🌍 Gravitational PE: $PE_g = mgh$

Part 1 of 7 — Gravitational Potential Energy

When you lift a ball above the ground, you're storing energy in it — energy that can be released when the ball falls. This stored energy is called gravitational potential energy. It depends on an object's mass, height, and the strength of gravity.

Defining Gravitational Potential Energy

The gravitational potential energy of an object near Earth's surface is:

$PE_g = mgh$

where:

• $m$ = mass of the object (kg)
• $g$ = acceleration due to gravity ($\approx 10$ m/s²)
• $h$ = height above the chosen reference level (m)

Properties

The Reference Level

What Is It?

The reference level (or reference point) is the position where $h = 0$ and therefore $PE_g = 0$. You get to choose it!

Rules for Choosing

• You can place the reference level anywhere
• Common choices: ground level, tabletop, lowest point in the problem
• Once chosen, keep it consistent throughout the problem
• Only changes in PE ($\Delta PE$) have physical meaning — the absolute value depends on your reference

Example

A ball is on a 3 m high table in a room with a 2 m deep basement:

Different reference levels give different PE values, but the change in PE between any two positions is always the same!

Changes in Gravitational PE

The change in gravitational PE is:

$\Delta PE_g = mg\Delta h = mg(h_f - h_i)$

Sign Convention

Key Insight

$\Delta PE_g$ is independent of the reference level you choose. This is why only changes in PE are physically meaningful.

Relation to Work by Gravity

$W_{\text{gravity}} = -\Delta PE_g$

When an object falls ($\Delta PE_g < 0$), gravity does positive work. When an object rises ($\Delta PE_g > 0$), gravity does negative work.

Gravitational PE Concepts 🎯

PE Calculations 🧮

Use $g = 10$ m/s².

1. What is the gravitational PE of a 5 kg object 8 m above the ground (in J)?

2. A 3 kg book is moved from a 1.2 m table to a 2.0 m shelf. What is the change in gravitational PE (in J)?

3. A 70 kg person descends 15 m in an elevator. What is the change in their gravitational PE (in J)?

Reference Level Practice 🔍

Exit Quiz — Gravitational PE ✅

🔄 Conservative Forces & Path Independence

Part 2 of 7 — Gravitational Potential Energy

Why can we define "potential energy" for gravity but not for friction? The answer lies in a special property of gravity: it's a conservative force. The work it does depends only on the starting and ending positions, not on the path taken.

What Is a Conservative Force?

A force is conservative if the work it does on an object depends only on the initial and final positions, not on the path taken between them.

Equivalent Definitions

A force is conservative if:

1. The work done is path-independent
2. The work done around any closed path (round trip) is zero
3. A potential energy function can be defined for it

Examples

Gravity: Path Independence

Key Demonstration

Consider moving a ball from point A (height $h_1$) to point B (height $h_2$):

Path 1: Straight up $W_g = -mg(h_2 - h_1)$

Path 2: Diagonal ramp $W_g = -mg(h_2 - h_1) \text{ (same!)}$

Path 3: Crazy winding path $W_g = -mg(h_2 - h_1) \text{ (still the same!)}$

No matter how the object gets from $h_1$ to $h_2$, gravity does the same work. Only the vertical displacement matters.

Round Trip

If an object starts and ends at the same height: $W_g = -mg(h_f - h_i) = -mg(0) = 0$

The work done by gravity over any closed path is zero. ✓

Friction: NOT Conservative

Why Friction Is Different

Friction always opposes motion, so:

• A longer path → more distance → more negative work by friction
• A shorter path → less distance → less negative work by friction

The work done by friction depends on the path length, not just the endpoints.

Round Trip with Friction

Push a box 5 m to the right and then 5 m back ($f_k = 10$ N): $W_f = -10(5) + (-10)(5) = -100 \text{ J} \neq 0$

Friction does net negative work on a round trip → energy is lost to heat → friction is non-conservative.

Consequence

Because friction is non-conservative, we cannot define a "friction potential energy." The energy lost to friction is converted to thermal energy and cannot be fully recovered.

Conservative Force Concepts 🎯

Conservative Force Calculations 🧮

Use $g = 10$ m/s².

