A-level Physics (Advancing Physics)/Stress, Strain & the Young Modulus/Worked Solutions

1. 10N of force are exerted on a wire with cross-sectional area 0.5mm². How much stress is being exerted on the wire?

0.5mm² = 0.5 x (10⁻³)²m² = 0.5 x 10⁻⁶m²

$\sigma ={\frac {10N}{0.5\times 10^{-6}m^{2}}}=20\ 000\ 000{\mbox{ Pa}}=20{\mbox{ MPa}}$

2. Another wire has a tensile strength of 70MPa, and breaks under 100N of force. What is the cross-sectional area of the wire just before breaking?

$70\times 10^{6}={\frac {100}{A}}$

$A={\frac {100}{70\times 10^{6}}}\approx 1.43\times 10^{-6}{\mbox{ m}}^{2}=1.43{\mbox{ }}\mu {\mbox{m}}^{2}$

3. What is the strain on a Twix bar (original length 10 cm) if it is now 12 cm long?

$E={\frac {12-10}{10}}={\frac {2}{10}}=0.2$

4. What is this strain, expressed as a percentage?

0.2 x 100 = 20%

5. 50N are applied to a wire with a radius of 1mm. The wire was 0.7m long, but is now 0.75m long. What is the Young's Modulus for the material the wire is made of?

$Y={\frac {({\frac {50}{\pi \times (1\times 10^{-}3)^{2}}})}{({\frac {0.75-0.7}{0.7}})}}\approx {\frac {16000000}{0.0714}}\approx 224000000{\mbox{ Pa}}=224{\mbox{ MPa}}$

6. Glass, a brittle material, fractures at a strain of 0.004 and a stress of 240 MPa. Sketch the stress-strain graph for glass.

7. (Extra nasty question which you won't ever get in an exam) What is the toughness of glass?

Toughness equals the area under the above graph. Since glass is brittle, we can assume that the gradient of the graph is constant, and, since the graph passes through the origin, it is a triangle. So:

$A_{triangle}={\frac {{\mbox{base }}\times {\mbox{ height}}}{2}}={\mbox{ Toughness }}={\frac {{\mbox{strain }}\times {\mbox{ stress}}}{2}}={\frac {0.004\times 240\times 10^{6}}{2}}=480000{\mbox{ Jm}}^{-3}=480{\mbox{ kJm}}^{-3}$