A 3mathrm~m$3\mathrm{~m}$ long wire of radius 3mathrm~mm$3\mathrm{~mm}$ shows an extension of 0.1mathrm~mm$0.1\mathrm{~mm}$ when loaded vertically by a mass of 50mathrm~kg$50\mathrm{~kg}$ in an experiment to determine Young's modulus. The value of Young's modulus of the wire as per this experiment is P times 10^11mathrm~Ncdotmathrmm^-2$P \times 10^{11}\mathrm{~N}\cdot\mathrm{m}^{-2}$, where the value of P$P$ is: (Take g = 3pimathrm~m/s^2$g = 3\pi\mathrm{~m/s}^{2}$)

### Related Formula Y = fractextStresstextStrain = fracF / ADelta L / L = fracmg Lpi r^2 Delta L$$Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F / A}{\Delta L / L} = \frac{mg L}{\pi r^2 \Delta L}$$ where, Y$Y$ = Young's modulus F = mg$F = mg$ = stretching force (load) A = pi r^2$A = \pi r^2$ = cross-sectional area of the wire L$L$ = original length of the wire Delta L$\Delta L$ = extension produced ### Core Logic Given parameters: - Original length of wire, L = 3mathrm~m$L = 3\mathrm{~m}$ - Radius of wire, r = 3mathrm~mm = 3 times 10^-3mathrm~m$r = 3\mathrm{~mm} = 3 \times 10^{-3}\mathrm{~m}$ - Mass loaded, m = 50mathrm~kg$m = 50\mathrm{~kg}$ - Acceleration due to gravity, g = 3pimathrm~m/s^2$g = 3\pi\mathrm{~m/s}^{2}$ - Extension, Delta L = 0.1mathrm~mm = 10^-4mathrm~m$\Delta L = 0.1\mathrm{~mm} = 10^{-4}\mathrm{~m}$ Substitute the values into the Young's modulus formula: Y = frac50 times (3pi) times 3pi times (3 times 10^-3)^2 times 10^-4$$Y = \frac{50 \times (3\pi) \times 3}{\pi \times (3 \times 10^{-3})^2 \times 10^{-4}}$$ ### Step 1: Detailed Computation Simplify the equation: Y = frac450pipi times 9 times 10^-6 times 10^-4$$Y = \frac{450\pi}{\pi \times 9 \times 10^{-6} \times 10^{-4}}$$ Cancelling pi$\pi$ from numerator and denominator: Y = frac4509 times 10^-10 = 50 times 10^10 = 5 times 10^11mathrm~N/m^2$$Y = \frac{450}{9 \times 10^{-10}} = 50 \times 10^{10} = 5 \times 10^{11}\mathrm{~N/m}^{2}$$ Comparing this with P times 10^11mathrm~N/m^2$P \times 10^{11}\mathrm{~N/m}^{2}$: P = 5$P = 5$ ### Pattern Recognition Sees: Standard Young's modulus measurement. Trap: Don't forget that the radius is in millimeters and the extension is in millimeters. Convert all units to SI systematically first. Calculation Tip: The variable g = 3pi$g = 3\pi$ is chosen deliberately to cancel the pi$\pi$ in the area formula. Keep your eye open for these mathematical shortcuts! ✓ ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Elasticity