🎯⭐ INTERACTIVE LESSON

Energy in Simple Harmonic Motion

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Energy in Simple Harmonic Motion - Complete Interactive Lesson

Part 1: Energy in Springs

⚡ KE and PE Exchange in SHM

Part 1 of 7 — Energy in Simple Harmonic Motion

In SHM, energy continuously transforms between kinetic and potential forms. The total mechanical energy remains constant (no friction), but the split between KE and PE changes throughout the motion.

Energy Forms in SHM

Mass-Spring System

$KE = \frac{1}{2}mv^2 \qquad PE = \frac{1}{2}kx^2$

Total Mechanical Energy

$E = KE + PE = \frac{1}{2}mv^2 + \frac{1}{2}kx^2 = \text{constant}$

Energy at Special Points

Position	KE	PE	Total E
$x = 0$ (equilibrium)	Maximum	Zero	$\frac{1}{2}mv_{\text{max}}^2$

At the endpoints, $v = 0$ so all energy is potential. At equilibrium, $x = 0$ so all energy is kinetic.

Energy vs. Position Graphs

What the Graphs Look Like

PE curve: Parabola $U = \frac{1}{2}kx^2$ (upward-opening, minimum at $x = 0$ )

Energy Exchange Quiz 🎯

Energy Calculations 🧮

A 0.40 kg block on a spring ( $k = 160$ N/m) oscillates with amplitude $A = 0.10$ m.

What is the total energy? (in J)
What is the maximum speed? (in m/s)
What is the PE when $x = 0.06$ m? (in J, round to 3 significant figures)

Energy Review 🔍

Exit Quiz — KE and PE Exchange ✅

Part 2: Energy in Pendulums

🔋 Total Energy $E = \frac{1}{2}kA^2$

Part 2 of 7 — Energy in Simple Harmonic Motion

The total mechanical energy of a mass-spring system depends only on the spring constant and the amplitude. This simple but powerful result connects initial conditions to all energy calculations.

Deriving Total Energy

Part 3: KE & PE Graphs in SHM

📐 Energy at Any Position

Part 3 of 7 — Energy in Simple Harmonic Motion

Using energy conservation, we can find the speed of an oscillating object at any position — not just at the endpoints or equilibrium.

The Master Energy Equation

At any displacement $x$ :

$\frac{1}{2}kA^2 = \frac{1}{2}kx^2 + \frac{1}{2}mv^2$

Part 4: Amplitude & Total Energy

🚀 Maximum Velocity $v_{\text{max}} = A\omega$

Part 4 of 7 — Energy in Simple Harmonic Motion

The maximum velocity is one of the most frequently tested quantities in AP Physics 1 SHM problems. It connects amplitude, angular frequency, and energy in a powerful way.

Deriving $v_{\text{max}}$

Part 5: Damped Oscillations

🌊 Damped Oscillations (Conceptual)

Part 5 of 7 — Energy in Simple Harmonic Motion

Real oscillating systems always lose energy to friction, air resistance, or other dissipative forces. This causes the amplitude to decrease over time — a process called damping.

What Is Damping?

Damping is the gradual loss of mechanical energy from an oscillating system, typically due to:

Friction (surface contact)
Air resistance (drag)
Internal friction (deformation of materials)

Effect on Motion

With damping:

Amplitude decreases with each cycle
Period stays approximately the same (for light damping)
Total energy decreases over time
The object eventually comes to rest at the equilibrium position

Energy Perspective

Without damping: $E = \frac{1}{2}kA^2 = \text{constant}$

Part 6: Problem-Solving Workshop

🛠️ Problem-Solving Workshop

Part 6 of 7 — Energy in Simple Harmonic Motion

Time to tackle comprehensive energy problems that combine everything: $E = \frac{1}{2}kA^2$ , speed at any position, , and damping concepts.

Part 7: Synthesis & AP Review

🎓 Synthesis & AP Review

Part 7 of 7 — Energy in Simple Harmonic Motion

This final part brings together all energy concepts in SHM — KE/PE exchange, total energy, speed at any position, maximum velocity, and damping — for a comprehensive AP exam review.

