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mechanics

Kinematics incl. polar coordinates, forces & constraints, momentum & mass flow, energy & stability, conservative forces, rigid-body rotation & gyroscopes, noninertial frames, fluids.

kinematics · polar coordinates · projectiles · circular motion · forces & constraints · momentum & collisions · mass flow & momentum transport · work, energy & power · potential energy & stability · conservative forces & the gradient · fluids & materials · fixed-axis rotation · rigid-body dynamics · noninertial frames

Kinematics

Definitions. \(\vv{v}=\dfrac{\dd\vv{s}}{\dd t}\), \(\vv{a}=\dfrac{\dd\vv{v}}{\dd t}=\dfrac{\dd^2\vv{s}}{\dd t^2}\).

Formal solution. \(\vv{v}(t)=\vv{v}_0+\int_0^t\vv{a}\dd t'\), \(\vv{r}(t)=\vv{r}_0+\int_0^t\vv{v}\dd t'\) — integrate component by component for any \(\vv{a}(t)\).

Graphs. Area under \(v\)–\(t\): displacement; gradient of \(v\)–\(t\): \(a\); area under \(a\)–\(t\): \(\Delta v\).

Relative velocity. \(\vv{v}_{A/B}=\vv{v}_A-\vv{v}_B\).

Constant acceleration. \(v=u+at\), \(s=ut+\tfrac12at^2\), \(v^2=u^2+2as\), \(s=\tfrac12(u+v)t\). In 2D, apply by components independently.

Derivative of a vector. The component of \(\dd\vv{A}/\dd t\) along \(\vv{A}\) changes its magnitude; the perpendicular component changes its direction. A vector of constant magnitude rotating at rate \(\omega\) has \(|\dd\vv{A}/\dd t|=A\omega\), perpendicular to \(\vv{A}\) — so \(v=r\omega\) for circular motion, and unit vectors attached to a moving point have nonzero derivatives.

Polar coordinates

Unit vectors. \(\uvec{r}\), \(\hat{\boldsymbol\theta}\) point radially outward and tangentially at the particle's position, so they rotate with it: \(\dd\uvec{r}/\dd t=\dot\theta\,\hat{\boldsymbol\theta}\), \(\dd\hat{\boldsymbol\theta}/\dd t=-\dot\theta\,\uvec{r}\).

Velocity. \(\vv{v}=\dot r\,\uvec{r}+r\dot\theta\,\hat{\boldsymbol\theta}\): radial speed plus tangential speed \(r\dot\theta\).

Acceleration. \[\vv{a}=(\ddot r-r\dot\theta^2)\,\uvec{r}+(r\ddot\theta+2\dot r\dot\theta)\,\hat{\boldsymbol\theta}.\] \(-r\dot\theta^2\) is the centripetal term (pure circular motion recovers \(a_r=-r\omega^2\), \(a_\theta=r\alpha\)); \(2\dot r\dot\theta\) is the Coriolis term — half from the rotation of the radial velocity, half from the growth of tangential speed as \(r\) changes. Beware: \(a_r\ne\ddot r\) whenever \(\dot\theta\ne0\).

Projectiles

Components. \(x=u\cos\theta\,t\), \(y=u\sin\theta\,t-\tfrac12gt^2\); \(v_x=u\cos\theta\), \(v_y=u\sin\theta-gt\).

Same launch/landing height. \(T=2u\sin\theta/g\), \(H=u^2\sin^2\theta/(2g)\), \(R=u^2\sin2\theta/g\).

Trajectory. \(y=x\tan\theta-\dfrac{gx^2}{2u^2\cos^2\theta}\).

Circular motion

Kinematics. \(\omega=2\pi f=2\pi/T\), \(v=r\omega\), \(a_c=v^2/r=\omega^2 r\) with \(\vv{a}_c\) toward the centre; \(F_c=mv^2/r=m\omega^2 r\).

