Dynamics & Statics

UPSC Mathematics Optional · Paper I · Complete Formulae

1. Statics — Forces & Equilibrium 2. Moments, Couples, Parallel Forces 3. Friction 4. Centre of Gravity 5. Virtual Work 6. Catenary 7. Rectilinear Motion & SHM 8. Projectile Motion 9. Central Orbits & Kepler's Laws 10. Moment of Inertia 11. Motion of a Rigid Body

§1 · Statics — Forces & Equilibrium

Resultant of Forces PYQ★★

Parallelogram Law: \(R^2=P^2+Q^2+2PQ\cos\theta\), \(\tan\alpha=\dfrac{Q\sin\theta}{P+Q\cos\theta}\).

Lami's Theorem: If three concurrent forces are in equilibrium, \(\dfrac{P}{\sin\alpha}=\dfrac{Q}{\sin\beta}=\dfrac{R}{\sin\gamma}\) where \(\alpha,\beta,\gamma\) are angles opposite to \(P,Q,R\).

Conditions for Equilibrium of Coplanar Forces

\(\sum F_x=0\), \(\sum F_y=0\), \(\sum M_O=0\) (moments about any point \(O\)).

For three forces in equilibrium: they must be concurrent or all parallel.

§2 · Moments, Couples & Parallel Forces

Moment & Couple PYQ★★

Moment of force \(F\) about point \(O\) = \(F\times\perp\text{ distance}\). Couple = two equal, opposite, non-collinear forces. Moment of couple = \(F\times d\) (constant, independent of pivot).

Resultant of Parallel Forces

Two parallel forces \(P\) and \(Q\) (same sense): Resultant \(R=P+Q\), divides \(PQ\) internally in ratio \(Q:P\).

Opposite sense: \(R=P-Q\), divides externally in ratio \(Q:P\).

§3 · Friction

Laws of Friction PYQ★★

Limiting friction \(F=\mu R\) where \(\mu\) = coefficient of friction, \(R\) = normal reaction.
Angle of friction \(\lambda\): \(\tan\lambda=\mu\). Cone of friction: semi-angle \(\lambda\).
On an inclined plane (angle \(\alpha\)): body on verge of slipping up if \(\alpha > \lambda\).
For body on inclined plane: min force to pull up = \(W\sin(\alpha+\lambda)/\cos(\theta-\lambda)\) where \(\theta\) is inclination of force.

§4 · Centre of Gravity

C.G. Formulae PYQ★★★

\(\bar x=\dfrac{\sum m_i x_i}{\sum m_i}\), \(\bar y=\dfrac{\sum m_i y_i}{\sum m_i}\).

Continuous: \(\bar x=\dfrac{\int x\,dm}{\int dm}\).

Solid hemisphere radius \(a\): \(\bar y=\dfrac{3a}{8}\) from centre.
Hemispherical shell: \(\bar y=\dfrac{a}{2}\).
Solid cone height \(h\): \(\bar y=\dfrac{h}{4}\) from base.
Arc of circle radius \(r\), angle \(2\alpha\): \(\bar x=\dfrac{r\sin\alpha}{\alpha}\).
Sector of circle: \(\bar x=\dfrac{2r\sin\alpha}{3\alpha}\).

§5 · Principle of Virtual Work

Principle of Virtual Work PYQ★★★

A system is in equilibrium iff the total virtual work done by all active forces in any virtual displacement is zero: \(\delta W = \sum F_i\cdot\delta r_i = 0\).

Powerful for systems with constraints — no need to find reactions.

§6 · Catenary

Common Catenary PYQ★★★

Equation: \(y=c\cosh\dfrac{x}{c}\) (vertex at \((0,c)\) above lowest point).

Arc length from vertex: \(s=c\sinh\dfrac{x}{c}\)
Tension at any point: \(T=wy=wc\cosh(x/c)\) where \(w\) = weight per unit length.
Tension at lowest point \(T_0=wc\).
Relation: \(y^2=s^2+c^2\), \(T^2=T_0^2+(ws)^2\).
Inclination \(\psi\) at arc length \(s\): \(\tan\psi=s/c\), \(\sin\psi=s/y\).
Intrinsic equation: \(s=c\tan\psi\).

Uniform Chain Hanging Under Gravity — Key Results

Span \(2l\), sag \(d\), parameter \(c\): \(l=c\sinh(l/c)\approx\) ... (solve numerically or by approximation).

For nearly taut chain (\(d\ll l\)): \(c\approx\dfrac{l^2}{2d}\), \(T_0=\dfrac{wl^2}{2d}\).

§7 · Rectilinear Motion & SHM

Equations of Motion PYQ★★

Use \(v\dfrac{dv}{dx}=\dfrac{d^2x}{dt^2}=f\) (acceleration). \(v=\dfrac{dx}{dt}\). Energy: \(\frac{1}{2}mv^2+V=E\).

Simple Harmonic Motion (SHM) PYQ★★★

\(\ddot{x}+\omega^2 x=0\). Solution: \(x=a\cos(\omega t+\phi)\).

Amplitude \(a\), period \(T=\dfrac{2\pi}{\omega}\), frequency \(n=\dfrac{\omega}{2\pi}\).
\(v^2=\omega^2(a^2-x^2)\). Max velocity \(=a\omega\) at \(x=0\). Max acceleration \(=a\omega^2\) at \(x=\pm a\).
Elastic string SHM: \(\omega^2=\lambda/(ma)\) where \(\lambda\)=modulus, \(a\)=nat. length.

