Value Calculations 🧮

A mass oscillates with amplitude $A = 0.20$ m and period $T = 4.0$ s.

1. What is the angular frequency $\omega$? (in rad/s, round to 3 significant figures)

2. What is the maximum speed $v_{\text{max}} = A\omega$? (in m/s, round to 3 significant figures)

3. What is the maximum acceleration $a_{\text{max}} = A\omega^2$? (in m/s², round to 3 significant figures)

Phase Relationship Check 🔍

Exit Quiz — Sinusoidal Motion ✅

🔄 Amplitude, Period, and Frequency Relationships

Part 5 of 7 — Connecting the Parameters

Amplitude, period, and frequency are three fundamental quantities that describe every SHM system. Understanding their relationships is crucial for AP Physics 1.

Key Definitions

Fundamental Relationships

$f = \frac{1}{T} \quad \text{and} \quad T = \frac{1}{f}$

$\omega = 2\pi f = \frac{2\pi}{T}$

The Independence of Amplitude

A critical AP concept: amplitude does NOT affect period or frequency in ideal SHM.

• A mass-spring with $A = 10$ cm has the same period as one with $A = 1$ cm (same $m$ and $k$).
• A pendulum swinging 5° has the same period as one swinging 10° (same $L$, small angles).

How Parameters Connect

Maximum Values

$v_{\text{max}} = A\omega = 2\pi f A = \frac{2\pi A}{T}$

$a_{\text{max}} = A\omega^2 = 4\pi^2 f^2 A = \frac{4\pi^2 A}{T^2}$

What Determines Period?

Doubling Experiments

Concept Quiz 🎯

Parameter Conversions 🧮

1. An oscillator has $f = 5.0$ Hz. What is its period? (in seconds)

2. An oscillator has $T = 0.25$ s. What is its angular frequency? (in rad/s, round to 3 significant figures)

3. A spring system has $k = 200$ N/m and $m = 0.50$ kg with amplitude $A = 0.10$ m. What is the maximum speed? (in m/s)

Parameter Review 🔍

Exit Quiz — Amplitude, Period, Frequency ✅

🛠️ Problem-Solving Workshop

Part 6 of 7 — Introduction to SHM

Time to put all your SHM knowledge to work! This workshop covers mass-spring systems, pendulums, sinusoidal relationships, and connecting parameters.

Problem-Solving Strategy for SHM

1. Identify the system — Is it a mass-spring or a pendulum?
2. Choose the correct period formula:
   • Spring: $T = 2\pi\sqrt{m/k}$
   • Pendulum: $T = 2\pi\sqrt{L/g}$
3. Identify what is given and what is asked for ($T$, $f$, $\omega$, $A$, $v_{\text{max}}$, $a_{\text{max}}$)
4. Use relationships: $f = 1/T$, $\omega = 2\pi f$, $v_{\text{max}} = A\omega$, $a_{\text{max}} = A\omega^2$
5. Remember key independences: amplitude does not affect $T$ or $f$

Common Formulas

$T = 2\pi\sqrt{\frac{m}{k}} \qquad T = 2\pi\sqrt{\frac{L}{g}}$

$v_{\text{max}} = A\omega \qquad a_{\text{max}} = A\omega^2 \qquad a = -\omega^2 x$

Problem 1 🎯

A 0.50 kg block on a spring oscillates with a period of 0.40 s and an amplitude of 0.12 m. What is the spring constant?

Problem 2 🧮

A pendulum clock keeps perfect time on Earth ($g = 9.8$ m/s²). It is taken to a planet where $g = 4.9$ m/s².

1. By what factor does the period change? (give as a decimal, round to 3 significant figures)

2. If the Earth period was 1.00 s, what is the new period? (in seconds, round to 3 significant figures)

3. Will the clock run fast or slow on the new planet? (type "fast" or "slow")

Problem 3 🎯

An object in SHM has a position given by $x(t) = 0.15\cos(10t)$ (in meters, with $t$ in seconds).

Problem 4 — Conceptual Reasoning 🔍

Two identical springs each have spring constant $k$. A mass $m$ is attached.

Problem 5 — Challenge 🧮

A 2.0 kg mass hangs from a vertical spring and stretches it 0.10 m to a new equilibrium. It is then pulled down an additional 0.05 m and released.

1. What is the spring constant $k$? (in N/m, use $g = 9.8$ m/s²)

2. What is the period of oscillation? (in seconds, round to 3 significant figures)

3. What is the maximum speed? (in m/s, round to 3 significant figures)

Exit Quiz — Problem Workshop ✅

🎓 Synthesis & AP Review

Part 7 of 7 — Introduction to SHM

This final part ties together restoring forces, mass-spring systems, pendulums, sinusoidal relationships, and parameter connections — everything you need for the AP exam.

Big Picture: SHM at a Glance

Definition

SHM occurs when a restoring force is proportional to displacement: $F = -kx$.

Two Model Systems

Kinematic Equations

$x(t) = A\cos(\omega t) \qquad v(t) = -A\omega\sin(\omega t) \qquad a(t) = -A\omega^2\cos(\omega t)$

Maximum Values

$v_{\text{max}} = A\omega \qquad a_{\text{max}} = A\omega^2$

The Defining Relationship

$a = -\omega^2 x$

Common AP Mistakes to Avoid

❌ "Doubling amplitude doubles the period" — Amplitude does NOT affect period.

❌ "Heavier pendulum bobs swing slower" — Mass does NOT affect pendulum period.

❌ "Velocity is maximum at maximum displacement" — Velocity is ZERO at the endpoints; it is maximum at equilibrium.

❌ "Acceleration is zero at maximum displacement" — Acceleration is MAXIMUM at the endpoints; it is zero at equilibrium.

❌ "Period of a spring depends on gravity" — Spring period depends on $m$ and $k$ only.

❌ "A pendulum on the Moon has the same period" — Moon has $g_{\text{Moon}} \approx g/6$, so $T$ increases by $\sqrt{6}$.

AP-Style Questions — Set 1 🎯

AP Calculation Practice 🧮

1. A block-spring system has $v_{\text{max}} = 3.0$ m/s and $a_{\text{max}} = 12$ m/s². What is the amplitude? (in m, round to 3 significant figures)

2. A pendulum has $L = 2.5$ m. How many complete oscillations does it make in 60 s? (round to the nearest whole number)

3. A mass on a spring has $k = 50$ N/m and $m = 2.0$ kg. At $x = 0.08$ m, what is the magnitude of the acceleration? (in m/s²)

Synthesis Review 🔍

Final Exit Quiz — Introduction to SHM ✅