Chapter 15: Potential Energy and Conservation of Energy

15.1 What Potential Energy Means

Potential energy is stored energy: energy associated with the position or configuration of an object or system, available to be converted into kinetic energy when the system is released. 'Potential' here means 'waiting to do work' — the energy is real but not yet expressed as motion. Every form of potential energy is associated with a force. Gravitational potential energy is associated with the gravitational force. Elastic potential energy is associated with the spring (elastic restoring) force. Electrical potential energy is associated with electric forces. In every case, the potential energy is stored by virtue of the configuration of the system relative to some reference point — and can be released by allowing the relevant force to act. Potential energy is always defined relative to a reference level or configuration. The absolute value of potential energy is not physically meaningful; only changes in potential energy matter. We choose the reference to make the problem most convenient — and the physics never depends on that choice. Key Point: Potential energy belongs to the system, not to a single object. Gravitational PE is shared between the object and Earth. Elastic PE is stored in the spring. This distinction becomes important when analyzing which forces are conservative and which systems obey conservation of mechanical energy. Physics: Foundations of Physical Science Chapter 15 — Potential Energy and Conservation of Energy

15.2 Gravitational Potential Energy

Near Earth's surface, where gravitational field strength g is approximately constant, the gravitational potential energy of an object of mass m at height h above the chosen reference level is:

U_g = mgh

The reference level (h = 0) is chosen by the problem-solver. Common choices: the floor,

the lowest point of the motion, the ground level. The choice does not affect any

calculated change in PE or any predicted speed, because ΔU_g = mgΔh depends only

on the change in height.

When an object rises, h increases, U_g increases, and work done against gravity is

stored as gravitational PE. When an object falls, h decreases, U_g decreases, and

gravitational PE converts to kinetic energy (or other forms).

A 5.0 kg book is lifted from the floor (reference) to a

shelf 2.0 m high. U_g = 5.0 × 9.81 × 2.0 = 98.1 J. If it falls off the shelf, that 98.1 J

converts entirely to kinetic energy (ignoring air resistance). v at floor = √(2 × 98.1 /

5.0) = 6.26 m/s.

Gravitational PE depends on height above the reference level, not on the path taken to reach that height. An object lifted straight up 3 m has the same

U_g as one carried on a 45° ramp to the same height. Only the vertical height change

matters.

15.3 Elastic Potential Energy

Elastic potential energy is the energy stored in a deformed spring (or any elastic material) due to the work done against the spring's restoring force. It is stored in the elastic deformation itself — in the stretched or compressed molecular bonds of the spring material. Unlike gravitational PE, elastic PE is always non-negative: whether you stretch or compress a spring, you store positive energy in it. The reference configuration is the

spring at its natural (undeformed) length, where U_s = 0.

U_s = ½ k x²

Because U_s depends on x², the stored energy grows rapidly with deformation:

doubling the compression quadruples the stored energy. This quadratic relationship means springs store disproportionately more energy at large deformations. Physics: Foundations of Physical Science Chapter 15 — Potential Energy and Conservation of Energy

A spring with k = 400 N/m is compressed 0.15 m. U_s = ½ ×

400 × (0.15)² = 200 × 0.0225 = 4.5 J.

15.4 Energy Stored in Springs

The work done by an external agent to compress or stretch a spring from its natural length by displacement x is equal to the elastic potential energy stored:

W_{on spring} = U_s = ½ k x²

This result requires integration (since the spring force is variable), but the geometric argument gives the same answer: the work equals the area of the triangle under the F-

x graph from 0 to x, which is ½ × (kx) × x = ½ kx².

When the spring is released, it converts stored elastic PE entirely to kinetic energy of the object attached to it (on a frictionless surface):

U_s →KE:

½ k x² = ½ m v²

⇒

v = x√(k/m)

A spring (k = 300 N/m) is compressed 0.10 m against a 0.200 kg ball on a frictionless

surface. Find the ball's speed when it leaves the spring.

