Question

The diameter of a metal ring is $D$ and the coefficient of linear expansion is $\alpha$. If the temperature of the ring is increased by $1^{o}C$, the circumference and the area of the ring will increase by

• A. $\pi D\alpha ,2\pi D\alpha$
• B. $2\pi D\alpha ,\pi D^2\alpha$
• C. $\pi D\alpha ,\frac{\pi D\alpha }{2}$
• D. $\pi D\alpha ,\frac{\pi D^2\alpha }{2}$

Answer 1

The correct option is D $\pi D\alpha ,\frac{\pi D^2\alpha }{2}$

Increase in Diameter, $\Delta D$ $=D\alpha$
So, increase in circumference $=\pi \Delta D$ $=\pi D\alpha$
Also, we note that coefficient of area expansion $=2\alpha$
So change in area $=$ initial area $\times 2\times \alpha$ $=\frac{\pi D^2}{4}$ $\times 2\times \alpha$ = $\frac{\pi D^2\alpha }{2}$