The co-efficient of thermal expansion of a rod is temperature dependent and is given by the formula α=a T, where a is a positive constant 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:

Question

The co-efficient of thermal expansion of a rod is temperature dependent and is given by the formula α=a T, where a is a positive constant 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: (A) √( ln 2/α) (B) √( ln 4/a) (C) (1/α) (D) (2/α)

Tardigrade Admin · Accepted Answer

As d l =α l d T ∴ ∫ limitsl2 l (d l/l)=a ∫ limits0T T d T ln 2=a (T2/2) ∴ T=[( ln 4/a)]1 / 2

Q. The co-efficient of thermal expansion of a rod is temperature dependent and is given by the formula $α = a T$ , where $a$ is a positive constant and $T$ in $^{\circ} C$ . If the length of the rod is $l$ at temperature $0^{\circ} C$ , then the temperature at which the length will be $2 l$ is:

$\frac{l n 2}{α}$

$\frac{l n 4}{a}$

$\frac{1}{α}$

$\frac{2}{α}$

As $d l = α l d T$
$∴ l \int 2 l \frac{d l}{l} = a 0 \int T T d T$
$ln 2 = a \frac{T ^{2}}{2}$
$∴ T = [\frac{l n 4}{a}]^{1/2}$

Q. The co-efficient of thermal expansion of a rod is temperature dependent and is given by the formula α=aT, where a is a positive constant 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:

αln2​​

aln4​​

α1​

α2​