Step 1: Stored energy: U_s = ½ × 300 × (0.10)² = 1.5 J

Step 2: Energy conservation: U_s = KE ⇒1.5 = ½ × 0.200 × v² ⇒v² = 15.0

Step 3: Speed: v = √15.0 = 3.87 m/s

Step 4: Check: Units: J = kg·m²/s² ⇒v in m/s ✔

15.5 Mechanical Energy

The mechanical energy of a system is the sum of its kinetic energy and all its potential energies:

E_mech = KE + U_g + U_s

(+ any other PE forms) Mechanical energy is the total energy associated with the motion and configuration of objects in a mechanical system. It does not include thermal energy, chemical energy, or other non-mechanical forms. When only conservative forces do work on a system, mechanical energy is conserved: it remains constant as the system evolves. Energy moves between KE and PE forms but the total stays the same. When nonconservative forces (friction, drag) act, mechanical Physics: Foundations of Physical Science Chapter 15 — Potential Energy and Conservation of Energy energy decreases — it is converted to thermal or other forms.

15.6 Conservative vs. Nonconservative Forces

A conservative force is one for which the work done depends only on the starting and ending positions — not on the path taken between them. Equivalently, a conservative force does zero work on any closed path (a path that returns to its starting point). A nonconservative force is one for which the work done depends on the path. Friction is the most common example: sliding along a longer, winding path involves more friction work than a straight path between the same two points. Force Conservative? Reason Associated Potential Energy? Gravity (near Earth) Yes Work depends only on height change

Yes: U_g = mgh

Spring force Yes Work depends only on displacement from equilibrium

Yes: U_s = ½kx²

Universal gravitation Yes Central force; path- independent Yes:

U_g

= GMm/r Electric force Yes Path-independent (Coulomb)

Yes: U_e = kq₁q₂/r

Kinetic friction No

More path = more work

No Air resistance No Depends on speed and path No Tension (ideal rope) No (does zero net work) Special case; zero work on connected system No

Table 15.1

This distinction is fundamental: potential energy can only be defined for conservative

forces. When we write U_g = mgh, we are using the fact that gravity is conservative

and its work depends only on height change. For friction, no analogous potential energy can be defined.

15.7 Conservation of Mechanical Energy

Physics: Foundations of Physical Science Chapter 15 — Potential Energy and Conservation of Energy When only conservative forces do work on a system, the total mechanical energy is conserved:

E_mech = KE + U = constant

KE_i + U_i = KE_f + U_f

The power of this principle is that it replaces multiple force and kinematic equations with a single energy equation. We do not need to know how an object moved along its path; we only need to know its kinetic and potential energies at the start and end.

A 500 kg roller coaster car starts from rest at the top of a 40.0 m hill. Find its speed at:

(a) the bottom (h = 0); (b) a point 15.0 m below the top.

Step 1: Reference: Bottom of hill, h = 0

Step 2: Initial state:

v_i = 0, h_i = 40.0 m. E_i = 0 + mgh_i = 500×9.81×40.0 =

196,200 J

Step 3: (a) At bottom:

E_f = ½mv_f² + 0 = 196,200

⇒

v_f = √(2×196,200/500) =

√784.8 = 28.0 m/s

Step 4: (b) At 15 m below top (h = 25.0 m): E_f = ½mv_f² + mg(25.0) = 196,200 ⇒

½mv_f² = 196,200

122,625 = 73,575 J ⇒v_f = 17.1 m/s

A 0.50 kg pendulum bob swings from rest at a height 0.30 m above the lowest point. Find its speed at the lowest point.

Step 1: Conservation: mgh_i + 0 = 0 + ½mv_f² ⇒v_f = √(2gh) = √(2×9.81×0.30) =

√5.886 = 2.43 m/s

Step 2: Note:

Mass cancelled! The speed at the bottom of a pendulum swing depends only on the height dropped, not the mass.

15.8 General Conservation of Energy

The law of conservation of energy, in its most general form, states that the total energy of an isolated system is constant. Energy can transform between forms — kinetic, potential, thermal, chemical, electromagnetic, nuclear — but the total never changes.

E_total = KE + U_g + U_s + E_thermal + E_chemical + … =

Physics: Foundations of Physical Science Chapter 15 — Potential Energy and Conservation of Energy constant When nonconservative forces act, mechanical energy is not conserved, but total energy still is. The mechanical energy 'lost' appears as thermal energy (heat from friction), sound, or deformation. No energy disappears; it just changes form.