Damped & Forced Oscillations

\(\ddot{x}+2k\dot{x}+\omega^2 x=F_0\cos pt\).

Underdamped (\(k<\omega\)): \(x=e^{-kt}(A\cos\mu t+B\sin\mu t)\), \(\mu=\sqrt{\omega^2-k^2}\).
Overdamped (\(k>\omega\)): exponential decay, no oscillation.
Resonance: when \(p=\omega\), amplitude grows.

§8 · Projectile Motion

Projectile Formulae PYQ★★★

Initial velocity \(u\), angle \(\alpha\) to horizontal. \(x=u\cos\alpha\cdot t\), \(y=u\sin\alpha\cdot t-\frac{1}{2}gt^2\).

Trajectory: \(y=x\tan\alpha-\dfrac{gx^2}{2u^2\cos^2\alpha}\) (parabola).
Time of flight: \(T=\dfrac{2u\sin\alpha}{g}\).
Range: \(R=\dfrac{u^2\sin 2\alpha}{g}\). Max range at \(\alpha=45°\): \(R_{max}=\dfrac{u^2}{g}\).
Max height: \(H=\dfrac{u^2\sin^2\alpha}{2g}\).
Velocity at time \(t\): \(v=\sqrt{(u\cos\alpha)^2+(u\sin\alpha-gt)^2}\).
Range on inclined plane (angle \(\beta\)): \(R=\dfrac{2u^2\sin(\alpha-\beta)\cos\alpha}{g\cos^2\beta}\).
Max range on incline: \(R_{max}=\dfrac{u^2}{g(1+\sin\beta)}\) at \(\alpha=\pi/4+\beta/2\).

§9 · Central Orbits & Kepler's Laws

Central Force Motion PYQ★★★

In polar coords with force \(F\) towards pole:

Angular momentum conservation (Areal velocity): \(h=r^2\dot\theta=\text{const}\). (Kepler II: equal areas in equal times.)

Pedal equation of orbit: \(F = \dfrac{h^2}{p^3}\dfrac{dp}{dr}\cdot\dfrac{1}{r}\cdot\dfrac{p}{r}\)... use \(u=1/r\):

\[F = \frac{h^2 u^2}{m}\left(u+\frac{d^2u}{d\theta^2}\right)\] (Binet's equation)

Kepler's Laws PYQ★★★

I: Planets move in ellipses with sun at one focus.
II: Areal velocity is constant (equal areas, equal times). Consequence of central force.
III: \(T^2\propto a^3\) (square of period ∝ cube of semi-major axis).

For inverse-square law \(F=\mu/r^2\): orbit is a conic. Ellipse if \(E<0\), parabola if \(E=0\), hyperbola if \(E>0\).

Orbit Formulae

Orbit \(\dfrac{l}{r}=1+e\cos\theta\) (focus-directrix). Semi-latus rectum \(l=h^2/\mu\). Eccentricity \(e=\sqrt{1+2Eh^2/\mu^2}\).

Velocity: \(v^2=\mu\left(\dfrac{2}{r}-\dfrac{1}{a}\right)\) (vis-viva equation).

§10 · Moment of Inertia

Key Moments of Inertia PYQ★★★

Thin rod, length \(2l\), mass \(M\): about centre \(=\dfrac{Ml^2}{3}\); about end \(=\dfrac{4Ml^2}{3}\).
Rectangular plate \(a\times b\): about axis \(\parallel a\) through centre \(=\dfrac{Mb^2}{12}\).
Solid sphere radius \(R\): about diameter \(=\dfrac{2MR^2}{5}\).
Spherical shell radius \(R\): about diameter \(=\dfrac{2MR^2}{3}\).
Disc radius \(R\): about axis \(=\dfrac{MR^2}{2}\); about diameter \(=\dfrac{MR^2}{4}\).
Solid cylinder radius \(R\): about axis \(=\dfrac{MR^2}{2}\).
Hoop radius \(R\): about axis \(=MR^2\); about diameter \(=\dfrac{MR^2}{2}\).

Parallel & Perpendicular Axes Theorems PYQ★★★

Parallel axis: \(I = I_G + Md^2\) where \(I_G\) is MOI about axis through C.G., \(d\) = distance between axes.

Perpendicular axis (lamina only): \(I_z=I_x+I_y\) where \(z\perp\) plane of lamina.

§11 · Motion of a Rigid Body

Rotation About Fixed Axis PYQ★★

\(\tau = I\alpha\) where \(\tau\)=torque, \(I\)=MOI about axis, \(\alpha=\ddot\theta\).

KE of rotation \(=\dfrac{1}{2}I\omega^2\). Angular momentum \(L=I\omega\).

Conservation of angular momentum: if net torque = 0, \(I\omega=\text{const}\).

Compound Pendulum PYQ★★★

Period: \(T=2\pi\sqrt{\dfrac{I}{Mgh}}\) where \(I\)=MOI about pivot, \(h\)=distance from pivot to C.G.

Equivalent simple pendulum length: \(l=\dfrac{I}{Mh}=\dfrac{k^2}{h}+h\) where \(k\)=radius of gyration about C.G.

Minimum \(T\) when \(h=k\) (pivot at radius of gyration from C.G.). Then \(l_{min}=2k\).