1. A 3 kg ball is carried from the ground to a height of 5 m via a winding staircase. What is the work done by gravity (in J)?

2. The same ball is then dropped back to the ground. What is the total work done by gravity for the entire round trip (in J)?

3. A 4 kg box is pushed 10 m across a rough floor ($\mu_k = 0.2$) and then pushed 10 m back. What is the total work done by friction (in J)?

Classify the Forces 🔍

Exit Quiz — Conservative Forces ✅

⬇️ Work Done by Gravity: $W_g = -\Delta PE_g$

Part 3 of 7 — Gravitational Potential Energy

There's a deep connection between the work gravity does and the change in gravitational potential energy. Understanding this relationship is essential for solving energy problems efficiently.

The Work-PE Relationship

For gravity (a conservative force):

$W_g = -\Delta PE_g = -(PE_f - PE_i) = PE_i - PE_f$

Equivalently:

$W_g = -mg(h_f - h_i) = mg(h_i - h_f)$

Why the Negative Sign?

The negative sign captures an important physical idea:

When PE decreases, gravity does positive work (energy is released). When PE increases, gravity does negative work (energy is stored).

Think of It Like a Bank Account

• $PE_g$ is your "gravitational savings account"
• Going up = depositing energy (positive $\Delta PE$, negative work by gravity)
• Going down = withdrawing energy (negative $\Delta PE$, positive work by gravity)

Worked Examples

Example 1: Free Fall

A 2 kg ball falls from $h = 10$ m to $h = 3$ m. ($g = 10$ m/s²)

$W_g = -\Delta PE_g = -mg(h_f - h_i) = -2(10)(3 - 10) = -20(-7) = 140 \text{ J}$

Gravity does +140 J of work. This energy goes into kinetic energy.

Example 2: Throwing Upward

A 0.5 kg ball is thrown upward from $h = 1$ m to $h = 6$ m.

$W_g = -mg(h_f - h_i) = -0.5(10)(6 - 1) = -25 \text{ J}$

Gravity does −25 J of work. The ball slows down as KE converts to PE.

Example 3: Projectile

A ball is launched at an angle. At the peak, it has risen 4 m. Mass = 1 kg.

$W_g = -mg\Delta h = -1(10)(4) = -40 \text{ J}$

The ball lost 40 J of KE going up (converting to PE).

Work by Gravity Concepts 🎯

Work by Gravity Calculations 🧮

Use $g = 10$ m/s².

1. A 3 kg ball is dropped from 15 m. What is the work done by gravity as it falls to the ground (in J)?

2. A 0.4 kg ball is thrown straight up and rises 8 m above its launch point. What is the work done by gravity during the ascent (in J)?

3. A 10 kg box slides 5 m down a $30°$ incline. What is the work done by gravity (in J)?

Work-PE Sign Analysis 🔍

Exit Quiz — Work Done by Gravity ✅

🔄 KE ↔ PE Conversions

Part 4 of 7 — Gravitational Potential Energy

One of the most elegant ideas in physics is that energy can transform between kinetic and potential forms. A falling ball converts PE to KE; a rising ball converts KE to PE. In this lesson, we'll master these energy conversions.

Falling Objects: PE → KE

When an object falls freely (no friction), gravitational PE converts entirely to kinetic energy:

$\Delta PE + \Delta KE = 0$ $mgh_i + \frac{1}{2}mv_i^2 = mgh_f + \frac{1}{2}mv_f^2$

Dropped from Rest ($v_i = 0$)

$mgh = \frac{1}{2}mv_f^2$ $v_f = \sqrt{2gh}$

Notice: The mass cancels! All objects fall at the same rate (Galileo's insight).

