Cables, springs, and pulleys
Cables, springs, and pulleys connect, store, and redirect forces. Each has a simple rule that turns a messy device into a solvable equilibrium problem.
Ideal cables
An ideal cable is weightless, cannot stretch, and carries only tension along its length. Its tension is the same at every point along a straight segment. Real cables sag under their own weight, forming a curve called a catenary, studied by Huygens, Leibniz, and Johann Bernoulli in 1691.
Linear springs
A linear elastic spring exerts a force proportional to its change in length: F = ks, where k is the stiffness in N/m and s the stretch or compression from its unstretched length. A 20 kg mass hung on a spring with k = 500 N/m stretches it by 196.2 / 500, about 0.392 m.
Springs in equilibrium problems
A spring's force depends on its stretched length, so geometry and equilibrium must be combined. If a spring with k = 800 N/m and unstretched length 0.40 m is stretched to 0.55 m, it pulls with 800 x 0.15 = 120 N along its axis.
Ideal pulleys
An ideal pulley is frictionless and weightless, so it changes a rope's direction without changing its tension. A single fixed pulley lets you lift a load by pulling down, but the pull still equals the load's weight.
Mechanical advantage of pulley systems
In a block and tackle, a single rope passes around several pulleys, and the load is supported by n rope segments, each carrying the same tension T. Equilibrium of the moving block gives nT = W, so the pull needed is W/n. With 4 supporting segments, lifting 2,000 N takes only 500 N, but the rope must be pulled 4 times as far.
Friction in real pulleys
Real pulleys have bearing friction, so tension changes slightly around each pulley, and efficiency falls as more pulleys are added. Engineers account for this with efficiency factors, often around 90 to 95% per sheave for good bearings.
Particles in 3D
A ring held by three cables in three dimensions needs three equations: \(\sum F_x = 0\), \(\sum F_y = 0\), and \(\sum F_z = 0\). Writing each cable force as magnitude times unit vector, from coordinates, sets up a 3 by 3 linear system for the three tensions.
Worked example
A 60 kg engine is lifted with a block and tackle in which 3 rope segments support the moving block. Ignoring friction, the pull needed is 60 x 9.81 / 3 = 196.2 N. To raise the engine 0.5 m, the mechanic pulls 1.5 m of rope. With 92% efficiency per pulley, the real pull would be noticeably higher.
Key takeaways
Ideal cables carry only tension, constant along a straight segment; real cables sag in a catenary. Linear springs exert F = ks, with s measured from the unstretched length. Ideal pulleys redirect tension without changing it. A block and tackle with n supporting segments needs a pull of W/n but n times the rope travel. Real pulleys lose some force to friction. Particles in 3D need three equilibrium equations, solved with Cartesian vectors.
Flashcard
Weightless, inextensible, tension only
Check your understanding
A block and tackle has 4 rope segments supporting a 2,000 N load. Ignoring friction, what pull is needed?