Energy instead of forces

Science · Forces & Motion · ages 21-22

Name ______________________ Date ____________

In the basic mechanical case, how is the Lagrangian built from T and V?
- T plus V
- T minus V
- T times V
A simple pendulum moves in a plane with a fixed length. How many generalised coordinates does it need?
Answer: ______________
Which pair of forces admits a potential for the basic recipe?
- Gravity and springs
- Friction and air drag
- Tension and friction
A fixed-pivot pendulum swings on a massless string. Why does string tension never enter the Lagrangian?
- The physical tension drops to zero throughout the swing, so no force acts along the string
- Ideal holonomic constraints do zero virtual work, so the generalised tension vanishes while the physical force stays present
- String tension is an applied nonconservative force, so it enters the generalised forces in the same way as sliding friction does
A planar simple pendulum has a fixed pivot, a point bob of mass m, and a massless string of fixed length l. theta is the angle from the downward vertical and omega is the angular rate. Which expression is the kinetic energy?
- m l times theta
- m g l cos theta
- ½ m l² ω²
Friction can be folded into the potential V inside the basic Lagrangian recipe.
Circle one: True False
A bead slides on a fixed hoop. How many coordinates does the Lagrangian need?
- Two, x and y
- Three, with tension included
- One angle along the hoop
A simple pendulum has one generalised coordinate, the angle theta. What else does the Lagrangian need before the equations of motion follow?
- The angular rate as well, since the kinetic energy depends on it
- Nothing else; one coordinate fixes the whole system by itself
- The string tension, which is needed to close the equations of motion

Answer key

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

Energy instead of forces W1-mt_9QPk1zI9jE-s1

T minus V · In the basic mechanical case the Lagrangian is kinetic minus potential energy in the chosen coordinates. Other systems can need a more general L, so this form is the standard case rather than a universal rule.
1 · One angle fixes the whole layout, so the degrees of freedom are one.
Gravity and springs · Gravity and ideal springs have path independent work, so a V exists for them.
Ideal holonomic constraints do zero virtual work, so the generalised tension vanishes while the physical force stays present · String tension is an ideal holonomic constraint force. It does zero virtual work on any fixed-time nudge along the arc, so its generalised contribution vanishes and it never enters script L. The physical tension is not zero; it holds the bob on its circle.
½ m l² ω² · Turning half m v squared into the angle gives one half m l squared omega squared, with l the string length and omega the angular rate.
False · Friction work depends on the path, so no potential V exists for it. In the Lagrangian formalism it enters as a generalised force instead.
One angle along the hoop · The hoop constraint leaves one independent way to move, so one angle does the job.
The angular rate as well, since the kinetic energy depends on it · One generalised coordinate fixes the configuration, but the state also needs the rate. The kinetic energy depends on the angular rate, so script L is built from theta and its rate. Tension is a constraint force and never enters.