Question

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

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

Answer 1

The correct option is B $\frac{F}{A\alpha \Delta T}$

$\begin{array}{c}\alpha =\text{coefficent of thermal expansion}\\ \text{A a srea of cross section}\\ \Delta T=\text{temperature increase}\\ \text{tength doesnot chonge}\end{array}$

$\begin{array}{c}\to \text{Which mean, when we increase temperature there.will be}\\ \text{expansion and due to Force applied there will be}\\ \text{compression, as length remaing same which mean}\\ \text{(increase in length)}=\text{(decrease in length)}\\ \text{due to rise in temp.}\end{array}$

$\begin{array}{c}(\Delta L{)}_{\text{due to temp}}=L\alpha \Delta T\\ \text{Now}\\ \begin{array}{c}y=\frac{FL}{A\Delta L}\\ \\ \\ \left(\Delta L\right)\text{due to Force}\end{array}=\frac{FL}{yA}\end{array}$

$\begin{array}{c}\text{(i)}=\left(i\right)\\ \begin{array}{c}L\alpha \Delta T=\frac{FL}{yA}\\ y=\frac{FK}{K\alpha \Delta TA}\end{array}\end{array}$

$y=\frac{F}{\alpha \Delta TA}$