Name: Numerical on Elevation in Boiling Point
Uploaded: 2019-07-19
Duration: 7 min 6 s
Description: Did you know that the boiling point of a liquid depends on the vapour pressure? This video shows how the boiling point of a liquid can be elevated by simple...

Question

A metal rod,

A

of

25 cm

length expands by,

0.050 cm

when, its temperature is raised from

0 ° C

to

100 ° C

. Another rod,

B

of a different metal of

40 cm

length expands by,

0.040 cm

for the same rise in temperature. A third rod,

C

of

50 cm

length is made up of pieces of rods,

A

and

B

placed end to end, expands by

0.03 cm

on heating from

0 ° C

to

50 ° C

. Find the lengths of each portion of the composite rod.

Accepted Answer

Given, metal rod, $A$ expands by $0.05 cm$ , when its temperature, $θ$ is raised from $0 ° C to 100 ° C$ .

We know that, $∆ l = l α ∆ θ$ .

here $∆ l =$ Change in length
$l =$ Initial length
$α =$ Coefficient of linear expansion
$∆ θ =$ Change in temperature

For rod, $A$ ,

$0.05 = 25 α_{A} (100 - 0)$

$\Rightarrow α_{A} = \frac{0.05}{2500} = 2 \times 10^{- 5} per ° C$

Metal rod, $B$ expands by $0.04 cm$ , when its temperature is raised from $0 ° C to 100 ° C$ .

$\Rightarrow 0.04 = 40 α_{B} (100 - 0)$

$\Rightarrow α_{B} = \frac{0.04}{4000} = 1 \times 10^{- 5} per ° C$ .

Now, a third rod, $C$ of $50 cm$ length is made up of pieces of rods, $A$ and $B$ .

Let's assume the length of $A$ in rod $C$ as $x$ and the length of $B$ in rod $C$ as $(50 - x) cm$ .

If, $∆ l_{1}$ and $∆ l_{2}$ are the change in lengths of $A$ and $B$ in rod $C$ then, the total change in length of rod $C$ is,

$∆ l = ∆ l_{1} + ∆ l_{2}$

$\Rightarrow 0.03 = x (2 \times 10^{- 5}) (50 - 0) + (50 - x) (10^{- 5}) (50 - 0)$

$\Rightarrow 0.03 = x (10^{- 3}) + (50 - x) (10^{- 5}) (50)$

$\Rightarrow 0.03 = x (10^{- 3}) + (50) (10^{- 5}) (50) - x (0.5 \times 10^{- 3})$

$\Rightarrow 0.03 = x (0.5 \times 10^{- 3}) + 0.025$

$\Rightarrow 5 \times 10^{- 3} = x (0.5 \times 10^{- 3})$

$∴ x = 10 cm$ and $50 - x = (50 - 10) cm = 40 cm .$

Therefore, the length of rod $A$ in the composite rod is, $10 cm$ and the length of rod $B$ in the composite rod is, $40 cm$ .

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