Unit 3 · Topic 3.3 Beta

Potential Energy

Potential energy is stored in the configuration of interacting objects and belongs to the system.

Practice 1: Creating RepresentationsPractice 2: Mathematical Routines

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Question set for this topic

Part 1 · Hook

Why this matters

Lift a bowling ball onto a high shelf and nothing seems to happen, until it falls. Wind up a toy car and it sits still, until you let go. The energy is stored in the arrangement of interacting objects, ready to become motion. That stored energy is potential energy, and it belongs to a system, not to one object.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. Work done by a force along a displacement d with angle θ is

  1. F d cos θ
  2. F d sin θ
  3. F/d
  4. F d²
Show the answer

W = F d cos θ.

  • Correct: F d cos θ:
  • F d sin θ:
  • F/d:
  • F d²:

2. Hooke's law gives the size of an ideal spring's force as

  1. kΔx
  2. ½kΔx²
  3. k/Δx
  4. mg
Show the answer

F_s = kΔx, proportional to the stretch or compression.

  • Correct: kΔx:
  • ½kΔx²:
  • k/Δx:
  • mg:

3. Newton's law of universal gravitation gives the force between two masses as

  1. Gm₁m₂/r²
  2. Gm₁m₂/r
  3. m₁m₂/(Gr²)
  4. Gm₁m₂r²
Show the answer

F_g = Gm₁m₂/r².

  • Correct: Gm₁m₂/r²:
  • Gm₁m₂/r:
  • m₁m₂/(Gr²):
  • Gm₁m₂r²:

Part 4 · See it

See it first

Left: a ball above a table, with Earth below, inside a dashed boundary labeled ball plus Earth; U_g = mgΔy from a chosen zero at the table top. Right: a block compressing a spring against a wall by Δx, inside a dashed boundary labeled block plus spring; U_s = ½k(Δx)².
Potential energy belongs to a system: ball and Earth (gravitational), block and spring (elastic). LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. Two objects interact through a conservative force (gravity, an ideal spring)the work the force does depends only on where the objects start and end
  2. The work depends only on the configurationthe system can store energy in its configuration: potential energy, with ΔU = −W_conservative
  3. The objects are pulled apart against gravity or a spring is squeezedthe system's potential energy increases: ΔU_g = mgΔy, U_s = ½k(Δx)²
  4. Only changes in U appear in physicsthe zero of potential energy can be placed wherever is convenient
  5. A force does more work on a longer path, like kinetic frictionit is nonconservative and has no potential energy

Part 6 · Key ideas

Key ideas

  • Potential energy is energy stored in the configuration of a system of interacting objects. One object alone has none.
  • Near Earth: ΔU_g = mgΔy. For an ideal spring: U_s = ½k(Δx)², with Δx measured from the relaxed length.
  • A conservative force does work that depends only on the start and end positions; for it, W = −ΔU. Kinetic friction is nonconservative.
  • The zero of potential energy is a choice. Values change with the choice; changes do not.
  • Far from a planet: U_G = −Gm₁m₂/r, zero at infinite separation and negative everywhere else.

Part 7 · Misconception

A common mistake

The wrong idea: The book on the shelf has 29 J of potential energy.

What actually happens: The book-Earth system has 29 J of gravitational potential energy (with the floor as zero). The energy is stored in the interaction, so it belongs to the system.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Diagram

A book, a shelf and a table

A 2.0 kg book, a shelf and a table1.50 m0.80 mtable topfloor

Use g = 9.8 m/s². Heights are measured to the bottom of the book.

1. Take the floor as the zero of gravitational potential energy. What is the gravitational potential energy of the book-Earth system with the book on the shelf? Give the value and its unit.

Type a number and its unit.

Show the answer

U_g = mgΔy = (2.0 kg)(9.8 m/s²)(1.50 m) = 29 J, measured from the floor.

  • Answer: 29 J

2. Now take the table top as the zero. What is the gravitational potential energy of the book-Earth system with the book on the shelf, in joules?

Type a number in J.

Show the answer

Δy = 1.50 m − 0.80 m = 0.70 m above the table top, so U_g = (2.0 kg)(9.8 m/s²)(0.70 m) = 14 J.

  • Answer: 14 J

3. The book is moved from the shelf to the table top. How does the change in the system's gravitational potential energy depend on the choice of zero?

  1. It is about −14 J for either choice of zero
  2. It is −29 J with the floor as zero and −14 J with the table as zero
  3. It is zero with the table as zero, because the book ends at zero
  4. It is +14 J, because the book moves to a new height
Show the answer

ΔU_g = mgΔy = (2.0 kg)(9.8 m/s²)(−0.70 m) = −14 J. Any zero gives the same change, which is why the choice of zero is free.

  • Correct: It is about −14 J for either choice of zero: Right: the change depends only on the change in height, −0.70 m.
  • It is −29 J with the floor as zero and −14 J with the table as zero: These are values of U_g on the shelf, not changes. A change subtracts two values measured from the same zero.
  • It is zero with the table as zero, because the book ends at zero: Ending at zero does not make the change zero; it started at +14 J.
  • It is +14 J, because the book moves to a new height: The book moves down, so U_g decreases: the change is negative.

4. A student writes "the book has 29 J of potential energy." Which correction is best?

  1. The potential energy belongs to the shelf, which holds the book up
  2. The potential energy belongs to the book-Earth system, not to the book alone
  3. The value should be 29 N, because it comes from the weight
  4. The book has 29 J of kinetic energy instead
Show the answer

Potential energy is energy of a system of interacting objects. The gravitational potential energy here belongs to the book-Earth system.

  • The potential energy belongs to the shelf, which holds the book up: The shelf's support is a contact force, not the interaction that stores gravitational potential energy.
  • Correct: The potential energy belongs to the book-Earth system, not to the book alone: Right: it comes from the interaction of the book and Earth.
  • The value should be 29 N, because it comes from the weight: Weight is a force in newtons; energy is in joules, a force times a distance.
  • The book has 29 J of kinetic energy instead: The book is at rest on the shelf, so it has no kinetic energy.

Data table

Energy stored in a compressed spring

A student compresses an ideal spring by different amounts and calculates the elastic potential energy of the block-spring system for each.

Compression and stored energy
Compression Δx (m)U_s (J)
0.0200.10
0.0400.40
0.0600.90
0.0801.6

5. When the compression doubles from 0.020 m to 0.040 m, the stored energy

  1. doubles
  2. stays the same, because the spring is the same
  3. increases by 0.10 J
  4. becomes 4 times as large
Show the answer

U_s = ½k(Δx)², so doubling Δx multiplies U_s by 4.

  • doubles: That would be true if U_s were proportional to Δx. The table shows a factor of 4.
  • stays the same, because the spring is the same: The spring constant stays the same, but the energy depends on the compression.
  • increases by 0.10 J: It increases by 0.30 J, from 0.10 J to 0.40 J.
  • Correct: becomes 4 times as large: Right: U_s ∝ (Δx)², and 2² = 4; the table shows 0.10 J to 0.40 J.

6. Use the 0.080 m row to find the spring constant, in N/m.

Type a number in N/m.

Show the answer

k = 2U_s/(Δx)² = 2(1.6 J)/(0.080 m)² = 5.0 × 10² N/m.

  • Answer: 500 N/m

Part 9 · Summary

Summary

Potential energy is stored in the configuration of interacting objects and belongs to the system. ΔU_g = mgΔy near Earth, U_s = ½k(Δx)² for a spring, U_G = −Gm₁m₂/r far away. Only conservative forces have one; the zero is a free choice because only changes matter.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections