Mechanical engineering cheat sheet

Forces, motion, heat and mechanical power.

Force and acceleration

For constant mass in an inertial reference frame, resultant force equals mass times acceleration.

ΣF = ma

Torque

Torque measures a force’s turning effect about an axis. The perpendicular moment arm matters.

τ = Fd⊥

Work and power

Work transfers energy through a force acting over displacement. Power is the rate of energy transfer.

For a constant force along the displacement: W = Fs; P = W/t.

Thermal balance

The first law relates a system’s internal-energy change to heat and work. State the sign convention.

With work done by the system positive: ΔU = Q − W.

Momentum and impulse

Linear momentum is p = mv. Impulse is the integral of force over time and equals the change in momentum for the defined system.

For a constant force, impulse J = FΔt.

Energy methods

Kinetic energy is ½mv² for translational motion. Work changes mechanical energy, while friction and other processes may transfer energy into internal forms.

An energy balance can avoid solving every instant of a motion.

Rotational motion

Angular velocity, moment of inertia and torque are rotational quantities. In a fixed-axis model with constant inertia, net torque is Iα.

Mass distribution matters: equal masses can have different moments of inertia.

Heat and work

Heat and work are modes of energy transfer across a system boundary. Internal energy is a property of the system, so it should not be described as “heat stored inside”.

State the sign convention before using a first-law energy balance.

Worked example: power

A machine supplies a constant 150 N force through 4 m in 3 s along the direction of motion. Work is 600 J and average power is 600/3 = 200 W.

Power is a rate; the same work performed in half the time requires twice the average power.

Common mistake: force guarantees work

Mechanical work depends on the component of force along displacement. A perpendicular force can change direction without doing work in an idealised model.

W = Fs cos θ for a constant force and straight displacement.