Banked turn, no friction. \(\tan\theta=v^2/(rg)\).

Vertical circle. Top: \(N+mg=mv^2/r\); bottom: \(N-mg=mv^2/r\); minimum top speed \(v=\sqrt{gr}\).

Forces & constraints

Newton. \(\sum\vv{F}=m\vv{a}\), valid in inertial frames — frames in which an isolated body stays at rest or moves uniformly (the first law defines them). Third law: forces come in equal-and-opposite pairs acting on different bodies. Weight \(\vv{W}=m\vv{g}\); gravitation \(F_g=Gm_1m_2/r^2\).

Method. Isolate each body with a free-body diagram; write \(\sum\vv F=m\vv a\) per body; write the constraint equations (fixed string length, contact with a surface) and differentiate them to relate the accelerations.

Contact forces. Normal \(N\) perpendicular to surface; tension along string; thrust/drag opposite relative motion.

Ropes. Tension is uniform only if the rope is massless and unloaded; in a rope of linear density \(\lambda\), tension varies so each element \(\lambda\,\dd x\) gets its share of \(m a\) (or its weight) — e.g. it grows from zero at the free end of a hanging rope to \(Mg\) at the support. A uniform rope of mass \(M\), length \(L\) whirled about one end carries \(T(r)=\tfrac{M\omega^2}{2L}(L^2-r^2)\).

Friction. Static \(f_s\le\mu_s N\); kinetic \(f_k=\mu_k N\); limiting \(f_s=\mu_s N\).

Hooke. \(F=-kx\) up to the elastic limit; spring energy \(E_e=\tfrac12kx^2\); mass on a spring executes SHM with \(\omega=\sqrt{k/m}\).

Drag. Viscous \(F_D=-Cv\): released body decays as \(v=v_0e^{-t/\tau}\), \(\tau=m/C\); quadratic \(F_D=\tfrac12C_D\rho A v^2\); terminal velocity when \(\sum F=0\).

Incline. Component \(mg\sin\theta\) down the slope, \(mg\cos\theta\) normal to it.

Apparent weight in a lift. \(N=m(g\pm a)\).

Canonical results. Atwood machine: \(a=\dfrac{(M_1-M_2)g}{M_1+M_2+I/R^2}\) (\(I\) of the pulley; drop \(I/R^2\) if light). Conical pendulum of length \(L\) at angle \(\alpha\): \(\cos\alpha=g/(\omega^2L)\) — faster spin, flatter cone, but never horizontal.

Momentum & collisions

Momentum/impulse. \(\vv{p}=m\vv{v}\), \(\vv{F}=\dfrac{\dd\vv{p}}{\dd t}\), \(\vv{J}=\int\vv{F}\dd t=\Delta\vv{p}\); average \(F=\Delta p/\Delta t\). Peak forces in collisions far exceed the average — extending \(\Delta t\) (bent knees, crumple zones) lowers the force.

Systems of particles. Internal forces cancel in third-law pairs, so \(\dfrac{\dd\vv{P}}{\dd t}=\sum\vv{F}_{\mathrm{ext}}\) with \(\vv{P}=M\vv{v}_{\mathrm{cm}}\): the centre of mass moves as a point particle under the external force alone.

Conservation. If \(\sum\vv{F}_{\mathrm{ext}}=0\), then \(\sum\vv{p}_i=\sum\vv{p}_f\).

Centre of mass. \(\vv{r}_{\mathrm{cm}}=\dfrac{\sum m_i\vv{r}_i}{\sum m_i}=\dfrac1M\int\vv{r}\dd m\), \(\vv{v}_{\mathrm{cm}}=\dfrac{\sum m_i\vv{v}_i}{M}\), \(M\vv{a}_{\mathrm{cm}}=\sum\vv{F}_{\mathrm{ext}}\).

Centre-of-mass frame. Total momentum vanishes in the frame moving at \(\vv{v}_{\mathrm{cm}}\); \(K_{\mathrm{lab}}=K_{\mathrm{cm}}+\tfrac12Mv_{\mathrm{cm}}^2\). In an elastic collision the C-frame speeds are unchanged — the velocity pair simply rotates through the scattering angle set by the interaction.

1D collisions. \(m_1u_1+m_2u_2=m_1v_1+m_2v_2\).

Elastic. Kinetic energy also conserved: \(\tfrac12m_1u_1^2+\tfrac12m_2u_2^2=\tfrac12m_1v_1^2+\tfrac12m_2v_2^2\); relative speed reverses: \(u_1-u_2=-(v_1-v_2)\).

Coefficient of restitution. \(e=\dfrac{v_2-v_1}{u_1-u_2}\).

Perfectly inelastic. Common velocity \(v=(m_1u_1+m_2u_2)/(m_1+m_2)\); \(K\) not conserved (the C-frame kinetic energy is the most a collision can dissipate).

Explosion. \(\vv{p}_{\mathrm{before}}=\vv{p}_{\mathrm{after}}\); \(K\) increases.

Mass flow & momentum transport

Variable-mass bodies. Apply \(\vv F_{\mathrm{ext}}=\dd\vv P/\dd t\) to a fixed collection of matter over \(t\to t+\Delta t\). For a body of instantaneous mass \(m\) ejecting mass at rate \(\dd m/\dd t\) with velocity \(\vv u\) relative to itself: \(m\dfrac{\dd\vv v}{\dd t}=\vv F_{\mathrm{ext}}+\vv u\dfrac{\dd m}{\dd t}\) — the last term is the thrust.

Rocket equation. Free space: \[v-v_0=u\ln\frac{M_0}{M};\] constant gravity: \(v-v_0=u\ln(M_0/M)-gt\). The gain depends on the mass ratio, hence staging.

Momentum flux. A stream of linear mass density \(\lambda\) moving at \(v\) transports momentum at rate \(\lambda v^2\). Brought to rest it pushes with \(F=\lambda v^2=\dfrac{\dd m}{\dd t}v\); rebounding at \(v'\), \(F=\lambda v(v+v')\) (elastic: \(2\lambda v^2\)). The same accounting, averaged over directions, gives the kinetic-theory gas pressure \(p=\tfrac13nm\avg{v^2}\).

Work, energy & power

Work–energy. \(W=\int\vv{F}\cdot\dd\vv{s}=Fs\cos\theta\); \(W_{\mathrm{net}}=\Delta K\), \(K=\tfrac12mv^2\) — obtained by integrating \(m\,\dd\vv v/\dd t\) along the path, in any number of dimensions. Forces perpendicular to \(\vv v\) (normal forces, magnetic forces) do no work.

Potential energies. Conservative force: \(W\) is path-independent, so \(U_b-U_a=-\int_a^b\vv F\cdot\dd\vv r\) defines \(U\) (zero point arbitrary). \(\Delta U_g=mg\Delta h\) near Earth; \(U_g=-GMm/r\); \(U_e=\tfrac12kx^2\); \(E_{\mathrm{mech}}=K+U\).

Non-conservative forces. \(W_{\mathrm{nc}}=\Delta E_{\mathrm{mech}}\). Friction converts macroscopic kinetic energy into internal (thermal) energy — total energy, with heat counted, is always conserved.

Efficiency. \(\eta=\dfrac{\text{useful output}}{\text{input}}\).

Power. \(P=\dfrac{\dd W}{\dd t}=\vv{F}\cdot\vv{v}\); average \(P=E/t\).

Potential energy & stability

Force from potential. \(F_x=-\dd U/\dd x\): force points downhill in \(U\). Equilibria sit where \(\dd U/\dd x=0\).

Stability. \(\dd^2U/\dd x^2\gt0\): stable (restoring force); \(\lt0\): unstable; \(=0\): neutral. For composite systems (e.g. balancing toys) write \(U\) of the configuration and test its curvature — stability requires the centre of mass to rise for any displacement.

