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3.2.3Work, Energy, and Power

Elastic Potential Energy — AP PHYSICS 1 Revision Notes & Topical Questions

5 min read 9 parts Lesson 6 of 14 1 quiz

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3.2.3 Elastic Potential Energy — AP PHYSICS 1 notes: Calculate the elastic potential energy stored in a spring with Us = ½k(Δx)2 and use it in energy problems.Elastic potential energy = the energy stored in a material such as a spring when it is stretched or compressed.Origin — w…

  • Change between extensions
  • Us against Δx
  • Us against (Δx)2
  • Horizontal frictionless spring
  • Maximum extension (amplitude A)
  • Equilibrium position

Section 1 of 9Energy stored in a spring

Elastic potential energy
Elastic potential energy the energy stored in a material such as a spring when it is stretched or compressed.
Origin
  • Origin work done against the spring force is stored in the spring–object system.
Release
  • Release the energy returns as kinetic energy.
    • (dart gun, pogo stick, trampoline)
Symmetry
  • Symmetry equal stretch and compression store equal energy.
Scalar
  • Scalar elastic potential energy has magnitude only.

Section 2 of 9The equation

1Us
½k(Δx)2
  • Us in J, k in N/m, Δx the stretch or compression from the natural length in m.
  • It is the area under the force–extension graph, a triangle of area ½ × Δx × kΔx.
  • Doubling the extension quadruples the energy, while the force only doubles.
2Change between extensions
  • ΔUs = ½k(x22 − x12), not ½k(x2 − x1)2.

Section 3 of 9Graphs

Us against Δx
  • Us against Δx a parabola with its minimum at zero extension.
    • (stable equilibrium at the natural length)
Gradient
  • Gradient at any point equals the spring force magnitude kΔx.
Us against (Δx)2
  • Us against (Δx)2 a straight line through the origin with gradient k/2, used to find k.

Section 4 of 9Springs and conservation of energy

1Horizontal frictionless spring
  • ½mv2 + ½k(Δx)2 = constant.
2Maximum extension (amplitude A)
  • Maximum extension (amplitude A) speed is zero and all energy is elastic, E = ½kA2.
3Equilibrium position
  • Equilibrium position all energy is kinetic, vmax = A√(k/m).
4Vertical spring
  • Vertical spring include mgy as well as ½k(Δx)2.
5Spring-launched projectile
  • Spring-launched projectile elastic energy equals kinetic energy when the spring reaches its natural length.
6Breaking wire
  • Breaking wire stored elastic energy becomes kinetic energy, ½k(Δx)2 = ½mv2, so a larger extension gives a larger speed.

Section 5 of 9Energy-transfer chains

Energy-transfer chains
Equation
  • Equation initial energies + work by external forces = final energies + thermal energy.
Block launched up a rough ramp until it stops
  • ½k(Δx)2 = mgh + fd.
Method
  • Method include only terms that exist at the initial and final instants.
    • (K = 0 when at rest)

Section 6 of 9Cautions

Deformation
  • Deformation use the change from the natural length, not the total length.
Units
  • Units convert cm to m before squaring.
Idealisation
  • Idealisation springs are ideal and massless, so the spring's own kinetic energy is ignored.

Section 7 of 9More from the specification

More from the specification
Elastic energy across reference frames
  • Elastic energy across reference frames spring strain energy depends solely on internal spring deformation, remaining invariant between different reference frames.
Reference frame
  • Coordinate system used by an observer.
Planet
  • A large body that orbits the Sun and whose gravity clears its orbit.
Stationary
  • Not moving.
Observer
  • Person or instrument measuring from within a frame.
Driver
  • The external system supplying the periodic force.
Gravitational potential
  • Work done per unit mass in bringing a test mass from infinity to the point.
Gravitational field
  • Region where mass experiences gravitational force.
Scalar quantity
  • A quantity with magnitude only, such as potential
Field strength
  • G = Fg/m.
Velocity
  • Rate of change of displacement.
Distance
  • Total length of path travelled, a scalar.
Satellite
  • A body that orbits a larger one; the Moon is the Earth's natural satellite.

Section 8 of 9Key terms

TermMeaning
Elastic potential energyEnergy stored in a deformed spring, ½k(Δx)2.
Spring constantStiffness k in N/m.
Natural lengthLength when no force acts.
AmplitudeMaximum displacement from equilibrium.
Conservation of energyTotal energy of an isolated system is constant.

Section 9 of 9Exam technique

1
Using ½kΔx instead of ½k(Δx)2.
2
Using the spring's total length instead of the extension.
3
Computing ΔU from (x2 − x1)2.
4
Forgetting that stretch and compression store energy equally.
5
Forgetting to convert cm to m.

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