Complete Energy Summary

Core Equations

$E = \frac{1}{2}kA^2 = \frac{1}{2}mv_{\text{max}}^2 = \frac{1}{2}m\omega^2 A^2$

KE curve: Inverted parabola (maximum at

x = 0

, zero at

x = \pm A

)

Total E: Horizontal line at

E = \frac{1}{2}kA^2

PE oscillates as $\cos^2(\omega t)$
KE oscillates as $\sin^2(\omega t)$
Both oscillate at twice the frequency of the position oscillation
When one is at maximum, the other is at minimum

For a pendulum, $PE = mgh$ where $h$ is the height above the lowest point:

At the bottom: $KE$ is max, $PE$ is min
At the sides: $KE = 0$ , $PE$ is max

At the endpoints ( $x = \pm A$ ), velocity is zero, so:

$E = KE + PE = 0 + \frac{1}{2}kA^2 = \frac{1}{2}kA^2$

At equilibrium ( $x = 0$ ), displacement is zero, so:

$E = KE + PE = \frac{1}{2}mv_{\text{max}}^2 + 0 = \frac{1}{2}mv_{\text{max}}^2$

$\frac{1}{2}kA^2 = \frac{1}{2}mv_{\text{max}}^2$

$v_{\text{max}} = A\sqrt{\frac{k}{m}} = A\omega$

What Total Energy Depends On

Factor	Effect on $E$
Double $A$	$E$ quadruples ( $E \propto A^2$ )
Double $k$	$E$ doubles ( $E \propto k$ )
Double $m$	No direct effect (but changes $v_{\text{max}}$ )
Change $g$	No effect (for springs)

Equivalent Expressions for Total Energy

The total energy can be written several ways:

$E = \frac{1}{2}kA^2 = \frac{1}{2}mv_{\text{max}}^2 = \frac{1}{2}m\omega^2 A^2$

Since $\omega^2 = k/m$ :

$E = \frac{1}{2}m\omega^2 A^2 = \frac{1}{2}m \cdot \frac{k}{m} \cdot A^2 = \frac{1}{2}kA^2 \checkmark$

For a Pendulum

$E = mgh_{\text{max}} = \frac{1}{2}mv_{\text{max}}^2$

where $h_{\text{max}}$ is the maximum height above the lowest point. For small angles:

$h_{\text{max}} = L(1 - \cos\theta_{\text{max}}) \approx \frac{L\theta_{\text{max}}^2}{2}$

Total Energy Quiz 🎯

Energy Drill 🧮

A 0.50 kg mass on a spring ( $k = 200$ N/m) has a total energy of 1.0 J. What is the amplitude? (in m)
With the same system, what is $v_{\text{max}}$ ? (in m/s)
A different spring has $k = 80$ N/m and $A = 0.25$ m. What is the total energy? (in J)

Round all answers to 3 significant figures.

Total Energy Concepts 🔍

Exit Quiz — Total Energy ✅

Solving for velocity at position $x$ :

$v = \pm\sqrt{\frac{k}{m}(A^2 - x^2)} = \pm\omega\sqrt{A^2 - x^2}$

Position	Speed
$x = 0$	$v_{\text{max}} = A\omega = A\sqrt{k/m}$
$x = \pm A$	$v = 0$
$x = \pm A/2$	$v = \frac{\sqrt{3}}{2}A\omega \approx 0.866 \cdot v_{\text{max}}$
$x = \pm A/\sqrt{2}$	$v = \frac{1}{\sqrt{2}}A\omega \approx 0.707 \cdot v_{\text{max}}$

At $x = A/\sqrt{2}$ :

$PE = \frac{1}{2}k(A/\sqrt{2})^2 = \frac{1}{4}kA^2 = E/2$
$KE = E/2$
This is where KE = PE (energy is split equally!)

Problem-Solving Strategy

Finding Speed at a Given Position

Find total energy: $E = \frac{1}{2}kA^2$
Find PE at position: $PE = \frac{1}{2}kx^2$
Find KE: $KE = E - PE$
Find speed: $v = \sqrt{2 \cdot KE/m}$

Finding Position at a Given Speed

Find total energy: $E = \frac{1}{2}kA^2$
Find KE: $KE = \frac{1}{2}mv^2$

Energy Bar Charts

At each position, you can draw a bar chart:

Total bar height is constant ( $= E$ )
KE bar shrinks as PE bar grows (moving away from equilibrium)
PE bar shrinks as KE bar grows (moving toward equilibrium)

Speed at a Position Quiz 🎯

Position-Energy Calculations 🧮

A 2.0 kg block on a spring ( $k = 200$ N/m) oscillates with amplitude $A = 0.20$ m.

