A steel rod has radius of 10 mm and a length of 1m. A 100 KN force stretches it along its length. Calculate the stress?

Question

Accepted Answer

The correct option is (2):

To calculate the stress in the steel rod, we first need to understand the definition of stress. Stress ($\sigma$) is defined as the force ($F$) applied per unit cross-sectional area ($A$) of the material. The formula for stress is given by:
$\sigma = \frac{F}{A}$

Given:
Radius of the steel rod, $r = 10 	ext{ mm} = 10 	imes 10^{-3} 	ext{ m} = 0.01 	ext{ m}$
Force applied, $F = 100 	ext{ KN} = 100 	imes 10^{3} 	ext{ N} = 1 	imes 10^{5} 	ext{ N}$

First, calculate the cross-sectional area ($A$) of the steel rod. Since it's a rod, its cross-section is circular.
$A = \pi r^2$
$A = \pi (0.01 	ext{ m})^2$
$A = \pi (1 	imes 10^{-4} 	ext{ m}^2)$
$A \approx 3.14159 	imes 10^{-4} 	ext{ m}^2$

Now, substitute the values of force and area into the stress formula:
$\sigma = \frac{1 	imes 10^{5} 	ext{ N}}{3.14159 	imes 10^{-4} 	ext{ m}^2}$
$\sigma \approx 0.3183 	imes 10^{9} 	ext{ N/m}^2$
$\sigma \approx 3.183 	imes 10^{8} 	ext{ N/m}^2$

Therefore, the stress in the steel rod is approximately $3.18 	imes 10^{8} 	ext{ Nm}^{-2}$. This value matches option (2). The length of the rod is not required for calculating stress.

The incorrect option is (1): $5	imes10^{4}Nm^{-2}$. This value is significantly lower than the calculated stress and would result from an incorrect calculation of either force or area, or an incorrect formula.
The incorrect option is (3): $7	imes10^{6}Nm^{-2}$. This option also represents a value much smaller than the actual stress, indicating a potential miscalculation of the units or magnitude.
The incorrect option is (4): $8.2	imes10^{6}Nm^{-2}$. Similar to options (1) and (3), this value is considerably different from the correct stress, suggesting errors in the application of the stress formula or unit conversions.