Mechanical Properties of Solids — JEE Mains Physics

Formulas (2)

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

Reference tables (1)

Watch out for (7)

• Pulled from both ends is not twice the tension
  → Stress, strain and the extension of a wire
• A diameter put in as a radius
  → Stress, strain and the extension of a wire
• Squared units
  → Stress, strain and the extension of a wire
• Forgetting to square the diameter
  → Comparing two wires by ratio
• Same volume moves the length too
  → Comparing two wires by ratio
• Reading a strain–stress slope as Y
  → What Young's modulus depends on, and reading it off a graph
• Thinking a thicker wire has a larger Y
  → What Young's modulus depends on, and reading it off a graph

Formulas (3)

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

Watch out for (6)

• The upper wire carries more than its own block
  → Wires in series and stacked loads
• Series wires share tension, not strain
  → Wires in series and stacked loads
• Testing only the upper wire
  → Breaking stress: the largest load a wire can hold
• An accelerating lift needs more than mg
  → Breaking stress: the largest load a wire can hold
• The full weight for an own-weight stretch
  → Tension from mechanics: own weight, sag, circles and pulleys
• Leaving out mg at the bottom of a vertical circle
  → Tension from mechanics: own weight, sag, circles and pulleys

Formulas (3)

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

Watch out for (6)

• A percentage is not a fraction
  → Bulk modulus and the change in volume
• Volume units
  → Bulk modulus and the change in volume
• The wrong body's bulk modulus
  → Compression at a depth in water
• Adding atmospheric pressure
  → Compression at a depth in water
• The area of the wrong face
  → Shear modulus, Poisson's ratio and the relations between the moduli
• Shear strain is an angle
  → Shear modulus, Poisson's ratio and the relations between the moduli

Formulas (2)

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

Watch out for (4)

• Tension proportional to length
  → Natural length from two loaded lengths
• Swapping the pairs in the l₀ formula
  → Natural length from two loaded lengths
• The load's work is not the energy stored
  → Energy stored in a stretched wire
• Using the lateral strain as the strain
  → Energy stored in a stretched wire