Generalised Coordinates and the Lagrangian · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Energy instead of forces

Science · Forces & Motion · ages 21-22
Name ______________________   Date ____________

  1. How is the Lagrangian built from T and V?

    • T plus V
    • T minus V
    • T times V

  2. A simple pendulum moves in a plane with a fixed length. How many generalised coordinates does it need?

    Answer: ______________

  3. Which pair of forces admits a potential for the basic recipe?

    • Gravity and springs
    • Friction and air drag
    • Tension and friction

  4. Why do constraint forces like tension not appear once good coordinates are chosen?

    • They become infinite
    • The coordinates allow no motion against them, so they do no work
    • Energy methods forbid all forces

  5. For a pendulum of length L, which expression is the kinetic energy in the angle?

    • m L times angle
    • m g L
    • ½ m L² ω²

  6. Friction can be folded into the potential V inside the basic Lagrangian recipe.

    Circle one:   True   False

  7. A bead slides on a fixed hoop. How many coordinates does the Lagrangian need?

    • Two, x and y
    • One angle along the hoop
    • Three, with tension included

  8. Could you write the Lagrangian of a simple pendulum in terms of its angle alone?

    • Yes, theta carries T and V
    • Only with two extra multipliers
    • No, tension must stay
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Answer key

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

Energy instead of forces W1-mt_9QPk1zI9jE-s1

  1. T minus V · The Lagrangian is kinetic minus potential energy in the chosen coordinates.
  2. 1 · One angle fixes the whole layout, so the degrees of freedom are one.
  3. Gravity and springs · Gravity and ideal springs have path independent work, so a V exists for them.
  4. The coordinates allow no motion against them, so they do no work · The angle only moves along the allowed arc, where tension has nothing to push against.
  5. ½ m L² ω² · Turning half m v squared into angle variables gives that rotational form.
  6. False · Friction work depends on the path, so no V exists for it and it needs separate treatment.
  7. One angle along the hoop · The hoop constraint leaves one independent way to move, so one angle does the job.
  8. Yes, theta carries T and V · The single angle fixes position and velocity, so T minus V in theta is complete.
Worksheet · LightMySky