E_{mech,i} = E_{mech,f} + E_{thermal}

The thermal energy produced by friction over displacement d with kinetic friction force

f_k is:

ΔE_thermal = f_k × d

15.9 Energy Losses and Dissipation

In real systems, some mechanical energy is always converted to thermal energy through friction, air resistance, inelastic deformation, or viscosity. This process is called energy dissipation. The dissipated energy is not destroyed; it raises the temperature of the materials involved and eventually radiates into the environment. Energy dissipation is irreversible: thermal energy does not spontaneously convert back to organized mechanical energy. This directionality — from ordered motion to disordered thermal energy — is one aspect of the second law of thermodynamics, which Chapter 33 explores. Process Dissipation Mechanism Energy Destination Sliding friction Atomic bond breaking/forming at interface Thermal energy in both surfaces Air resistance Viscous drag on fluid Thermal energy in air Bouncing ball Inelastic deformation of ball Thermal energy in ball material Electric current in wire Electron-ion collisions Thermal energy (Joule heating) Sound in collision Compression waves in air/material Thermal energy (eventually)

Table 15.2

Physics: Foundations of Physical Science Chapter 15 — Potential Energy and Conservation of Energy

A 3.0 kg block slides down a 5.0 m, 30° ramp with μ_k = 0.20. Find the speed at the

bottom.

Step 1: Height fallen:

h = 5.0 sin 30° = 2.5 m

⇒

ΔU_g = mgh = 3.0×9.81×2.5 =

73.6 J

Step 2: Normal force: N = mg cos 30° = 3.0×9.81×0.866 = 25.5 N

Step 3: Friction work: W_f = μ_k N d = 0.20×25.5×5.0 = 25.5 J (dissipated)

Step 4: Energy equation:

KE_i + U_i = KE_f + U_f + ΔE_thermal

⇒0 + 73.6 = KE_f

+ 0 + 25.5 ⇒KE_f = 48.1 J

Step 5: Speed: v_f = √(2×48.1/3.0) = √32.1 = 5.66 m/s

15.10 Solving Problems with Energy Methods

The energy method replaces the step-by-step force and kinematic analysis with a global accounting of energy at two states. The procedure is:

Step 1: Choose the system (object + Earth for gravity; object + spring for elastic PE).

Step 2: Identify initial and final states (when the problem begins and ends).

Step 3: List all energy forms at each state: KE, U_g, U_s.

Step 4:

Check for nonconservative forces (friction, applied forces not from conservative fields). If none, use conservation of mechanical energy. If present, include

ΔE_thermal = f_k d.

Step 5: Write the energy equation and solve for the unknown.

Step 6: Check units, sign, and whether the answer is physically reasonable.

Energy methods eliminate the need to find acceleration, normal force, or tension — unless those are specifically asked for. When the problem asks for a speed or height and gives enough information about energies, energy methods are almost always faster than force-kinematic analysis.

15.11 Comparing Force-Based and Energy-Based Analysis

Both the force-kinematic approach (Chapters 8–12) and the energy approach (Chapters 14–15) are valid. The choice of method depends on what the problem asks and what information is given. Feature Force-Kinematic Method Energy Method What it tracks Acceleration, velocity at each instant Energy at start and end states Physics: Foundations of Physical Science Chapter 15 — Potential Energy and Conservation of Energy Requires knowing Force at each point along path Work done or PE stored Works well when Path details, time, or internal forces needed Speed or height at end is needed Handles variable forces Difficult (need integration) Naturally (potential energy handles it) Handles friction

Directly as F = μN × a

ΔE_thermal = f_k d

Gives direction of motion? Yes (from acceleration direction) No (energy is scalar; must infer direction)

Table 15.3

Ideally, mastery of both methods allows you to choose the most efficient approach for each problem. A classic strategy: use energy to find the speed at the bottom of a ramp, then use Newton's laws to find the normal force at the bottom if the ramp curves (circular motion).

15.12 Common Mistakes in Energy Conservation Problems

Mistake 1: Applying conservation of mechanical energy when friction acts

Conservation of E_mech requires that only conservative forces do work. If friction, air

resistance, or applied forces are present, E_mech is not conserved. Use E_mech,i =

E_mech,f + ΔE_thermal, including the dissipated energy term.

Mistake 2: Choosing an inconvenient reference level

While any reference level gives the correct ΔU and the correct final speed, a poorly

chosen reference can make the algebra messy. Choose the reference at the lowest point of the motion so that no potential energy values are negative.

Mistake 3: Forgetting to square v in ½mv²

KE = ½mv²: the velocity is squared. Students sometimes write ½mv instead of ½mv²,

especially when rearranging. Solve for v² first, then take the square root.

