Kinematic constraints

A constraint is a geometric statement that stays true for the whole motion: the rope keeps its length, the surfaces stay in contact, the rim does not slide. Write it as an equation between coordinates measured from fixed points — an origin that rides on one of the moving objects gives derivatives that do not mean what you want them to. After that the kinematics is calculus: differentiate once for a relation between velocities, twice for a relation between accelerations. Force never enters. The counting is the payoff. Two objects free to move along a line carry two unknown accelerations; one rope between them supplies one equation, leaving a single independent motion for Newton's second law to determine later.

An ideal rope is inextensible and stays taut, so its two straight segments add to a constant L — the rope length less the fixed arc over the pulley. With x₁ and x₂ measured from the pulley along each segment, x₁ + x₂ = L. Differentiating gives v₁ + v₂ = 0, that is v₁ = −v₂, and then a₁ = −a₂: whatever one block gains, the other loses, in those coordinates. That minus sign is bookkeeping rather than physics. It follows from measuring both coordinates away from the same point, and it flips if one coordinate is measured the other way. The equation dies the instant the rope goes slack or stretches, so state that idealisation when you use it.

When a pulley itself moves, count how many rope segments run to it. Fix one end of the rope to the ceiling, pass it under a movable pulley carrying the load, then back up over a fixed pulley to your hand: the load hangs on two segments, so every metre you take in is shared between them and the load rises half a metre. With every coordinate measured down from the ceiling, 2yₚ + y_free = L, so vₚ = −½ v_free and aₚ = −½ a_free. Take the free end in at 0.60 m s⁻¹ and the load rises at 0.30 m s⁻¹. Thread the same rope a different way and the ratio changes, so re-derive the constraint for each arrangement instead of remembering a factor.

A rope is not always parallel to the motion it controls. A winch a height h above the water reels in a boat at horizontal distance x, so ℓ² = x² + h², and because h is fixed, ℓ(dℓ/dt) = x(dx/dt). Written another way, v cos θ = |dℓ/dt| with θ the angle between rope and velocity: the rope shortens at the rate of the velocity component along it, so the boat outruns the reel-in rate by a factor 1/cos θ. With h = 4.0 m, x = 3.0 m and a reel-in rate of 1.5 m s⁻¹, the rope is 5.0 m long and the boat moves at 2.5 m s⁻¹. As x → 0 the predicted speed diverges, which is where the straight-taut-rope model fails, not the boat.

Two bodies that stay in contact without overlapping or separating must have equal velocity components along their common normal. The tangential components may differ, and that difference is the sliding. A rigid rod is the same statement with the line taken along the rod, so v_A · n̂ = v_B · n̂ for its two ends. Take a 5.0 m rod with its foot on the floor and its top against a wall: x² + y² = L² gives x vₓ + y vᵧ = 0, so a foot sliding out at 0.40 m s⁻¹ when x = 3.0 m and y = 4.0 m sends the top down at 0.30 m s⁻¹. A block on a moving wedge obeys the same rule along the incline's normal: its velocity relative to the wedge stays parallel to the surface.

A wheel rolling without slipping lays its rim onto the ground, so the arc Rθ equals the distance the centre travels: s = Rθ. Differentiating that one geometric fact gives v_cm = ωR and a_cm = αR, with ω = dθ/dt and α = dω/dt. Read at the contact, the same condition says that point is instantaneously at rest, which puts the centre at ωR and the top of the rim at 2ωR. With R = 0.35 m and ω = 12 rad s⁻¹ the centre moves at 4.2 m s⁻¹ and the top at 8.4 m s⁻¹; α = 8.0 rad s⁻² gives a_cm = 2.8 m s⁻². If the surface moves as well, the condition applies to the relative velocity there: v_cm − v_surface = ωR. For a wheel confined to one plane this relation integrates back to a relation between positions; for a ball free to roll over a plane it does not, and the constraint restricts velocities only.