M · Mechanics — Reference Library

Kinematics

Position, velocity and acceleration describe motion without asking what causes it.

M1.1Position, Velocity, Acceleration

Velocity is the rate of change of position; acceleration is the rate of change of velocity. Both are vectors.

Instantaneous rate of change of position

v = \frac{d r}{d t}

DefinitionM1.1.1 Instantaneous rate of change of velocity

a = \frac{d v}{d t}

DefinitionM1.1.2

M1.2Constant Acceleration

With acceleration constant, integrating once gives velocity as a function of time and integrating again gives position.

Velocity and position under constant acceleration

v = v_{0} + a t, x = x_{0} + v_{0} t + \frac{1}{2} a t^{2}

ResultM1.2.1

M1.3Projectile Motion

With air resistance neglected, horizontal and vertical motion are independent: constant velocity horizontally, constant acceleration g downward vertically.

Trajectory of an object launched into free fall

x = v_{0} cos θ t, y = v_{0} sin θ t - \frac{1}{2} g t^{2}

ResultM1.3.1

M1.4Relative Motion

Velocities measured in different reference frames moving at constant velocity relative to one another add as vectors.

Velocity of one object relative to another

v_{A C} = v_{A B} + v_{B C}

ResultM1.4.1

Forces

Newton's three laws connect force to motion, and give every force an equal and opposite partner.

M2.1The Three Laws of Motion

Newton's first law: an object retains constant velocity unless a net force acts on it — inertia. His second and third laws are stated below.

Net force equals mass times acceleration

F_{net} = m a

LawM2.1.1 Forces between two objects are equal and opposite

F_{A B} = - F_{B A}

LawM2.1.2

M2.2Free-Body Diagrams

A free-body diagram isolates one object and shows every external force acting on it as a vector — weight, normal force, tension, applied force, friction — drawn separately from any force it exerts on something else. Summing components along chosen axes converts the diagram directly into Newton's second law.

M2.3Friction

Friction opposes relative sliding, or the tendency to slide, between two surfaces in contact.

Force resisting sliding between two surfaces

f_{s} \leq μ_{s} N, f_{k} = μ_{k} N

LawM2.3.1

M2.4Circular Motion

An object moving on a circular path at any speed accelerates toward the centre, even if its speed is constant.

Acceleration and force directed toward the centre of a circular path

a_{c} = \frac{v ^{2}}{r}, F_{c} = \frac{m v ^{2}}{r}

ResultM2.4.1

Energy

Energy is conserved in every closed system; work is the bookkeeping that tracks it moving between forms.

M3.1Work

Work is the energy transferred to or from an object by a force acting through a displacement.

Energy transferred by a force through a displacement

W = \int F \cdot d s = F d cos θ

DefinitionM3.1.1

M3.2Kinetic and Potential Energy

Kinetic energy is the energy of motion. Potential energy is stored energy that depends on position, released as work when that position changes.

Net work equals the change in kinetic energy

W_{net} = Δ K = \frac{1}{2} m v_{f}^{2} - \frac{1}{2} m v_{i}^{2}

ResultM3.2.1 Energy stored by height or by spring deformation

U_{g} = m g h, U_{s} = \frac{1}{2} k x^{2}

ResultM3.2.2

M3.3Conservation of Energy

In a system with only conservative forces doing work, the total mechanical energy stays constant.

Kinetic plus potential energy stays constant

K_{i} + U_{i} = K_{f} + U_{f}

PrincipleM3.3.1

M3.4Power

Power is the rate at which energy is transferred or work is done.

Rate of doing work

P = \frac{d W}{d t} = F \cdot v

DefinitionM3.4.1

Momentum

Momentum, like energy, is conserved in an isolated system — a bookkeeping tool that survives even violent, energy-dissipating collisions.

M4.1Impulse

Impulse is the effect of a force acting over time; it changes an object's momentum by exactly that amount.

Force integrated over time equals change in momentum

J = \int F d t = Δ p

ResultM4.1.1

M4.2Conservation of Momentum

In an isolated system — no net external force — total momentum does not change, however violently the parts of the system interact with each other.

Total momentum of an isolated system is unchanged

p_{total, i} = p_{total, f}

PrincipleM4.2.1

M4.3Collisions

In an elastic collision, kinetic energy is conserved along with momentum. In an inelastic collision it is not — kinetic energy is lost to heat, sound, or deformation. In a perfectly inelastic collision the objects stick together.

Common final velocity after a perfectly inelastic collision

v_{f} = \frac{m _{1} v _{1 i} + m _{2} v _{2 i}}{m _{1} + m _{2}}

ResultM4.3.1

M4.4Centre of Mass

The centre of mass is the mass-weighted average position of a system; it moves as though all the system's mass were concentrated there and all external force applied there.

Mass-weighted average position of a system

r_{c m} = \frac{1}{M} \sum_{i} m_{i} r_{i}

DefinitionM4.4.1

Rotation

Every linear quantity has a rotational counterpart; the same equations apply once mass becomes moment of inertia and force becomes torque.