Mistake 4: Treating the spring equation as F = kx instead of U = ½kx²

The spring force is F = kx. The energy stored is U_s = ½kx² (the area under the force-

displacement graph). Using F = kx in an energy equation gives the wrong answer.

Mistake 5: Forgetting that reference height cancels in Δh

If you choose h = 0 at the starting point, U_g,i = 0 and U_g,f = mgh_f. If you choose h

Physics: Foundations of Physical Science Chapter 15 — Potential Energy and Conservation of Energy

= 0 at the end, U_g,i = mgh_i and U_g,f = 0. Either gives the same equation; only Δh

appears in the final result.

Energy conservation is about the total system. Include potential energy for all conservative forces in the system — both gravitational PE and spring PE if a spring is involved. Leaving out one term gives a wrong answer, not a simpler one.

Chapter Summary

Potential energy is stored energy associated with the configuration of a system. Conservation of mechanical energy — valid when only conservative forces act — is one of the most powerful tools in physics. •

Gravitational PE: U_g = mgh; reference at chosen height; only Δh matters.

•

Elastic PE: U_s = ½kx²; always ≥0; reference at natural spring length.

•

Mechanical energy: E_mech = KE + U_g + U_s; conserved when only

conservative forces do work. • Conservative force: work is path-independent; has an associated potential energy.

•

When friction acts: E_mech,i = E_mech,f + f_k d (dissipated thermal energy).

• General conservation: total energy (including thermal, chemical, etc.) is always conserved. • Energy method procedure: identify states, list energies, apply energy equation, solve for unknown. • Energy methods are preferred when speed or height is unknown and path details are not needed.

Key Terms

potential energy

Stored energy due to the position or configuration of a system; can

be converted to kinetic energy.

gravitational potential energy (U_g)

Energy stored due to an object's height above a

reference level

U_g = mgh.

elastic potential energy (U_s)

Energy stored in a deformed spring or elastic material:

U_s = ½kx².

Physics

Foundations of Physical Science

Chapter 15 — Potential Energy and Conservation of Energy

reference level

The chosen position at which potential energy is defined to be zero;

the physics (energy differences) is independent of this choice.

mechanical energy

The sum of kinetic energy and all potential energies in a system:

E_mech = KE + U.

conservative force

A force for which work is path-independent; has an associated

potential energy; examples

gravity, spring force.

nonconservative force

A force for which work is path-dependent; no associated

potential energy; examples

kinetic friction, air resistance.

conservation of mechanical energy

E_mech is constant when only conservative forces

do work

KE_i + U_i = KE_f + U_f.

energy dissipation

The conversion of mechanical energy to thermal energy by nonconservative forces (friction, drag); irreversible.

thermal energy (ΔE_thermal)

Energy produced by friction over distance d: ΔE_thermal

= f_k d; not available to do mechanical work.

Concept Review Questions

1. What is potential energy? Give two examples from everyday life where potential

energy is stored and released.

2. Why does the choice of reference level for gravitational PE not affect the

outcome of a problem?

3. A spring is compressed twice as much. By what factor does the stored elastic PE

increase? Explain.

4. State the condition under which conservation of mechanical energy applies. Give

one example where it does and one where it does not.

5. What is the difference between a conservative and a nonconservative force?

Why can't we define a potential energy for friction?

6. When friction acts on a sliding object, where does the mechanical energy 'go'? Is

it destroyed?

7. A ball is thrown upward. Describe the energy transformations from launch to

maximum height to return.

8. Why does the mass cancel in the pendulum problem (Worked Example 15.3)?
9. Compare the force method and the energy method for solving mechanics

problems. When is each preferred? Physics: Foundations of Physical Science Chapter 15 — Potential Energy and Conservation of Energy

10. A block slides down a frictionless ramp and then a rough surface. How do you

account for the rough surface in the energy equation?

Potential Energy Problems

11. Find the gravitational PE of: (a) a 2.0 kg book 1.5 m above the floor; (b) a 70.0

kg person 10.0 m above ground; (c) a 1200 kg car at the top of a 50.0 m hill.

12. A spring (k = 200 N/m) is stretched 0.25 m. Find: (a) elastic PE; (b) force at that

extension.

13. A 0.50 kg ball is dropped from a height of 8.0 m. Find: (a) U_g at 8.0 m; (b) KE at

4.0 m; (c) speed at 4.0 m; (d) speed at the ground.

