The coefficient of thermal expansion of a rod is temperature-dependent and is given by the formula α =aT , where a is 3.465× 10- 5 ° C- 2 and T in oC . If the length of the rod is l at temperature 0oC , then the temperature at which the length will be 2l is [Given, ln(2)=0.693 ]

Question

The coefficient of thermal expansion of a rod is temperature-dependent and is given by the formula α =aT , where a is 3.465× 10- 5 ° C- 2 and T in oC . If the length of the rod is l at temperature 0oC , then the temperature at which the length will be 2l is [Given, ln(2)=0.693 ] (A) 12 oC  (B) 20 oC (C) 200 oC (D) 100 oC

Tardigrade Admin · Accepted Answer

As; dl=α ldT ∴ displaystyle ∫ l2 l(d l/l)=a displaystyle ∫ 0TTdT ln2=a(T2/2) ∴ T=[(ln 4/a)](1/2)=[(2 × 0.693/3.465 × 10- 5)](1/2)=√(2/5) × 105=200 oC

$12^{o} C$

$20^{o} C$

$200^{o} C$

$100^{o} C$

As; $d l = α l d T$
$∴ \int_{l}^{2 l} \frac{d l}{l} = a \int_{0}^{T} T d T$
$l n 2 = a \frac{T ^{2}}{2}$
$∴ T = [\frac{l n 4}{a}]^{\frac{1}{2}} = [\frac{2 \times 0.693}{3.465 \times 1 0 ^{- 5}}]^{\frac{1}{2}} = \frac{2}{5} \times 1 0^{5} = 200^{o} C$

Q. The coefficient of thermal expansion of a rod is temperature-dependent and is given by the formula α=aT , where a is 3.465×10−5∘C−2 and T in oC . If the length of the rod is l at temperature 0oC , then the temperature at which the length will be 2l is [Given, ln(2)=0.693 ]

12oC

20oC

200oC

100oC

As; dl=αldT ∴∫l2l​ldl​=a∫0T​TdT ln2=a2T2​ ∴T=[aln4​]21​=[3.465×10−52×0.693​]21​=52​×105​=200oC

$12^{o} C$

$20^{o} C$

$200^{o} C$

$100^{o} C$

As; $d l = α l d T$
$∴ \int_{l}^{2 l} \frac{d l}{l} = a \int_{0}^{T} T d T$
$l n 2 = a \frac{T ^{2}}{2}$
$∴ T = [\frac{l n 4}{a}]^{\frac{1}{2}} = [\frac{2 \times 0.693}{3.465 \times 1 0 ^{- 5}}]^{\frac{1}{2}} = \frac{2}{5} \times 1 0^{5} = 200^{o} C$