Unit 4 · Topic 05

Gravitational and elastic potential energy

Potential energy organizes the work of certain interactions through system configuration. Its changes are physical; its zero value is a conventional reference.

Unit 4Potential energyOpen navigation

Concept 01

A system property

Essential The minimum you should retain

Potential energy belongs to the configuration of interacting objects.

UnderstandInterpret and connect

It should not be described as a substance stored inside an isolated particle.

DeepenFormulation and conditions

A function U represents the work of a conservative interaction through changes between states.

ExploreConnections for further study

Declaring the system boundary clarifies which interaction is included in U.

Concept 02

Approximately uniform gravity

Essential The minimum you should retain

Near Earth's surface, ΔU_g=mg(y_f-y_i).

UnderstandInterpret and connect

Changing the y=0 level changes absolute U_g values but not ΔU_g.

DeepenFormulation and conditions

U_g=mgy+C in an approximately uniform gravitational field.

ExploreConnections for further study

The expression mgy is not a universal law for planetary distances.

Mathematical relation

Gravitational potential in a uniform field

ΔUg=mg(yf−yi),Ug=mgy+C \Delta U_g=mg(y_f-y_i),\quad U_g=mgy+C
Represents

The change and one reference choice for gravitational potential energy near Earth's surface.

Physical interpretation

Changing C alters neither ΔU_g nor motion.

DeepenVariables, conditions, and checks

Variables

U_g
system gravitational potential energy; usual unit: J
y
height on the chosen axis; usual unit: m
C
reference constant; usual unit: J

Conditions of application

  • g is approximately uniform.

Dimensional check

kg·m/s²·m=J.

Errors it helps prevent

  • Treating mgy as a universal gravity law.
  • Believing the zero of U is unique.
Two vertical axes show the same states A and B with different zero levels.same Δyreference 1reference 2y=0y'=0ABAB

Moving the reference level changes U values, but mgΔy between two states stays the same.

Concept 03

Ideal spring

Essential The minimum you should retain

For an ideal spring, Us=12kx2 U_s=\frac12kx^2 when x is measured from natural length.

UnderstandInterpret and connect

Equal compression and extension have equal U_s because x is squared.

DeepenFormulation and conditions

The general form includes an additive constant C that changes neither force nor ΔU.

ExploreConnections for further study

Gravity and a spring may simultaneously contribute to the system's total potential energy.

Worked example

A compressed spring launches a glider

A 0.25 kg glider starts from rest against an ideal k=160 N/m spring compressed 0.12 m, with no losses.

Given
  • m=0.25 kg
  • k=160 N/m
  • x=0.12 m
Target

Find the speed when the spring returns to natural length.

  1. Initial state

    U_s,i=(1/2)kx² and K_i=0.

  2. Final state

    At natural length, U_s,f=0.

  3. Conservation

    (1/2)kx²=(1/2)mv².

  4. Calculation

    v=x sqrt(k/m)=0.12sqrt(160/0.25)≈3.04 m/s.

Conclusion

The speed is approximately 3.04 m/s.

Mathematical relation

Elastic potential energy

Us=12kx2+C U_s=\frac12kx^2+C
Represents

Potential energy of an ideal spring as a function of deformation.

Physical interpretation

U_s(+x)=U_s(-x) for the same reference.

DeepenVariables, conditions, and checks

Variables

U_s
elastic potential energy; usual unit: J
k
spring constant; usual unit: N/m
x
deformation from natural length; usual unit: m

Conditions of application

  • Ideal spring.

Dimensional check

(N/m)m²=J.

Errors it helps prevent

  • Using total spring length instead of deformation.

U_s depends on x²

Elastic-potential-energy parabola symmetric about natural length.U(-a)x=0U(+a)
  • U_s=(1/2)kx²

Compression -a and extension +a have equal elastic potential energy.

Concept 04

The zero is conventional

Essential The minimum you should retain

The absolute value of U depends on the chosen reference.

UnderstandInterpret and connect

Adding the same constant everywhere does not change energy differences.

DeepenFormulation and conditions

Predictions depend on ΔU or combinations such as E-U.

ExploreConnections for further study

Reference freedom lets us choose a convenient zero without changing the physics.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Saying potential energy is inside an isolated particle.

It is a property of the configuration of the interacting system.

Believing that changing the zero of U changes the physics.

Adding a constant to U changes neither ΔU, force, nor predictions.