Question

The radius of a ring is R and its coefficient of linear expansion is $\alpha $. If the temperature ring increases by$\theta $ then its circumference will increase by
A. $\pi {R^2}\alpha \theta $
B. $2\pi R\theta $
C. $\pi {R^{}}\alpha \dfrac{\theta }{2}$
D. $\pi {R^{}}\alpha \dfrac{\theta }{4}$

Answer 1

Hint – In such questions, we need to remember the basic concept of linear expansion of a rod on change in temperature and then convert the length into the circumference of the circle.
Formula used - $\dfrac{{\Delta L}}{L} = \alpha \Delta T$

Complete step-by-step solution -

The ring is generally metallic so when we heat the ring its diameter increases and its not only about a ring. If you heat a metallic disc its diameter increases and it becomes thick as well. This is the thermal expansion of the ring.
Given,
Radius of ring=R
Coefficient of linear expansion=$\alpha $
Temperature change=$\theta $
We know that, $L = 2\pi R$
$\dfrac{{\Delta L}}{L} = \alpha \Delta T$
$\dfrac{{\Delta L}}{L} = \alpha \Delta T$, this is the formula for linear expansion where $\Delta L\,{\text{and }}\Delta T$ is the change in length and change in temperature. This formula is helpful in most of the problems.
$
  \Delta L = L\alpha \Delta T \\
   = 2\pi R\alpha \Delta T = 2\pi R\theta \\
$
$ = 2\pi R\theta $
Hence the correct answer is $2\pi R\theta $.
Hence, the correct option is B.

Note - In such a type of question we must take care in the application of basic formula of linear expansion and good conceptual knowledge is also required for determining the change in the length of the ring. Doing this will solve your problem and will give you the right answer.