A bead slides frictionlessly along a curved wire. As the bead moves along the wire, how much work does the wire's reaction force do on the bead?
APositive work — the wire supports the bead against gravity and therefore transfers energy to it
BNegative work — the wire resists the bead's motion and acts like friction
CZero — the reaction force is always perpendicular to the bead's velocity along the wire
DIt depends on whether the wire curves upward or downward at that point
The wire's reaction force constrains the bead to stay on the wire — it is always directed perpendicular to the wire at the bead's current position, and the bead's velocity is always tangent to the wire. Since force and velocity are perpendicular, their dot product is zero and no work is done. This is the fundamental theorem: smooth holonomic constraints produce reaction forces perpendicular to the allowed motion, contributing nothing to the energy budget. This is why energy methods can ignore constraint forces entirely.
Question 2 Multiple Choice
A structural support at a joint in a bridge truss prevents translation in two directions but allows free rotation. What type of support is this, and how many unknown reaction components must be solved for?
AA roller — it prevents translation in one direction, so there is one unknown reaction
BA fixed (clamped) support — it prevents all motion, so there are three unknowns (two forces, one moment)
CA pin (hinge) — it prevents two-directional translation but allows rotation, so there are two unknown force components
DA rope — it provides tension along its length, one unknown
A pin or hinge support prevents translation in both the horizontal and vertical directions (two constrained DOF) but allows rotation freely (one unconstrained DOF). Therefore it exerts two force components (one horizontal, one vertical) and no moment. This is the rule: the number of reaction force/moment components equals the number of constrained degrees of freedom. A fixed support (no translation, no rotation) adds a moment reaction for the constrained rotational DOF, giving three unknowns.
Question 3 True / False
In Lagrangian mechanics, constraint forces are automatically eliminated from the equations of motion when generalized coordinates are chosen to satisfy the constraints.
TTrue
FFalse
Answer: True
This is the central power of the Lagrangian formulation. By choosing generalized coordinates whose values automatically respect the constraints (e.g., the angle of a pendulum rather than the Cartesian x, y coordinates of the bob), every virtual displacement is consistent with the constraints. Constraint forces do no virtual work on such displacements, so they vanish from the equations. You work only with the degrees of freedom that actually move, and you never need to find the constraint forces explicitly — unless you specifically need them (e.g., to check if a string goes slack).
Question 4 True / False
A constraint force does positive work on a body whenever the body is accelerating in the direction of the constraint force.
TTrue
FFalse
Answer: False
Constraint forces are always perpendicular to the motion allowed by the constraint — this is a geometric property of the constraint, not a function of acceleration. Even if the body accelerates (because other forces act on it), the constraint force direction is dictated by the constraint geometry, not the acceleration direction. For example, a block accelerating down a ramp still has a normal force perpendicular to the ramp surface, doing zero work. Work is force dotted with velocity, and velocity is always tangent to the constrained motion — which is perpendicular to the reaction force.
Question 5 Short Answer
Why do constraint forces not appear in energy-based analyses such as the work-energy theorem or Lagrangian mechanics, even though they are real physical forces that appear in free-body diagrams?
Think about your answer, then reveal below.
Model answer: Constraint forces do no work during motion that is consistent with the constraint, because they are always perpendicular to the allowed displacement. Work is the integral of F·ds, and since the constraint force is perpendicular to ds at every instant, the integral is zero. Energy methods track energy changes through work, so forces that do no work are invisible to them. In Newton-Euler analysis, constraint forces must be found explicitly because ΣF = ma cares about all forces regardless of their work. In the Lagrangian formulation, generalized coordinates are chosen to satisfy constraints, so all virtual displacements are constraint-compatible and constraint forces drop out of the principle of virtual work entirely.
This distinction explains why Lagrangian mechanics is often dramatically simpler than Newton-Euler analysis for constrained systems. A double pendulum written in Cartesian coordinates requires 6 equations and 4 constraint forces; written in two angles (generalized coordinates), it requires just 2 equations with no constraint forces. The physics is identical; the choice of coordinates determines how much bookkeeping is needed.