Mechanical Properties of Solids — JEE Mains Physics Formulas and Traps

Stress, Strain and Young's Modulus

Stress, strain and the extension of a wire

Young's modulus

Y = \frac{F/A}{\Delta L/L} \qquad \Delta L = \frac{FL}{AY}

Comparing two wires by ratio

Two wires

\frac{\Delta L_1}{\Delta L_2} = \frac{F_1}{F_2}\cdot\frac{L_1}{L_2}\cdot\left(\frac{d_2}{d_1}\right)^{2}\cdot\frac{Y_2}{Y_1}

What Young's modulus depends on, and reading it off a graph

Graph (y against x)	Slope	What it gives
Stress against strain	$Y$	The steepest line has the largest Y
Strain against stress	$1/Y$	The shallowest line has the largest Y The axes are swapped from the usual plot, so the order reverses.
Load against extension	$AY/L$	$Y = \text{slope} \times L/A$
Extension against load	$L/(AY)$	$Y = L/(A \times \text{slope})$ A line at 45° has slope 1 only in the units printed on the axes.
Extension/load against length	$1/(AY)$	$Y = 1/(A \times \text{slope})$
Y against the length or radius of the wire	Zero, a flat line	Y does not depend on the wire's size

Write the slope from

\Delta L = FL/(AY)

before reading any number off the axes.

Common traps

Pulled from both ends is not twice the tension

Two people pulling the ends with F each give a tension F, the same as a wall at one end and one person pulling with F. Using 2F doubles the answer.

A diameter put in as a radius

A = πd²/4. Putting the diameter into πr² makes the area four times too large and the extension four times too small.

Squared units

1 mm² is 10⁻⁶ m² and 1 cm² is 10⁻⁴ m². Converting only the length (10⁻³, 10⁻²) is the commonest slip.

Forgetting to square the diameter

A diameter ratio of 2 is an area ratio of 4. Using 2 leaves the answer off by a factor of 2.

Same volume moves the length too

If a wire of fixed volume is made four times thicker in area, it becomes four times shorter. Both changes go into ΔL ∝ FL/A.

Reading a strain–stress slope as Y

With strain on the y-axis the slope is 1/Y. The steepest line there is the softest material, not the stiffest.

Thinking a thicker wire has a larger Y

A thicker wire stretches less under the same load because its stiffness AY/L is larger. Its Y, a property of the material, is unchanged.

Loaded Wires, Combinations and Breaking Stress

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Wires in series and stacked loads

Series wires

\Delta L = T\left(\frac{L_1}{A_1Y_1} + \frac{L_2}{A_2Y_2}\right) \qquad Y_{eq} = \frac{2Y_1Y_2}{Y_1 + Y_2}

Breaking stress: the largest load a wire can hold

Breaking load

T_{max} = \sigma_b A \qquad a_{max} = \frac{\sigma_b A}{m} - g

Tension from mechanics: own weight, sag, circles and pulleys

Tensions to use

\Delta L_{own} = \frac{MgL}{2AY} \qquad L_{max} = \frac{\sigma_b}{\rho g} \qquad 2T\theta = mg

Common traps

The upper wire carries more than its own block

It holds every block and wire below it. Giving it only the block attached to it is the commonest wrong option.

Series wires share tension, not strain

Joined end to end, the wires have the same tension. Their strains and extensions differ unless their areas, lengths and moduli match.

Testing only the upper wire

The upper wire carries more, but it is often thicker. Test every wire against its own limit; the thinner lower wire often breaks first.

An accelerating lift needs more than mg

Going up with acceleration a, the tension is m(g + a). Setting σ_b A = ma forgets the weight.

The full weight for an own-weight stretch

Only the top of the rod carries the full weight; the bottom carries none. The stretch uses Mg/2, so the full weight doubles the answer.

Leaving out mg at the bottom of a vertical circle

At the lowest point the string supports the weight and supplies the centripetal force: T = mg + mv²/r, not mv²/r.

Bulk Modulus, Shear Modulus and Poisson's Ratio

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Bulk modulus and the change in volume

Bulk modulus

B = -\frac{\Delta P}{\Delta V / V} \qquad \Delta V = \frac{\Delta P\,V}{B} \qquad \Delta\rho = \frac{\rho\,\Delta P}{B}

Compression at a depth in water

At a depth h

\frac{\Delta V}{V} = \frac{\rho g h}{B} \qquad h = \frac{B}{\rho g}\cdot\frac{\Delta V}{V}

Shear modulus, Poisson's ratio and the relations between the moduli

Shear and the relations

\eta = \frac{F/A}{x/h} \qquad Y = 2\eta(1 + \sigma) = 3B(1 - 2\sigma)

Common traps

A percentage is not a fraction

A 0.2% fall in volume is ΔV/V = 2 × 10⁻³, not 0.2. Divide by 100 before using the formula.

Volume units

1 litre is 10⁻³ m³, and 1 m³ is 10⁶ cm³ or 10⁹ mm³. An answer asked in mm³ needs the cube of the length conversion.

The wrong body's bulk modulus

A rubber ball taken down is squeezed by the water, but it is the ball that shrinks, so its own B goes in the formula.

Adding atmospheric pressure

The ball already felt atmospheric pressure at the surface. Only the extra pressure ρgh changes its volume.

The area of the wrong face

A is the face the force acts on. A slab pushed on its narrow face has A = side × thickness, and the height is the side, not the thickness.

Shear strain is an angle

The strain is θ = x/h in radians, not the displacement x. Find θ first, then multiply by h if the question asks how far the top moves.

Hooke's Law and Elastic Potential Energy

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Natural length from two loaded lengths

Natural length

T = k(l - l_0) \qquad l_0 = \frac{T_2l_1 - T_1l_2}{T_2 - T_1}

Energy stored in a stretched wire

Elastic energy

u = \tfrac{1}{2}\,(\text{stress})(\text{strain}) = \tfrac{1}{2}Y\varepsilon^{2} \qquad U = \tfrac{1}{2}F\,\Delta L

Common traps

Tension proportional to length

T₁/T₂ = l₁/l₂ is wrong: the tension follows the extension l − l₀. Setting up T = k(l − l₀) for both readings avoids it.

Swapping the pairs in the l₀ formula

Check the formula by putting T₁ = 0: it must give l₀ = l₁. The other arrangement fails this test.

The load's work is not the energy stored

A load that stretches a wire by ΔL loses mgΔL of potential energy, but the wire stores only ½mgΔL. The other half leaves as heat or oscillation.

Using the lateral strain as the strain

u = ½Yε² needs the lengthwise strain. When the question gives the sideways strain, divide it by Poisson's ratio first.

Mechanical Properties of Solids formulas

Stress, Strain and Young's Modulus

Stress, strain and the extension of a wire

Comparing two wires by ratio

What Young's modulus depends on, and reading it off a graph

Common traps

Loaded Wires, Combinations and Breaking Stress

Wires in series and stacked loads

Breaking stress: the largest load a wire can hold

Tension from mechanics: own weight, sag, circles and pulleys

Common traps

Bulk Modulus, Shear Modulus and Poisson's Ratio

Bulk modulus and the change in volume

Compression at a depth in water

Shear modulus, Poisson's ratio and the relations between the moduli

Common traps

Hooke's Law and Elastic Potential Energy

Natural length from two loaded lengths

Energy stored in a stretched wire

Common traps

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