Noether's Theorem: Symmetry and Conserved Quantities · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

One symmetry, one conserved quantity

Science · Forces & Motion · ages 22-24
Name ______________________   Date ____________

  1. A spinning skater pulls her arms in while no net external torque acts about the vertical axis. What happens to her rate of spin?

    • It rises, as her moment of inertia shrank
    • It falls, because she stopped pushing
    • It stays the same, because nothing pushed her

  2. Which continuous symmetry of the action gives conservation of momentum?

    • Rotation of the setup
    • Rescaling all lengths by a factor
    • Translation of the setup in space

  3. About a fixed inertial origin and axis, the angular momentum component of a system stays constant whenever no net external torque acts about that axis.

    Circle one:   True   False

  4. You check an action and find it unchanged under any continuous shift in time, t → t + ε. Which quantity is conserved?

    • Angular momentum
    • Energy
    • Electric charge

  5. A system of two particles on a line has L = ½m₁ẋ₁² + ½m₂ẋ₂² − V(x₁ − x₂). This L is invariant under the simultaneous shift x₁ → x₁ + ε, x₂ → x₂ + ε. Which quantity is conserved?

    • The total momentum m₁ẋ₁ + m₂ẋ₂
    • The separation x₁ − x₂
    • The kinetic energy ½m₁ẋ₁² + ½m₂ẋ₂²

  6. A top spins about a fixed axis with initial angular momentum 10 N·m·s. A constant bearing friction torque of −0.01 N·m about that axis opposes the spin for 5 minutes. What is the angular momentum change over that interval?

    • It holds exactly fixed
    • It grows to compensate for the friction
    • It drifts downward by 3 N·m·s

  7. A wheel has moment of inertia 2 kg·m² and angular velocity 3 rad/s about a fixed axis, and its angular momentum component is L = Iω. Type the value of L in kg·m²/s.

    Answer: ______________

  8. A skater spins on ice about a fixed vertical axis through a fixed origin, with no net external torque acting about that axis. A student says she spins faster because pulling her arms in applies a torque that pushes her around. What is wrong with this?

    • Fixed L_z = Iω with smaller I means larger ω, no net external torque
    • Angular momentum is not defined for a skater in this situation at all, ever
    • Nothing: a torque is exactly what speeds her up
LightMySky · lightmysky.comW1-mt_kAoOn8syBK-s1

Answer key

For grown-ups. Fold this page away before handing over the rest.

One symmetry, one conserved quantity W1-mt_kAoOn8syBK-s1

  1. It rises, as her moment of inertia shrank · With no net external torque about the vertical axis, L_z = Iω stays fixed, so a smaller I forces a larger ω.
  2. Translation of the setup in space · Shifting every coordinate by the same amount leaves the action unchanged, and that symmetry conserves total momentum.
  3. True · About a fixed origin and axis, no net external torque means nothing changes that component of L.
  4. Energy · Continuous time-translation symmetry is the symmetry behind energy conservation; a one-hour delay alone would establish at most a discrete period.
  5. The total momentum m₁ẋ₁ + m₂ẋ₂ · The shift leaves L unchanged, so Σ ∂L/∂xᵢ = 0; the unforced Euler-Lagrange equation gives d/dt(Σ pᵢ) = 0, so the total momentum m₁ẋ₁ + m₂ẋ₂ is conserved.
  6. It drifts downward by 3 N·m·s · With constant torque, dL/dt = −0.01 N·m, so over 300 s the change is ΔL = τ Δt = −3 N·m·s, a 30 percent drop from the initial 10 N·m·s.
  7. 6 · Multiply moment of inertia by angular velocity: 2 kg·m² times 3 rad/s is 6 kg·m²/s.
  8. Fixed L_z = Iω with smaller I means larger ω, no net external torque · No net external torque acts about the axis, so the faster spin comes from a smaller I at fixed L_z = Iω; internal torques only transfer angular momentum between parts.
Worksheet · LightMySky