What is the total energy? (in J)
What is the speed at $x = 0.12$ m? (in m/s)
At what displacement is the speed 1.0 m/s? (in m)

Round all answers to 3 significant figures.

Energy at Any Position Review 🔍

Exit Quiz — Energy at Any Position ✅

From Energy Conservation

At equilibrium ( $x = 0$ ), all energy is kinetic:

$\frac{1}{2}mv_{\text{max}}^2 = \frac{1}{2}kA^2$

$v_{\text{max}} = A\sqrt{\frac{k}{m}} = A\omega$

From the Velocity Equation

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

The maximum value of $|\sin(\omega t)|$ is 1, so:

$v_{\text{max}} = A\omega$

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

How $v_{\text{max}}$ Depends on Parameters

Change	Effect on $v_{\text{max}}$
Double $A$	Doubles
Double $\omega$ (or $f$ )	Doubles
Double both $A$ and $\omega$	Quadruples
Double $m$ (spring)	Decreases by $\sqrt{2}$
Double $k$ (spring)	Increases by $\sqrt{2}$

Connecting $v_{\text{max}}$ to Energy

$E = \frac{1}{2}mv_{\text{max}}^2 = \frac{1}{2}kA^2$

This means:

$v_{\text{max}} = \sqrt{\frac{2E}{m}}$

Speed at Any Position (revisited)

$v(x) = v_{\text{max}}\sqrt{1 - \frac{x^2}{A^2}}$

This elegant form shows that:

At $x = 0$ : $v = v_{\text{max}}$
At $x = A$ :

Maximum Acceleration Connection

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

So $\omega = a_{\text{max}}/v_{\text{max}}$ — a useful relationship for AP problems!

Maximum Velocity Quiz 🎯

Maximum Velocity Calculations 🧮

A 0.25 kg block on a spring ( $k = 100$ N/m) oscillates with $A = 0.08$ m. Find $v_{\text{max}}$ . (in m/s)
An oscillator has $v_{\text{max}} = 5.0$ m/s and $\omega = 10$ rad/s. Find the amplitude. (in m)
A mass has $v_{\text{max}} = 2.0$ m/s and $m = 0.50$ kg. Find the total energy. (in J)

Round all answers to 3 significant figures.

$v_{\text{max}}$ Concepts 🔍

Exit Quiz — Maximum Velocity ✅

With damping: $E$ decreases over time. Since $E \propto A^2$ , the amplitude also decreases. The "lost" mechanical energy is converted to thermal energy (heat).

Types of Damping

1. Underdamped

The system oscillates with decreasing amplitude
Most common in AP problems
Example: a pendulum in air

2. Critically Damped

The system returns to equilibrium as quickly as possible without oscillating
Example: car shock absorbers (ideally)

3. Overdamped

The system returns to equilibrium slowly without oscillating
Takes longer than critical damping
Example: a door closer set too tight

What the Graphs Look Like

Type	Motion
Underdamped	Oscillations with decaying envelope
Critically damped	Smooth exponential return (fastest)
Overdamped	Slow exponential return (no oscillation)

On the AP exam, you mostly need to recognize underdamped motion and understand that energy is gradually converted to thermal energy.

Damping Concepts Quiz 🎯

Energy Loss in Damped Systems

After Each Cycle

If a damped oscillator loses a fraction $f$ of its energy each cycle:

After 1 cycle: $E_1 = (1-f)E_0$
After 2 cycles: $E_2 = (1-f)^2 E_0$
After $n$ cycles: $E_n = (1-f)^n E_0$

Amplitude Decay

Since $E \propto A^2$ :

$A_n = A_0(1-f)^{n/2}$

Quality Factor (Conceptual)

The Q-factor measures how "good" an oscillator is at maintaining its energy:

High Q → low damping → many oscillations before stopping
Low Q → high damping → few oscillations before stopping

Examples: tuning fork (high Q), pendulum in honey (low Q)

Damped Energy Calculations 🧮

A damped oscillator starts with amplitude $A_0 = 0.20$ m and loses 10% of its energy each cycle.