Energy diagrams. Motion is confined to \(U(x)\le E\); turning points where \(U=E\). Comparing \(E\) with \(U(\infty)\) separates bound from unbound motion; a local minimum between barriers traps the particle.

Small oscillations. Near a minimum \(x_0\), \(U\approx U(x_0)+\tfrac12k(x-x_0)^2\) with \(k=U''(x_0)\): every smooth bound system is harmonic at small amplitude, with \[\omega=\sqrt{\frac{U''(x_0)}{m}}.\] This is how molecular vibration frequencies follow from the interatomic potential's curvature.

Conservative forces & the gradient

Gradient form. In three dimensions \(\vv F=-\nabla U\), i.e. \(F_x=-\partial U/\partial x\), etc.; \(\dd U=-\vv F\cdot\dd\vv r\). \(\nabla U\) is perpendicular to the equipotential surfaces and points along the steepest increase, so force lines cross equipotentials at right angles (full machinery: multivariable calculus).

Test for a conservative force. Equivalent statements: \(\oint\vv F\cdot\dd\vv r=0\) around every closed path ⇔ \(W\) path-independent ⇔ \(\vv F=-\nabla U\) exists ⇔ \(\nabla\times\vv F=\vv 0\) everywhere (component conditions \(\partial F_y/\partial x=\partial F_x/\partial y\), cyclic). Any central force \(F(r)\,\uvec r\) passes; velocity-dependent forces (friction, drag) fail.

Stokes' theorem. \(\oint_C\vv F\cdot\dd\vv r=\int_S(\nabla\times\vv F)\cdot\dd\vv A\) — curl is circulation per unit area, which is why zero curl kills every closed-loop work integral.

Fluids & materials

Fluids. Density \(\rho=m/V\); pressure \(p=F/A\); hydrostatic \(p=p_0+\rho gh\); upthrust \(F_B=\rho_{\mathrm{fluid}}\,g\,V_{\mathrm{displaced}}\).

Materials. Stress \(\sigma=F/A\), strain \(\varepsilon=\Delta L/L\), Young modulus \(E_Y=\sigma/\varepsilon=FL/(A\,\Delta L)\); elastic energy density \(=\tfrac12\sigma\varepsilon\).

Fixed-axis rotation

Angular variables. \(s=r\theta\), \(\omega=\dd\theta/\dd t\), \(\alpha=\dd\omega/\dd t\); \(v=r\omega\), \(a_t=r\alpha\), \(a_c=r\omega^2\).

Constant angular acceleration. \(\omega=\omega_0+\alpha t\), \(\theta=\omega_0 t+\tfrac12\alpha t^2\), \(\omega^2=\omega_0^2+2\alpha\theta\), \(\theta=\tfrac12(\omega_0+\omega)t\).

Torque/dynamics. \(\boldsymbol{\tau}=\vv{r}\times\vv{F}\), \(\tau=rF\sin\theta=Fd_\perp\); \(\sum\tau=I\alpha\), \(I=\sum m_ir_i^2=\int r^2\dd m\). Gravity torques a body as if its whole weight acted at the centre of mass: \(\boldsymbol\tau=\vv R_{\mathrm{cm}}\times M\vv g\).

Rotational work/energy. \(K_{\mathrm{rot}}=\tfrac12I\omega^2\), \(W=\int\tau\dd\theta\), \(P=\tau\omega\).

Angular momentum. \(\vv{L}=\vv{r}\times\vv{p}\); rigid body \(L=I\omega\); \(\sum\tau=\dd L/\dd t\), \(J_\theta=\int\tau\dd t=\Delta L\). No external torque \(\Rightarrow L\) conserved (internal forces along the lines joining particles contribute no net torque).

