Chapter 5: Mechanics and Energy for Technology
Motion
For motion in a straight line with constant acceleration : , , . A body falling freely accelerates at . Graphs: the gradient of a displacement–time graph is velocity; the gradient of a velocity–time graph is acceleration; the area under a velocity–time graph is displacement.
Forces
Newton’s laws: (1) a body stays at rest or moves uniformly unless a net force acts; (2) ; (3) forces occur in equal and opposite pairs. Mass (kg) is the amount of matter; weight (N) is the gravitational force. Friction opposes motion; its limiting value is , where is the normal reaction; friction is useful (brakes, grip) and harmful (wear, heat), and is reduced by lubrication, bearings, and smooth surfaces.
Moments and equilibrium
The moment of a force about a point is (d is the perpendicular distance). A body in equilibrium has no net force and no net moment: clockwise moments equal anticlockwise moments. Example: a lever has an effort of 50 N applied 1.2 m from the pivot, balancing a load 0.3 m from the pivot on the other side: load N. The centre of gravity is the point where the weight acts; a structure is stable if its centre of gravity is low and above a wide base.
Work, energy, and power
Work: (joules).
Kinetic energy: . Gravitational potential energy: . Elastic energy in a spring: .
Conservation of energy: energy is neither created nor destroyed, only transformed. In real machines some becomes heat through friction.
Power: (watts).
Efficiency .
Examples: lifting 100 kg through 5 m in 10 s needs a power of W. A machine with 500 J input and 350 J useful output has efficiency 70%.
Simple machines
Machines make work easier by changing the size or direction of a force; they do not reduce the work. Key ratios:
Examples: levers (three classes), pulleys (VR equals the number of rope sections supporting the load), inclined planes, wheel and axle, screws, and gears. Example: a pulley system with four supporting ropes (VR = 4) lifts a 400 N load with an effort of 125 N: MA , efficiency . For gears, the speed ratio is inverse to the tooth ratio: a driver of 20 teeth turning a driven gear of 60 teeth gives a 3:1 reduction (speed ÷ 3, torque × 3, ignoring losses).
Pressure and fluids
Density . Pressure (pascals). In a fluid at depth , (gauge). Example: at 10 m depth in water, kPa above atmospheric. Pascal’s principle: pressure applied to an enclosed fluid is transmitted undiminished, which is the basis of hydraulic jacks and brakes: . A 100 N force on a 10 cm² piston gives N on a 200 cm² piston. Archimedes’ principle: the upthrust on a body equals the weight of the fluid it displaces. A body floats if its average density is less than that of the fluid.
Common mistakes
Confusing mass and weight.
Taking moments about different points in one equation.
Using the wrong direction for friction.
Forgetting that a machine’s efficiency is always below 100%.
Practice questions
A car accelerates from rest at for 8 s. Find the speed and the distance. [24 m s⁻¹; 96 m]
A motor lifts a 50 kg load through 12 m in 20 s. Find the power output. [294 W]
A hydraulic press has pistons of areas 20 cm² and 500 cm². What force on the large piston results from 80 N on the small one? [2000 N]