What fraction of the original energy remains after 3 cycles? (as a decimal, round to 3 significant figures)
What is the amplitude after 3 cycles? (in m, round to 3 significant figures)
After how many complete cycles is the energy reduced to less than half? (give the smallest integer)

Damping Review 🔍

Exit Quiz — Damped Oscillations ✅

Energy Problem-Solving Strategy

Identify what you know: $m$ , $k$ , $A$ , $v_{\text{max}}$ , $E$ , position, speed, etc.
Write the energy conservation equation: $\frac{1}{2}kA^2 = \frac{1}{2}kx^2 + \frac{1}{2}mv^2$
Use the appropriate form:
- Need $v_{\text{max}}$ ? → $v_{\text{max}} = A\omega = A\sqrt{k/m}$
Check: Does the answer make physical sense?

Quick Reference

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

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

Problem 1 🎯

A 0.30 kg block attached to a spring ( $k = 120$ N/m) is pulled 0.15 m from equilibrium and released from rest on a frictionless surface.

Problem 2 🧮

A pendulum bob ( $m = 0.50$ kg) swings through its lowest point with a speed of 1.2 m/s.

What is the kinetic energy at the lowest point? (in J, round to 3 significant figures)
What is the maximum height above the lowest point? (in m, round to 3 significant figures, use $g = 9.8$ m/s²)
What is the total mechanical energy of the system? (in J, round to 3 significant figures)

Problem 3 — Energy Ratios 🎯

A block on a spring oscillates with amplitude $A$ and total energy $E$ .

Problem 4 — Conceptual Problem Solving 🔍

A mass-spring system oscillates on a frictionless surface with energy $E$ and amplitude $A$ . Various changes are described below.

Challenge Problem 🧮

A 1.0 kg mass on a spring ( $k = 400$ N/m) has total energy $E = 8.0$ J.

What is the amplitude? (in m)
At what position is the speed 3.0 m/s? (in m, round to 3 significant figures)
At what position is $KE = 2PE$ ? (in m, round to 3 significant figures)

Exit Quiz — Problem Workshop ✅

$\frac{1}{2}kA^2 = \frac{1}{2}kx^2 + \frac{1}{2}mv^2$

$v = \omega\sqrt{A^2 - x^2} \qquad v_{\text{max}} = A\omega$

Energy at Special Positions

Position	KE	PE	Speed
$x = 0$	$E$	$0$	$v_{\text{max}} = A\omega$
$x = \pm A$	$0$	$E$	$0$
$x = \pm A/\sqrt{2}$	$E/2$
$x = \pm A/2$	$3E/4$	$E/4$	$(\sqrt{3}/2)v_{\text{max}}$

Key Proportionalities

$E \propto A^2$ (double amplitude → 4× energy)
$E \propto k$ (double spring constant → 2× energy)
$v_{\text{max}} \propto A$ and $v_{\text{max}} \propto \omega$
Energy oscillates at frequency $2f$

Common AP Mistakes to Avoid

❌ "Energy is proportional to amplitude" — No, $E \propto A^2$ !

❌ "Speed is maximum at the endpoints" — Speed is ZERO at endpoints, maximum at equilibrium.

❌ "KE = PE at x = A/2" — Actually, $KE = PE$ at $x = A/\sqrt{2} \approx 0.707A$ .

❌ "Total energy depends on mass" — $E = \frac{1}{2}kA^2$ has no mass! (Mass affects $v_{\text{max}}$ , not .)

❌ "Damping changes the period" — Light damping barely affects the period; it mainly reduces amplitude.

❌ "Energy is lost in damped motion" — Total energy is conserved; mechanical energy converts to thermal energy.