Example

A ball is dropped from 20 m ($g = 10$ m/s²):

$v_f = \sqrt{2(10)(20)} = \sqrt{400} = 20 \text{ m/s}$

Thrown Upward: KE → PE

When an object is thrown straight up, kinetic energy converts to potential energy until the object momentarily stops at the top:

$\frac{1}{2}mv_i^2 = mgh_{\max}$ $h_{\max} = \frac{v_i^2}{2g}$

Example

A ball is thrown upward at 30 m/s ($g = 10$ m/s²):

$h_{\max} = \frac{(30)^2}{2(10)} = \frac{900}{20} = 45 \text{ m}$

At Intermediate Heights

At any height $h$ during the flight:

$\frac{1}{2}mv_i^2 = mgh + \frac{1}{2}mv^2$ $v = \sqrt{v_i^2 - 2gh}$

General Case: Any Starting Conditions

For an object with initial speed $v_i$ at height $h_i$ that reaches height $h_f$ with speed $v_f$ (no friction):

$\frac{1}{2}mv_i^2 + mgh_i = \frac{1}{2}mv_f^2 + mgh_f$

Key Insight: Direction Doesn't Matter

Energy is a scalar. The direction of velocity doesn't affect KE. So:

• A ball thrown upward at 10 m/s from 5 m height
• A ball thrown horizontally at 10 m/s from 5 m height
• A ball thrown downward at 10 m/s from 5 m height

All three have the same speed when they reach the ground (assuming no air resistance). They have the same initial KE and PE, and the same final PE (ground level).

Energy Tracking Table

KE ↔ PE Conversion Concepts 🎯

KE ↔ PE Calculations 🧮

Use $g = 10$ m/s².

1. A ball is dropped from 31.25 m. What is its speed just before hitting the ground (in m/s)?

2. A ball is thrown upward at 24 m/s. What maximum height does it reach (in m)?

3. A 2 kg ball is thrown upward at 10 m/s from a 15 m tall building. What is its speed when it hits the ground (in m/s)?

Round all answers to 3 significant figures.

Energy Conversion Analysis 🔍

Exit Quiz — KE ↔ PE Conversions ✅

📊 Energy Bar Charts

Part 5 of 7 — Gravitational Potential Energy

Energy bar charts (also called LOL diagrams) are a powerful visual tool for tracking energy transformations. They show how energy is distributed among different forms at each stage of a process. The AP exam frequently uses and asks about these diagrams.

Reading Energy Bar Charts

An energy bar chart has:

• Bars representing each form of energy (KE, PE$_g$, PE$_s$, thermal, etc.)
• Height of each bar = amount of energy in that form
• Multiple columns = different moments in time (initial, final, etc.)
• Total height stays constant (if no non-conservative forces)

Example: Dropped Ball

The PE bar shrinks to zero while the KE bar grows to the same height. Total energy is conserved!

Drawing Energy Bar Charts

Steps

1. Identify the initial and final states
2. Choose a reference level for PE$_g = 0$
3. Calculate each form of energy at each state
4. Draw bars proportional to energy values
5. Check that total energy is conserved (or account for non-conservative work)

Example: Ball Thrown Upward (1 kg, $v_i = 20$ m/s, $g = 10$ m/s²)

Initial state (ground level, $h = 0$):

• KE$_i = \frac{1}{2}(1)(20)^2 = 200$ J
• PE$_i = 0$ J

At maximum height ($h_{\max} = 20$ m):

• KE$_f = 0$ J
• PE$_f = 1(10)(20) = 200$ J

Energy accounting:

At half the max height ($h = 10$ m):

• PE = $1(10)(10) = 100$ J
• KE = $200 - 100 = 100$ J
• Both bars are equal!

Bar Charts with Friction

When friction is present, some energy is converted to thermal energy ($E_{\text{thermal}}$). The total energy is still conserved, but now includes a thermal bar:

Example: Block Sliding Down a Rough Ramp

A 2 kg block slides 5 m down a $30°$ rough ramp ($\mu_k = 0.2$, $g = 10$ m/s²):

• $\Delta h = 5\sin(30°) = 2.5$ m
• Initial PE = $2(10)(2.5) = 50$ J
• $N = mg\cos(30°) = 2(10)(0.866) = 17.3$ N
• $f_k = 0.2(17.3) = 3.46$ N
• $W_f = -3.46(5) = -17.3$ J → $E_{\text{thermal}} = 17.3$ J

Energy Bar Chart Concepts 🎯

Bar Chart Calculations 🧮

A 4 kg ball is thrown upward at 15 m/s from ground level. Use $g = 10$ m/s².