Translation + rotation. \(\vv L=I_{\mathrm{cm}}\boldsymbol\omega+\vv R_{\mathrm{cm}}\times M\vv V\) (spin + orbital); \(\tau_{\mathrm{cm}}=I_{\mathrm{cm}}\alpha\) holds about the centre of mass even when the CM accelerates; \(K=\tfrac12MV_{\mathrm{cm}}^2+\tfrac12I_{\mathrm{cm}}\omega^2\), and the work–energy theorem splits the same way. Striking a pivoted body at its centre of percussion, a distance \(l=I_{\mathrm{pivot}}/(MR_{\mathrm{cm}})\) from the pivot (\(R_{\mathrm{cm}}\) = pivot–CM distance), produces no pivot reaction — the bat's sweet spot.

Rolling without slipping. \(v_{\mathrm{cm}}=R\omega\), \(a_{\mathrm{cm}}=R\alpha\), \(K=\tfrac12Mv^2+\tfrac12I\omega^2\). Rolling down an incline: \(a=\dfrac{g\sin\theta}{1+I/(MR^2)}\). The contact point is instantaneously at rest, so static friction does no work.

Pendulums as rigid bodies. Physical pendulum \(T=2\pi\sqrt{I/(Mgl)}\) (see oscillations); a reversible (Kater) pendulum with equal periods about two pivots a distance \(l_1+l_2\) apart gives \(T=2\pi\sqrt{(l_1+l_2)/g}\) — a precision route to \(g\) with no moment-of-inertia measurement.

Moments of inertia.

Parallel axis. \(I=I_{\mathrm{cm}}+Md^2\).

Static equilibrium. \(\sum F_x=\sum F_y=0\) and \(\sum\tau=0\) about any point.

Centre of mass & stability. \(x_{\mathrm{cm}}=\sum m_ix_i/M\); stability improves with low COM and wide base; tipping when the line of action of the weight crosses the pivot edge.

Rigid-body dynamics

Angular velocity vector. \(\vv v=\boldsymbol\omega\times\vv r\); angular velocities add vectorially. Finite rotations do not commute and are not vectors; infinitesimal ones do — which is why \(\boldsymbol\omega\) is one. Chasles' theorem: any rigid-body motion = translation of the centre of mass + rotation about it.

Inertia tensor. In general \(\vv L\) is not parallel to \(\boldsymbol\omega\) (a rotating skew rod's \(\vv L\) sweeps a cone, requiring torque even at constant \(\omega\)): \(L_i=\sum_jI_{ij}\omega_j\) with \(I_{xx}=\sum m(y^2+z^2)\), \(I_{xy}=-\sum mxy\), etc. — symmetric, so diagonalizable.

Principal axes. Every body has three orthogonal axes with \(\vv L=I_1\omega_1\uvec e_1+I_2\omega_2\uvec e_2+I_3\omega_3\uvec e_3\); \(K_{\mathrm{rot}}=\tfrac12(I_1\omega_1^2+I_2\omega_2^2+I_3\omega_3^2)=\tfrac12\boldsymbol\omega\cdot\vv L\). Spin about a principal axis makes \(\vv L\parallel\boldsymbol\omega\) — no wobble, no bearing loads.

Gyroscope. Torque changes the direction of a large spin angular momentum: uniform precession at \[\Omega=\frac{\tau}{L_s}=\frac{Mgl}{I_s\omega_s},\] independent of the tilt angle (valid for \(\omega_s\gg\Omega\)). A gyroscope released from rest dips and nutates — a fast small oscillation (frequency \(\approx I_s\omega_s/I_\perp\)) superposed on the mean precession; uniform precession is the special initial condition without it.

Applications. Precession of the equinoxes: sun and moon torque Earth's equatorial bulge, precessing the spin axis with a 26 000-yr period (moon's contribution ≈ twice the sun's). Gyrocompass: a spinning rotor on a rotating platform swings its spin axis into alignment with the platform's rotation axis — on Earth, toward true north.

