Question

The two bars below are made of the same metal. Bar 2 is half as long as bar 1. Each is heated so that its temperature goes up by 100 degrees Celsius.
How do their changes in length compare?

• A. The bars change length by the same amount.
• B. Bar 1 changes length by twice as much as bar 2
• C. Bar 2 changes length by twice as much as bar 1.
• D. bar 1 changes length by four times as much as bar 1.
• E. bar 1 changes length by just slightly more than bar 1 (much less than twice as much).

Answer 1

The correct option is B Bar 1 changes length by twice as much as bar 2

Coefficient of thermal expansion (linear) $\alpha$ is given by ,

$\alpha =\Delta l/\Delta t\times l$ ,

where $l=$ initial length ,

$\Delta l=$ change in length ,

$\Delta t=$ change in temperature ,

given that $\alpha$ is same for both bars because they are made of same material, and $\Delta t={100}^{o}C$ is also same for both the bars ,

for bar 1 , $\alpha =\Delta l_1/\Delta t\times 2l$ , ..............eq1

for bar 2 , $\alpha =\Delta l_2/\Delta t\times l$ , ..............eq2

therefore , by dividing eq1 by eq2 , we get

$\Delta l_1/\Delta l_2=2$ ,

or $\Delta l_1=2\Delta l_2$ ,

change in length of bar 1 will be twice of bar 2 .