Mechanics

Revision notes

Mechanics connects motion to forces and energy. For constant acceleration, useful equations include v = u + at, s = ut + \(\dfrac{1}{2}at^{2}\) and v² = u² + 2as. Vector components allow two-dimensional motion to be treated independently when appropriate, such as horizontal and vertical projectile motion without air resistance.

Newton's second law may be written F = \(\dfrac{dp}{dt}\), reducing to F = ma for constant mass. Impulse equals change in momentum. Work done by a force transfers energy; for a constant force parallel to displacement W = Fs. In uniform circular motion the velocity changes direction continuously, requiring centripetal acceleration a = \(\dfrac{v^{2}}{r}\) and a resultant inward force \(\dfrac{mv^{2}}{r}\).

Vectors, kinematics and motion graphs

Mechanics begins with careful definitions. Displacement, velocity, acceleration, force and momentum are vectors; distance, speed, mass, energy and power are scalars. For constant acceleration, the SUVAT equations connect displacement, initial and final velocity, acceleration and time. They are only valid when acceleration is constant, so do not apply them blindly to motion with changing drag or thrust.

On a displacement-time graph, gradient is velocity. On a velocity-time graph, gradient is acceleration and signed area is displacement. Curved graphs indicate changing gradients. For projectiles, horizontal and vertical motions can be analysed independently when air resistance is neglected: horizontal velocity is constant while vertical acceleration is approximately g.

Forces and Newton's laws

The resultant force determines acceleration according to F = ma for constant mass. Newton's first law describes the zero-resultant case; Newton's second links resultant force to rate of change of momentum; Newton's third concerns equal and opposite forces acting on different bodies.

Free-body diagrams are essential. Draw only forces acting on the chosen object. Resolve forces into perpendicular components before applying equilibrium or acceleration equations. Weight is mg and acts through the centre of mass in a uniform gravitational field.

Work, energy and power

Work done is energy transferred by a force. For a constant force parallel to displacement, W = Fs; more generally the component of force along the displacement matters. Kinetic energy is \(\dfrac{1}{2}mv^{2}\) and gravitational potential-energy change near Earth's surface is mgΔh. The work-energy principle links net work to change in kinetic energy.

Power is the rate of energy transfer, P = \(\dfrac{E}{t}\), and for steady motion under a force parallel to velocity, P = Fv. Efficiency compares useful output with total input and never exceeds 1 for a real system.

Momentum and impulse

Momentum p = mv is conserved in a closed system. Impulse is change in momentum and equals the area under a force-time graph. In collisions, momentum may be conserved while kinetic energy is not; energy can transfer to internal energy, deformation and sound.

For rebounds, signs matter. Choose a positive direction before substituting. A change from +v to -v is larger than simply stopping from +v to zero.

Circular motion

An object in uniform circular motion has constant speed but changing velocity, so it accelerates towards the centre. The centripetal acceleration is \(\dfrac{v^{2}}{r}\) = ω²r and the required resultant force is \(\dfrac{mv^{2}}{r}\). 'Centripetal force' is not an extra type of force; it is the name for the inward resultant supplied by real forces such as tension, gravity or friction.

Resistive forces and terminal motion

When drag increases with speed, a falling object can reach terminal speed when drag plus any other upward forces balance weight. The resultant force then becomes zero and acceleration becomes zero, but velocity remains non-zero.

Exam technique

Start with a diagram and define a positive direction. Check whether acceleration is actually constant before using SUVAT. In force questions, identify the physical force supplying any required centripetal resultant. In collision problems, conserve momentum first and discuss kinetic-energy changes separately.

Moments, couples and equilibrium

The moment of a force about a point is the force multiplied by the perpendicular distance from the point to its line of action. For an object in equilibrium, the resultant force is zero and the sum of clockwise moments about any point equals the sum of anticlockwise moments. A couple is a pair of equal, opposite, parallel forces not in line; its torque is one force multiplied by the perpendicular distance between them. The weight of a uniform object acts at its centre of mass.

Angular speed

An object moving in a circle of radius r at speed v has angular speed ω = \(\dfrac{v}{r}\) = \(\dfrac{2\pi }{T}\), measured in rad s⁻¹. Centripetal acceleration can be written a = \(\dfrac{v^{2}}{r}\) = ω²r.

Density

Density is mass per unit volume, ρ = \(\dfrac{m}{V}\), measured in kg m⁻³ (1 g cm⁻³ = 1000 kg m⁻³).