Machines and mechanical advantage
Machines are frames with moving parts that transform an input force into a larger or more useful output force. Statics finds that output by treating the machine as momentarily at rest.
What a machine is
A machine is a structure with moving parts designed to transmit and change forces: pliers, scissors, bolt cutters, car jacks, cranes, and robotic grippers. In statics, we analyse a machine at an instant when it is in equilibrium, holding a load or just about to move.
Mechanical advantage
The mechanical advantage (MA) is the ratio of output force to input force. A tool with MA = 4 turns a 50 N squeeze into a 200 N grip. Ideal mechanical advantage ignores friction; actual mechanical advantage, measured in practice, is always somewhat lower.
Pliers
For pliers, take moments about the pivot pin on one handle-jaw member. If your hand squeezes with 50 N at 90 mm from the pin and the jaws grip 30 mm from the pin, then the jaw force is 50 x 90 / 30 = 150 N, an MA of 3. Gripping closer to the pivot increases the force.
Work and distance
Machines do not create energy. Ignoring friction, input work equals output work, so a larger output force comes with a smaller output movement. The pliers' jaws move only a third as far as your hand, which is the price of tripling the force.
Efficiency
Efficiency is useful output work divided by input work, always less than 100% because of friction. A jack with an ideal MA of 100 and an efficiency of 60% has an actual MA of 60. Engineers choose bearings, lubrication, and geometry to keep efficiency high.
Compound machines
Linking machines multiplies their advantages. Bolt cutters use two levers in series: if each stage multiplies force by 6, the overall MA is 36, letting a 300 N squeeze produce about 10.8 kN at the blades.
Self-locking machines
Some machines hold a load without input force because friction resists reversal, such as most screw jacks and worm gears. Self-locking is essential for safety in car jacks and lifts but usually means efficiency below about 50%.
Worked example
A pair of garden shears has blades that cut 40 mm from the pivot and handles gripped 240 mm from the pivot. Squeezing with 120 N on each handle gives a cutting force of 120 x 240 / 40 = 720 N at the blades, an MA of 6. Cutting a thick branch nearer the pivot, at 20 mm, doubles the force to 1,440 N.
Key takeaways
A machine transmits and transforms forces through moving parts, analysed in statics at an instant of equilibrium. Mechanical advantage is output force over input force; actual MA is below ideal MA because of friction. Find a machine's output by taking moments about pivots, as with pliers. Larger forces come with smaller movements, since work is conserved. Efficiency is output work over input work. Compound machines multiply their advantages, and self-locking machines hold loads safely at the cost of efficiency.
Flashcard
Structure with moving parts transforming force
Check your understanding
Pliers are squeezed with 50 N at 90 mm from the pin; the jaws grip 30 mm from the pin. What is the grip force?