1. What is the initial KE (in J)?

2. What is the PE at maximum height (in J)?

3. At what height is KE = PE (in m)?

Round all answers to 3 significant figures.

Bar Chart Interpretation 🔍

A ball is dropped from a height and bounces back to a lower height.

Exit Quiz — Energy Bar Charts ✅

🛠️ Problem-Solving Workshop

Part 6 of 7 — Gravitational Potential Energy

Time to apply everything: $PE_g = mgh$, conservative forces, the work-PE relationship, and energy bar charts. These problems integrate multiple concepts and represent the level of difficulty you'll see on the AP exam.

Energy Problem-Solving Strategy

The 5-Step Energy Method

1. Define the system — what objects are included?
2. Choose initial and final states — where does the problem start and end?
3. Choose a reference level — where is $PE_g = 0$?
4. Write the energy equation:
   • No friction: $KE_i + PE_i = KE_f + PE_f$
   • With friction: $KE_i + PE_i = KE_f + PE_f + E_{\text{thermal}}$
5. Solve for the unknown

When to Use Energy vs. Forces

Energy methods are best when:

• You know positions but not time
• The path is complicated but endpoints are clear
• You want to find speed at a point
• Friction converts energy to heat

Worked Example: Roller Coaster

A 500 kg roller coaster car starts from rest at the top of a 40 m hill and rolls down to a 15 m hill. No friction. ($g = 10$ m/s²)

Step 1: System = car + Earth

Step 2: Initial = top of first hill; Final = top of second hill

Step 3: Reference = ground level

Step 4: $KE_i + PE_i = KE_f + PE_f$ $0 + 500(10)(40) = \frac{1}{2}(500)v_f^2 + 500(10)(15)$ $200{,}000 = 250v_f^2 + 75{,}000$

Step 5: $250v_f^2 = 125{,}000 \Rightarrow v_f^2 = 500 \Rightarrow v_f = \sqrt{500} \approx 22.4$ m/s

Note: Mass canceled! $v_f = \sqrt{2g(h_i - h_f)} = \sqrt{2(10)(25)} = \sqrt{500}$

Workshop Problems 🎯

Workshop Calculations 🧮

Use $g = 10$ m/s².

1. A skier starts from rest at the top of a 50 m hill and reaches the bottom of a 10 m hill at 20 m/s. How much energy per kg was lost to friction (in J/kg)?

2. A 0.5 kg ball is thrown upward at 12 m/s from a 10 m tall building. What is its speed when it reaches a height of 17.2 m above the ground (in m/s)?

3. A ball is released from rest on a frictionless track at height $h$. It passes through a valley and up to height $0.6h$. What is its speed at $0.6h$ in terms of $h$? Compute the numerical coefficient: $v = \sqrt{? \cdot h}$ (answer the coefficient to 3 significant figures, using $g = 10$ m/s²)

Energy Analysis 🔍

Exit Quiz — Workshop ✅

🎓 Synthesis & AP Review

Part 7 of 7 — Gravitational Potential Energy

This final lesson integrates all gravitational PE concepts: reference levels, conservative forces, work-PE relationships, KE ↔ PE conversions, and energy bar charts. Get ready for AP-level questions!

Key Equations & Concepts

Big Ideas

• PE is energy stored due to position in a gravitational field
• Only changes in PE are physically meaningful
• Gravity is conservative: work is path-independent
• Energy transformations: KE ↔ PE (conservative) and KE → thermal (non-conservative)

AP-Style Conceptual Questions 🎯

AP-Style Calculations 🧮

Use $g = 10$ m/s².

1. A roller coaster starts at rest at height 25 m and passes over a 10 m hill. What is its speed at the top of the 10 m hill (in m/s, to 1 decimal)?

2. A 3 kg ball is thrown upward at 16 m/s from a 5 m balcony. What is its maximum height above the ground (in m)?

3. A pendulum bob ($m = 0.5$ kg) is released from a height 0.45 m above its lowest point. What is its speed at the lowest point (in m/s)?

Round all answers to 3 significant figures.

AP Review — Conceptual Analysis 🔍

Final AP Exit Quiz — Gravitational PE ✅