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
$\begin{aligned} & a)\pi {{R}^{2}}\alpha \theta \\\ & b)2\pi R\alpha \theta \\\ & c)\pi R\alpha \dfrac{\theta }{2} \\\ & d)none \\\ \end{aligned}$

Answer 1

The change in the length of any material due to the change in its temperature can be calculated using the coefficient of linear expansion formula. Find the change in length of the ring due to the change in temperature. Next, the circumference can be written in terms of the length. Thus, we can calculate the change in the circumference easily.

Formula used:
$\Delta L=L\alpha \Delta T$

Complete answer:
Let us assume the change in the length, actual length of the ring as $\Delta L,L$ respectively. Now, we can write the change in the length as
$\Delta L=2\pi r$where r is the radius of the ring initially when there is no temperature change.
If the temperature changes by a value of $\theta$, the change in the length will be,
$\begin{aligned} & \Delta L=L\alpha \Delta T \\\ & \Rightarrow L=2\pi r \\\ & \Rightarrow \Delta L=2\pi r\alpha \Delta T \\\ & \Rightarrow \Delta T=\theta \\\ & \Rightarrow \Delta L=2\pi r\alpha \theta \\\ \end{aligned}$
Therefore, the change in the circumference of the ring if the temperature rises will be
$\Delta L=2\pi r\alpha \theta$.

So, the correct answer is “Option B”.

Additional Information:
Thermal expansion is the tendency of matter to change its shape, area or volume due to change in temperature. It doesn’t include any phase transitions. When a substance is heated, molecules in it start to vibrate and move more creating more distance between them. Substances with high melting point will have lower thermal expansion. Coefficient of thermal expansion measures the fractional change in size per degree change in temperature at constant pressure. There are three types of expansions, linear expansion, areal expansion and volume expansion. Linear expansion is basically the change in the length, areal expansion is the change in the area and the volume expansion is the change in the volume when temperature is increased or decreased.

Note:
The linear expansion is considered if the temperature increases when pressure is constant. Recently, we’ve discovered that the coefficient of linear expansion can be negative too. In the above question, the length of the material is increased due to increase in temperature, therefore it comes under linear expansion.