M5.1Angular Kinematics

Angular position, velocity and acceleration describe rotation exactly as their linear counterparts describe straight-line motion.

Rate of change of angular position and velocity

ω = \frac{d θ}{d t}, α = \frac{d ω}{d t}

DefinitionM5.1.1

M5.2Moment of Inertia

Moment of inertia is the rotational counterpart of mass — a measure of how hard it is to change an object's rotation, dependent on how its mass is distributed relative to the axis.

Rotational analogue of mass

I = \sum_{i} m_{i} r_{i}^{2} = \int r^{2} d m

DefinitionM5.2.1 Moment of inertia about an axis offset from the centre of mass

I = I_{c m} + M d^{2}

ResultM5.2.2

M5.3Torque

Torque is the rotational counterpart of force: it measures the twisting effect of a force about a chosen axis.

Twisting effect of a force about an axis

τ = r \times F, τ = r F sin θ

DefinitionM5.3.1 Net torque equals moment of inertia times angular acceleration

τ_{net} = I α

ResultM5.3.2

M5.4Angular Momentum

Angular momentum is the rotational counterpart of momentum, and like momentum it is conserved whenever no net external torque acts.

Rotational analogue of momentum

L = r \times p = I ω

DefinitionM5.4.1 Total angular momentum is unchanged with no net external torque

L_{i} = L_{f} (τ_{net,ext} = 0)

PrincipleM5.4.2

Equilibrium and Elasticity

An object at rest has zero net force and zero net torque; push it past its elastic limit and these equations stop applying.

M6.1Conditions for Equilibrium

An object is in static equilibrium when it has neither linear nor angular acceleration.

Net force and net torque both vanish

\sum F = 0, \sum τ = 0

PrincipleM6.1.1

M6.2Centre of Gravity

The centre of gravity is the point where the total weight of an object can be considered to act. In a uniform gravitational field it coincides exactly with the centre of mass.

M6.3Stress and Strain

Stress is force per unit area within a material; strain is the resulting fractional deformation.

Force per area, and fractional deformation

stress = \frac{F}{A}, strain = \frac{Δ L}{L _{0}}

DefinitionM6.3.1

M6.4Elastic Moduli

An elastic modulus is the ratio of stress to strain for a given type of deformation, constant while the material remains within its elastic limit.

Ratio of tensile stress to strain

Y = \frac{F / A}{Δ L / L _{0}}

DefinitionM6.4.1

Gravitation

Every pair of masses attracts, following the same inverse-square form as the force between charges.

M7.1Universal Gravitation

Every pair of point masses attracts one another with a force along the line joining them.

Attractive force between two masses

F_{g} = G \frac{m _{1} m _{2}}{r ^{2}}

LawM7.1.1

M7.2Gravitational Field and Potential Energy

A mass sets up a gravitational field around it; another mass placed in that field acquires potential energy that depends on separation.

Force per unit mass around a source mass

g = \frac{F _{g}}{m} = - \frac{GM}{r ^{2}} r

DefinitionM7.2.1 Potential energy of two masses at separation r

U_{g} = - \frac{GM m}{r}

ResultM7.2.2

M7.3Orbits

A satellite in a circular orbit is in free fall, with gravity supplying exactly the centripetal force needed.

Speed needed to maintain a circular orbit

v = \frac{GM}{r}

ResultM7.3.1

M7.4Kepler's Laws

Kepler's first law: planetary orbits are ellipses with the Sun at one focus. His second law: a line from the Sun to a planet sweeps out equal areas in equal times. The third law, below, follows directly from Newton's law of gravitation.

Orbital period related to orbital size

T^{2} = \frac{4 π ^{2}}{GM} a^{3}

ResultM7.4.1

Fluids

A fluid at rest carries pressure that grows with depth; a fluid in motion trades that pressure for speed.

M8.1Pressure and Depth

Pressure in a static fluid increases with depth, as each layer supports the weight of the fluid above it.

Pressure increase with depth in a static fluid

P = P_{0} + ρ g h

ResultM8.1.1

M8.2Buoyancy

A fluid exerts a net upward force on any object submerged in it, because pressure is greater on the object's lower surface than on its upper surface.

Upward force from fluid displaced by a submerged object

F_{B} = ρ_{fluid} g V_{displaced}

ResultM8.2.1

M8.3Continuity

For an incompressible fluid flowing through a pipe of varying cross-section, mass is conserved.

Flow speed and area related for incompressible flow

A_{1} v_{1} = A_{2} v_{2}

PrincipleM8.3.1

M8.4Bernoulli's Equation

Along a streamline, the sum of pressure, kinetic energy density and gravitational potential energy density is constant — energy conservation applied to flowing fluid.

Pressure, speed and height conserved along a streamline

P + \frac{1}{2} ρ v^{2} + ρ g h = constant

PrincipleM8.4.1