Question

If the stress in brass rod is $x$ , then the value of $\dfrac{x} {5}$ is:

Answer 1

Let us first see what stress and strain is-
Stress is known as the force encountered by the object that causes the object to change, while strain is known as the shift in an object’s form as stress is applied.
Here first we have to find the elongation, and then we have to add all the forces. After that we find the strain and at last we find the stress.
Complete step by step solution:
The elongation of the rod is directly proportional to the tensile force and the rod length and inversely proportional to the elasticity module and the cross-sectional area.
An elongation in a strain is given by
$\Delta l = \dfrac{{Fl}} {{AY}} \\\ \Rightarrow F = \dfrac{{AY\,\Delta l}} {l} \\\$

So,
${F_s} = \dfrac{{{A_s}{Y_s}\Delta {l_s}}} {{{l_s}}} \\\ {F_b} = \dfrac{{{A_b}{Y_b}\Delta {l_b}}} {{{l_b}}} \\\$

Now, $\Delta l$ is same for all the rods,
Adding force on all the rods, we get-
${F_b} + {F_s} + {F_b} = F = 5000\,N$

$\dfrac{{{A_b}{Y_b}\Delta {l_b}}} {{{l_b}}} + \dfrac{{{A_s}{Y_s}\Delta {l_s}}} {{{l_s}}} + \dfrac{{{A_b}{Y_b}\Delta {l_b}}} {{{l_b}}} = 5000$ ...... (1)
${A_s} = {A_b} = 1 \times {10^{ - 4}}{m^2} \\\ {I_b} = 0.2m,{I_s} = 0.3m,{Y_b} = 1 \times {10^{11}}N/{m^2};{Y_s} = 2 \times {10^{11}}N/{m^2} \\\$

Putting all the values in equation (1), we get-
$\Delta l = 0.03mm$

So, strain in brass rod is
$\dfrac{{\Delta l}} {{{I_b}}} = \dfrac{{0.03mm}} {{0.2m}} = 1.5 \times {10^{ - 4}}$

As stress = Y× strain
$= {10^{11}} \times 1.5 \times {10^{ - 4}} = 15MPa$

Additional information:
Stress is a force per unit area that works on a rock. If it is necessary to exceed the intensity of the object that is under stress, stress may trigger pressure. Strain is a shape or size change caused by applied forces.
Stresses are of two types-
a. Tensile stress
b. Compressive stress
Strains are also of two types-
a. Tensile strain
b. Compressive strain
Strain is dimensionless.
Modulus of elasticity- the ratio between a stress or force per unit area acting to deform the body and the resulting fractional deformation induced by the stress is represented by each of the elasticity coefficients of a body.

Note:
Here it is important to remember the formula for elongation and strain. The formula for elongation is confusing, so, we have to be careful while solving the question. Also the force on $b$ is added twice.