Question

A rod of certain length is acted by two equal forces as shown in the figure. The coefficient of thermal expansion of the rod is \[\alpha \]and area of cross section is A. When the temperature of the rod is increased by $\Delta T$. The length of the rod does not change. The young’s modulus Y will be.

A. $\dfrac{F}{{2A\alpha \Delta T}} \\ $
B. $\dfrac{F}{{A\alpha \Delta T}} \\ $
C. $\dfrac{{2F}}{{A\alpha \Delta T}} \\ $
D. $\dfrac{F}{{3A\alpha \Delta T}}$

Answer 1

Hint:As we know the young’s modulus is stress divided by strain so start with the relation between the thermal stress and length of the rod. Then try to find out thermal stress and hence we will get the required relation between force applied, area of cross section, increase in temperature coefficient of thermal expansion and young’s modulus.

Formula used:
\[\Delta l=\dfrac{Fl}{AY}\]
Where Y is the young’s modulus of the rod, l is the length of the rod, F is the force on the rod and A is its cross sectional area.
$\Delta l=\alpha l\Delta T$
$\Delta l$ is the change in length or the elongation, l is the original length and $\Delta T$ is the change in temperature

Complete step by step solution:
Here two equal forces F are applied on the rod from the opposite direction which will cause compression in the rod and temperature is increased by $\Delta T$ which will cause expansion in the rod.
Now we have, first change in length caused by forces applied,
$\Delta l = \dfrac{{Fl}}{{Ay}}$...........(equation 1)
Second change in length caused by increase in temperature,
$\Delta l = l\alpha \Delta T$.........(equation 2)

Since the length of the rod does not change this implies change in length caused by both the force and temperature balance each other. Then we can equate the equations 1 and 2.
By equating both the equation, we get;
\[\dfrac{{Fl}}{{Ay}} = l\alpha \Delta T\]
By further solving we get;
$\therefore y = \dfrac{F}{{A\alpha \Delta T}}$

Hence the correct answer is option B.

Note: In case the change in length of the rod is not same in both the cases that is by force applied and increase in temperature then we cannot equate both the equations. Also be careful about the direction in which the force is applied on the rod if this changes the whole equation will get changed.