02 / CHAPTER 07
Tension, pulleys, and constraints
Use geometry and idealized connectors to couple several bodies.
ANCHOR EQUATION
Tension constraint
In symbols: a₁ = a₂
An ideal inextensible connector couples the motion of its endpoints.
Constraints transfer acceleration information
An ideal taut, massless, inextensible rope forces its endpoints to change length together. A pulley changes force direction; it does not erase the geometric constraint connecting the coordinates.
Differentiate the rope length
For two masses y₁ and y₂ hanging from a fixed pulley, the constant rope length is L = y₁ + y₂ + constant. Differentiating gives v₁ + v₂ = 0 and differentiating again gives a₁ + a₂ = 0. The Newton equations m₁g − T = m₁a and T − m₂g = m₂a then combine: (m₁ − m₂)g = (m₁ + m₂)a.
With m₁ = 3 kg and m₂ = 2 kg, a = (1/5)9.81 = 1.96 m/s². T = m₁(g − a) = 3(7.85) = 23.5 N, which lies below the heavier weight and above the lighter weight.
Real ropes add dynamics
Rope mass, pulley inertia, stretch, and axle friction make tensions differ on the two sides. The ideal result is a limit in which those effects are negligible compared with the body weights.
Work it: For 4 kg and 1 kg masses, derive a and T symbolically, then calculate them. Check that a approaches g as one mass becomes negligible.