AP-Style Questions — Set 1 🎯

AP Calculation Practice 🧮

A spring ( $k = 800$ N/m) stores 4.0 J of energy when compressed. What is the compression distance? (in m)
A 0.20 kg block oscillates with $k = 80$ N/m and $A = 0.05$ m. What is the speed at $x = 0.03$ m? (in m)
A damped oscillator starts with $A_0 = 0.30$ m and amplitude decreases to $0.15$ m. What fraction of the original energy remains? (as a decimal)

Round all answers to 3 significant figures.

Comprehensive Energy Review 🔍

Final Exit Quiz — Energy in SHM ✅

Find PE:

PE = E - KE

Find position:

x = \sqrt{2 \cdot PE/k}

x = A/2

v = v_{\text{max}}\sqrt{3/4} = v_{\text{max}}(\sqrt{3}/2) \approx 0.866 v_{\text{max}}

Need

v

at position

x

? →

v = \omega\sqrt{A^2 - x^2}

Need energy? →

E = \frac{1}{2}kA^2 = \frac{1}{2}mv_{\text{max}}^2

Energy in Simple Harmonic Motion

Energy vs. Time

Pendulum Energy

What Total Energy Depends On

Equivalent Expressions for Total Energy

For a Pendulum

Special Cases

Energy Fractions

Problem-Solving Strategy

Finding Speed at a Given Position

Finding Position at a Given Speed

Energy Bar Charts

From Energy Conservation

From the Velocity Equation

Multiple Forms

How $v_{\text{max}}$ Depends on Parameters

Connecting $v_{\text{max}}$ to Energy

Speed at Any Position (revisited)

Maximum Acceleration Connection

Types of Damping

1. Underdamped

2. Critically Damped

3. Overdamped

What the Graphs Look Like

Energy Loss in Damped Systems

After Each Cycle

Amplitude Decay

Quality Factor (Conceptual)

Energy Problem-Solving Strategy

Quick Reference

Energy at Special Positions

Key Proportionalities

Common AP Mistakes to Avoid

Energy in Simple Harmonic Motion

Energy in Simple Harmonic Motion - Complete Interactive Lesson

Part 1: Energy in Springs

⚡ KE and PE Exchange in SHM

Energy Forms in SHM

Mass-Spring System

Total Mechanical Energy

Energy at Special Points

Energy vs. Position Graphs

What the Graphs Look Like

Part 2: Energy in Pendulums

🔋 Total Energy E=12kA2E = \frac{1}{2}kA^2E=21​kA2

Deriving Total Energy

Part 3: KE & PE Graphs in SHM

📐 Energy at Any Position

The Master Energy Equation

Part 4: Amplitude & Total Energy

🚀 Maximum Velocity vmax=Aωv_{\text{max}} = A\omegavmax​=Aω

Deriving vmaxv_{\text{max}}v

Part 5: Damped Oscillations

🌊 Damped Oscillations (Conceptual)

What Is Damping?

Effect on Motion

Energy Perspective

Part 6: Problem-Solving Workshop

🛠️ Problem-Solving Workshop

Part 7: Synthesis & AP Review

🎓 Synthesis & AP Review

Complete Energy Summary

Core Equations

Energy vs. Time

Pendulum Energy

What Total Energy Depends On

Equivalent Expressions for Total Energy

For a Pendulum

Special Cases

Energy Fractions

Problem-Solving Strategy

Finding Speed at a Given Position

Finding Position at a Given Speed

Energy Bar Charts

From Energy Conservation

From the Velocity Equation

Multiple Forms

How vmaxv_{\text{max}}vmax​ Depends on Parameters

Connecting vmaxv_{\text{max}}vmax​ to Energy

Speed at Any Position (revisited)

Maximum Acceleration Connection

Types of Damping

1. Underdamped

2. Critically Damped

3. Overdamped

What the Graphs Look Like

Energy Loss in Damped Systems

After Each Cycle

Amplitude Decay

Quality Factor (Conceptual)

Energy Problem-Solving Strategy

Quick Reference

Energy at Special Positions

Key Proportionalities

Common AP Mistakes to Avoid

🔋 Total Energy $E = \frac{1}{2}kA^2$

🚀 Maximum Velocity $v_{\text{max}} = A\omega$

Deriving $v_{\text{max}}$

How $v_{\text{max}}$ Depends on Parameters

Connecting $v_{\text{max}}$ to Energy