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

Tardigrade Admin · Accepted Answer

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

$12^{\circ} C$

$20^{\circ} C$

$200^{\circ} C$

$100^{\circ} C$

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

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 ∘C. If the length of the rod is l at temperature 0∘C, then the temperature at which the length will be 2l is [Given, ln(2)=0.693]

12∘C

20∘C

200∘C

100∘C

As; dl=αldT ∴l∫2l​ldl​=a0∫T​TdT ln2=a2T2​ ∴T=[aln4​]21​=[3.465×10−52×0.693​]21​=52​×105​=200∘C

$12^{\circ} C$

$20^{\circ} C$

$200^{\circ} C$

$100^{\circ} C$

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