Wedges and belt friction
Wedges use friction and inclined surfaces to split, lift, and hold. Belts and ropes wrapped around drums use friction to multiply holding forces exponentially.
Wedges
A wedge is a simple machine with two inclined faces, used to split wood, lift heavy objects slightly, or hold things in place. Analysing a wedge requires free-body diagrams of the wedge and the body it pushes, with friction at every contact surface acting opposite to the impending sliding.
Self-locking wedges
If friction is large enough relative to the wedge angle, a wedge stays in place after the driving force is removed. A doorstop works this way. A wedge with a small angle is more likely to be self-locking, because friction more easily exceeds the component of force pushing it out.
Screws as wedges
Screw threads behave like a wedge wrapped around a cylinder. Fine-pitch screws with low lead angles are usually self-locking, which is why bolts stay tight and screw jacks hold cars, while low-friction ball screws are not and need brakes.
Belt friction
When a flat belt or rope wraps around a fixed drum and is about to slip, the tensions on its two sides satisfy \(T_2 = T_1 e^{\mu\beta}\), where \(T_2\) is the larger tension, \(\mu\) the coefficient of friction, and \(\beta\) the wrap angle in radians. This is often called the Euler-Eytelwein or capstan equation, after work by Leonhard Euler and Johann Eytelwein in the 1700s.
Exponential growth
Because the wrap angle sits in an exponent, extra turns multiply holding power enormously. With \(\mu = 0.3\), one full turn (\(\beta = 2\pi\)) gives a ratio of \(e^{1.885} \approx 6.6\); three turns give \(e^{5.65} \approx 285\). A sailor holding 100 N can restrain about 28 kN with three turns around a bollard.
Belts and brakes
Belt drives in engines and machines transmit power by friction, with a tight side and a slack side whose tension ratio cannot exceed the capstan limit without slipping. Band brakes wrap a band around a drum, using the same exponential effect to stop rotation with modest lever forces.
Choosing the right formula
Use mu N for flat contact, the incline analysis for wedges and screws, and the capstan equation for flexible belts and ropes over curved surfaces. Mixing them up leads to wildly wrong answers, especially with ropes, where friction is far stronger than intuition suggests.
Worked example
A climber's rope passes over a smooth-looking rock edge with a 90-degree wrap (pi/2 radians) and mu = 0.4. The tension ratio is e to the power (0.4 x 1.571) = e to the 0.628, about 1.87. A belayer holding 400 N on one side can hold up to about 750 N on the other before the rope slips over the rock.
Key takeaways
Wedges split, lift, and hold, and are analysed with free-body diagrams that include friction on every contact face; small-angle wedges and fine screws are often self-locking. Flexible belts and ropes on drums obey the capstan equation T2 = T1 e^(mu beta), with beta in radians, so holding power grows exponentially with wrap angle: three turns at mu = 0.3 multiply force about 285 times. Belt drives and band brakes rely on this effect.
Flashcard
Two inclined faces
Check your understanding
A rope wraps 3 full turns around a bollard with mu = 0.3. Roughly how many times larger can the load-side tension be?