Question bank — Generalized coordinates — choosing them, degrees of freedom

False. A coordinate need only be independent and complete — an angle (radians), a ratio, or even a charge qualifies. See Configuration space and phase space where axes routinely mix dimensions.

True, provided it is independent of the others. An equation $f(\dots ,t)=0$ lets you solve for one variable in terms of the rest, killing one number.

False. Time is a parameter, not a coordinate you may freely dial; the constraint $f(\mathbf{r},t)=0$ still removes one DOF. See Constraints — holonomic vs non-holonomic.

False. Only independent equations count; if one follows from the others it removes nothing.

True. Rigidity fixes all inter-particle distances, leaving 3 for position and 3 for orientation — see Rigid body kinematics — Euler angles.

False in general. That formula holds for holonomic systems; non-holonomic (velocity) constraints can restrict motion without reducing the count of position coordinates.

False — it has 5. The bond-length equation removes one, and spinning about the bond axis moves nothing for point atoms, so that "rotation" is not a DOF.

False. DOF is a property of the system, not the coordinate choice; both describe a 1-DOF system, but $\theta$ bakes in the constraint so it is cleaner.

True. Each bead's confinement to the axis is two equations ($y_i=z_i=0$), so $k=4$ and $n=3(2)-4=2$.

False. A spring is a force, not a constraint; it changes dynamics and energy but leaves the count of coordinates untouched.

The wire's angle is forced to be $\omega t$ — it is not free. Only $r$ is independent, so the system has 1 DOF; the angle is set by time.

The ground plays no role for a swinging bob — there is no $y=0$ constraint. Only the rod length constrains it, so $k=1$ and $n=1$.

Pure rolling is typically non-holonomic — it constrains $\dot{x}=R\dot{\theta }$ (a velocity relation) that cannot be integrated to $f(x,\theta )=0$. The configuration space keeps both $x$ and $\theta$; see Constraints — holonomic vs non-holonomic.

The constraints also grow without bound and cancel all but 6. The correct count is 6, found by direct reasoning (position + orientation), not by naive subtraction.

That is actually correct for three general fixed distances — $9-3=6$ (3 translation + 3 rotation). The trap is assuming it "should" be 5 like the dumbbell; three non-collinear atoms have a genuine third rotational DOF.

Time appears explicitly only because the constraint is rheonomic; $t$ is the independent parameter of evolution, never a $q$ you vary at fixed instant.

Rotating two point atoms about the line joining them produces an identical configuration — the "spin" angle is physically invisible, so it is not an independent coordinate.

Because the constraint equation then holds automatically for every value of the coordinate, so it disappears and you work only with truly free variables — feeding cleanly into Lagrangian mechanics — the Lagrangian L = T - V.

A dependent constraint carries no new information; subtracting it double-counts a restriction and gives a wrong (too small) DOF.

Because the number of rigidity relations is enormous and highly redundant; reasoning "where + how oriented" directly yields the 6 independent numbers without wrestling the redundancy.

At any fixed instant it is still one equation relating the coordinates, so it pins one variable; time only shifts which configuration is allowed, not how many are free.

The spring energy depends only on $s$, so the equations decouple — $X$ evolves freely while $s$ carries all the interaction, simplifying the Euler–Lagrange equations.

It restricts velocities (differential relations) that can't be integrated into position equations, so it forbids certain motions without forbidding any configuration — the position-coordinate count stays high.

3 — the raw $(x,y,z)$; there is nothing to subtract, $n=3(1)-0=3$.

0. Three independent constraints fully lock it: $n=3-3=0$, a degenerate "system" that cannot move.

2. One holonomic equation removes one number from three, leaving a 2-parameter surface to roam.

It is algebraically the same equation scaled — not independent — so it removes nothing; still $n=1$.

Yes — a fully constrained (rigidly fixed) system has $n=0$; it has exactly one allowed configuration and no freedom to move.

$n=3(2)-3=3$. They move together as one point; the three equality constraints collapse six numbers to three.

Still 1. The constraint $x^2+y^2=\ell (t{)}^2$ is rheonomic but holonomic, removing one DOF; the angle $\theta$ remains the sole free coordinate.