Torque-free precession. A symmetric body spun slightly off its symmetry axis wobbles about the fixed \(\vv L\): the axis circles at \(\approx(I_3/I_\perp)\,\omega_3\) in space (a tossed thin coin, \(I_3=2I_\perp\), wobbles twice as fast as it spins), while an observer on the body sees the slower rate \(\Omega_b=\dfrac{I_3-I_\perp}{I_\perp}\,\omega_3\). For Earth \((I_3-I_\perp)/I_\perp\approx1/305\) predicts a ~300-day polar wobble (observed ≈ 430 d — Earth is elastic).

Euler's equations. In principal axes rotating with the body: \[I_1\dot\omega_1+(I_3-I_2)\,\omega_2\omega_3=\tau_1\quad\text{(and cyclic)}.\] Consequence: free rotation about the axes of largest or smallest \(I\) is stable, about the intermediate axis unstable — the tennis-racket flip.

Noninertial frames

Galilean invariance. Frames in uniform relative motion see the same accelerations and forces: \(\vv r'=\vv r-\vv Vt\) leaves \(\vv a\), and Newton's laws, unchanged.

Accelerating frames. In a frame accelerating at \(\vv A\), Newton's laws hold if every mass feels a fictitious force \(\vv F_{\mathrm{fict}}=-m\vv A\) — uniform, proportional to mass, indistinguishable from a gravitational field \(\vv g_{\mathrm{eff}}=\vv g-\vv A\). A plumb line or pendulum in an accelerating vehicle settles at \(\tan\theta=A/g\).

Equivalence principle. Gravitational mass = inertial mass (Eötvös, Dicke: to \(10^{-11}\)), so uniform acceleration and uniform gravity are locally indistinguishable and free fall cancels gravity locally. Consequences: light climbing a height \(h\) is red-shifted, \(\Delta\nu/\nu=-gh/c^2\) — clocks lower in a gravitational potential run slower.

Tides. Only nonuniform fields survive free fall: the residual (tidal) field across a freely falling Earth \(\sim 2GM_{\mathrm{moon}}R_e/r^3\), stretching along the Earth–moon line and compressing transversely — hence two tidal bulges and two tides a day. Moon's effect ≈ twice the sun's; aligned (new/full moon) they give spring tides, in quadrature neap tides.

Rotating frames. For any vector, \(\left(\dfrac{\dd\vv B}{\dd t}\right)_{\mathrm{in}}=\left(\dfrac{\dd\vv B}{\dd t}\right)_{\mathrm{rot}}+\boldsymbol\Omega\times\vv B\). Applying it twice: \(\vv a_{\mathrm{in}}=\vv a_{\mathrm{rot}}+2\boldsymbol\Omega\times\vv v_{\mathrm{rot}}+\boldsymbol\Omega\times(\boldsymbol\Omega\times\vv r)+\dot{\boldsymbol\Omega}\times\vv r\), so the fictitious force at constant \(\Omega\) is \[\vv F_{\mathrm{fict}}=-2m\,\boldsymbol\Omega\times\vv v_{\mathrm{rot}}-m\,\boldsymbol\Omega\times(\boldsymbol\Omega\times\vv r):\] the Coriolis force (velocity-dependent, perpendicular to \(\vv v_{\mathrm{rot}}\)) and the centrifugal force (\(m\Omega^2\rho\) outward from the axis).

On the rotating Earth. Apparent \(g\) includes the centrifugal term (\(g'=g-\Omega^2R_e\) at the equator) and sets the plumb direction; a rotating liquid surface is the paraboloid \(z=\Omega^2\rho^2/(2g)\). Horizontal motion feels \(F=2m\Omega v\sin\lambda\) deflecting right in the northern hemisphere — air spirals counterclockwise into lows (N hemisphere), and a mass dropped from height deflects east by \(\tfrac13g\Omega t^3\cos\lambda\). The Foucault pendulum's plane precesses with period \(24\,\mathrm{h}/\sin\lambda\).