14. A spring (k = 500 N/m) is compressed 0.12 m. Find: (a) U_s; (b) speed of a 0.30

kg ball when released from rest on a frictionless surface.

15. A 60.0 kg skier descends 80.0 m vertically. Ignore friction. Find: (a) ΔU_g; (b) final

KE; (c) final speed.

Conservation of Energy Problems

Use conservation of E_mech unless stated otherwise. Show energy at initial and final

states.

16. A 2.0 kg ball rolls off a 1.2 m shelf from rest. Find its speed just before hitting

the floor.

17. A roller coaster (m = 800 kg) starts from rest at 35.0 m height. Find the speed at:

(a) h = 0; (b) h = 12.0 m; (c) h = 35.0 m (top of a second hill).

18. A pendulum of length 1.5 m is released from 20.0° from vertical. Find the speed

at the lowest point. (Hint: height h = L

L cosθ.)

19. A 5.0 kg block slides 6.0 m down a rough 25° incline (μ_k = 0.15). Find the

speed at the bottom using the energy method.

20. A 0.40 kg ball is thrown upward at 12.0 m/s. Using energy methods, find: (a)

maximum height; (b) speed at 4.0 m; (c) speed when it returns to the throw point. Physics: Foundations of Physical Science Chapter 15 — Potential Energy and Conservation of Energy

21. A 10.0 kg block slides from rest down a frictionless curved track and then onto a

rough flat surface (μ_k = 0.30). The block starts 3.0 m above the flat surface.

How far does it slide on the flat surface before stopping?

Spring Energy Problems

22. A spring (k = 350 N/m) is compressed 0.08 m. It launches a 0.15 kg ball

horizontally on a frictionless surface. Find the ball's speed.

23. A 2.0 kg block rests on a spring (k = 800 N/m) and is pressed down 0.20 m.

When released, how high does the block rise above the compressed position?

24. A spring (k = 1200 N/m) fires a 0.10 kg ball upward. The spring is compressed

0.15 m. How high does the ball rise above the launch point?

25. A 0.50 kg block on a frictionless surface is connected to a spring (k = 250 N/m).

It is pulled 0.12 m from equilibrium and released. Find: (a) PE at maximum extension; (b) KE at equilibrium; (c) speed at equilibrium.

26. A spring-loaded gun fires a 0.040 kg dart horizontally. The spring (k = 600 N/m)

is compressed 0.10 m. Find: (a) initial spring PE; (b) dart's muzzle speed; (c) horizontal range if fired from 1.5 m height.

Mixed Mechanical Energy Analysis

27. A 1.0 kg ball is dropped onto a spring (k = 2000 N/m) from 0.50 m above it.

Find the maximum compression of the spring. (Set reference at maximum compression point.)

28. A 500 kg car rolls from rest down a 30.0 m, 15° hill with μ_k = 0.10. Find: (a)

energy dissipated by friction; (b) speed at bottom; (c) what fraction of the original PE was dissipated?

29. A 0.20 kg block slides from rest down a frictionless quarter-circle of radius 0.80

m. At the bottom it slides onto a rough horizontal surface (μ_k = 0.25). How far

does it slide before stopping?

30. (Challenge) A 3.0 kg block is pushed against a spring (k = 500 N/m, compressed

0.15 m) at the bottom of a 30° frictionless ramp. When released, how far up the ramp does it travel before momentarily stopping? Physics: Foundations of Physical Science Chapter 15 — Potential Energy and Conservation of Energy

31. (Challenge) Tarzan (m = 80 kg) swings on a 10.0 m vine from rest at an initial

angle of 40° from vertical. What is his speed at the lowest point? At what angle from vertical does he momentarily stop on the other side?

Chapter 15 Checkpoint

I can… Section Calculate gravitational

PE using U_g = mgh with a

clearly stated reference 15.2

Calculate elastic PE using U_s = ½kx²

15.3–15.4 State the condition for conservation of mechanical energy 15.7

Apply KE_i + U_i = KE_f + U_f to find speed or height

15.7

Include energy dissipation: E_mech,i = E_mech,f + f_k d

15.8–15.9 Identify conservative and nonconservative forces 15.6 Choose a convenient reference level and explain why it doesn't change the result 15.2 Select the energy method when appropriate vs. the force method 15.11 Identify and correct the five most common energy conservation mistakes 15.12