Virtual Work & Simple Machines Notes

Last reviewed 16 Sept 2026 · 6 min read

Principle of virtual work

A virtual displacement is an imaginary, infinitesimally small displacement consistent with the constraints of a system, applied without changing the forces. The work of real forces through such displacements is virtual work.

FormulaPrinciple of virtual work

A system of rigid bodies with ideal (frictionless) constraints is in equilibrium if and only if the total virtual work done by the active forces is zero for every virtual displacement consistent with the constraints:

δ W = \sum F_{i} δ s_{i} + \sum M_{j} δ θ_{j} = 0

Why it helps: reactions at ideal supports do no work through compatible displacements, so they drop out. It is ideal for linkages, lifting mechanisms and finding a single unknown reaction (release that support, give it a virtual displacement, and write one equation). It is also the basis of the kinematic (mechanism) method of plastic analysis and of the unit load method for deflections.

Worked ExampleReaction of a beam by virtual work

A simply supported beam AB of span 6 m carries 30 kN at 2 m from A. Find $R_{B}$ .

Solution. Release B and lift it by $δ$ (beam rotates about A). The load point rises $δ \times 2/6 = δ /3$ .

$R_{B} δ - 30 \times \frac{δ}{3} = 0$ → $R_{B} = 10 kN$ ✔

Simple machines — definitions

A machine enables a small effort $P$ to overcome a larger load $W$ (or to move a load conveniently).

DefinitionKey quantities

Mechanical advantage $M A = \frac{W}{P}$
Velocity ratio $V R = \frac{distance moved by effort}{distance moved by load} = \frac{y}{x}$ (depends only on geometry)
Output = $W x$ ; Input = $P y$
Efficiency $η = \frac{output}{input} = \frac{W x}{P y} = \frac{M A}{V R}$
Ideal machine ( $η = 100%$ ): ideal effort $P_{i} = W / V R$ ; ideal load $W_{i} = P \times V R$
Effort lost in friction $= P - P_{i}$ ; load lost in friction $= W_{i} - W$

Law of a machine

Experiments show effort varies linearly with load:

P = mW + C

$m$ = slope; $C$ = effort needed to overcome friction with no load.

Maximum mechanical advantage (as $W \to \infty$ ): $M A_{ma x} = \frac{1}{m}$ .
Maximum efficiency: $η_{ma x} = \frac{1}{m V R}$ .

Reversibility and self-locking

A machine is reversible if the load, when the effort is removed, runs the machine backwards; otherwise it is irreversible (self-locking).

FormulaCondition for self-locking

A machine is self-locking when its efficiency is less than 50%. At exactly 50% it is on the border. Screw jacks and worm gears are usually designed to be self-locking so that a raised load stays up.

Proof in brief: if frictional loss when lifting is $F$ (in load-distance terms), reversing requires $W x > F$ ; with $F = P y - W x$ this becomes $2 W x > P y$ → $η > 50%$ for reversibility.

Velocity ratios of common machines

Machine	Velocity ratio
Lever, effort arm $a$ , load arm $b$	$a / b$
First system of pulleys ( $n$ movable pulleys)	$2^{n}$
Second system (block and tackle), $n$ pulleys in both blocks	$n$ (number of rope segments supporting the load)
Third system of pulleys ( $n$ pulleys)	$2^{n} - 1$
Weston differential pulley block (radii $R$ and $r$ of the upper block)	$\frac{2 R}{R - r}$
Wheel and axle (wheel diameter $D$ , axle $d$ )	$D / d$
Differential wheel and axle (axles $d_{1} > d_{2}$ )	$\frac{2 D}{d _{1} - d _{2}}$
Simple screw jack (lever arm $L$ , pitch $p$ )	$\frac{2 π L}{p}$
Differential screw jack (pitches $p_{1} > p_{2}$ )	$\frac{2 π L}{p _{1} - p _{2}}$
Worm and worm wheel (single-start worm, $T$ teeth on wheel, effort wheel radius $R$ , load drum radius $r$ )	$\frac{R T}{r}$
Single purchase winch crab	$\frac{L T _{2}}{r T _{1}}$ ( $L$ handle length, $r$ drum radius, $T_{1}$ pinion teeth, $T_{2}$ spur wheel teeth)
Double purchase winch crab	$\frac{L}{r} \cdot \frac{T _{2}}{T _{1}} \cdot \frac{T _{4}}{T _{3}}$

For a multi-start worm with $n$ starts, divide the VR by $n$ .

Downloaded from civilgyantests.com · Not for resale. Verify every code clause and date against the official source.