Stress, Strain and the Young Modulus

Section: Deformation of Solids | Syllabus: Cambridge AS Level Physics (9702)

Stress, Strain and the Young Modulus

To compare materials independent of their dimensions, we use stress and strain.

Figure 6.7: Effect of Dimensions

((a) Larger cross-sectional area extends less for same force. (b) Longer wire extends more for same force.)

Definitions

Stress (σ): Force per unit cross-sectional area.
σ = (F/A) (Unit: Pa or N m^-2) Strain (): Extension per unit original length.
= (x/L) (Dimensionless / %) Young Modulus (E): Measure of stiffness. Ratio of stress to strain (within limit of proportionality).
E = (σ/) = (FL/Ax) (Unit: Pa or GPa) Worked Example: Harp String

Question: A nylon string (A = 8.11 × 10^-7 m^2, L = 0.477 m) extends by 6.2 mm under 67 N tension. Calculate the Young Modulus.

Answer

E = (FL/Ax) = 67 × 0.477(6.2 × 10^-3) × (8.11 × 10^-7) = 6.4 × 10^9 Pa (6.4 GPa).

Experimental Skills 6.2: Finding Young Modulus

Using a long suspended wire to measure small extensions.

Figure 6.8 & 6.10: Young Modulus Apparatus

(Show wire clamped over a pulley with a marker and scale, OR Searle's apparatus with reference wire and vernier scale. Label: Micrometer, Test Wire, Load.)

Measure diameter with a micrometer (multiple readings/orientations) to find Area.
Measure original length with a rule.
Measure extension (using marker or vernier scale) for various loads.
Plot Force vs Extension (or Stress vs Strain). Gradient helps find E.

Experimental Skills 6.3: Oscillating Rod

Another method involves measuring the period of oscillation (T) of a loaded rod. T^2 ∝ 1/E.

Figure 6.11: Oscillating Rod

(Cantilever rod clamped at one end, mass at other end oscillating vertically.)

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Stress, Strain and the Young Modulus