A hot body is being cooled in air according to Newton's law of cooling, the rate of fall of temperature being K times the difference of its temperature with respect to that of surroundings. The time after which the rate of heat loss of the body will be half of the maximum rate of heat loss is <math> <mfrac> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </mrow> </mfrac> <msub> <mrow> <mrow> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">g</mi> </mrow> </mrow> <mrow> <mrow> <mi>e</mi> </mrow> </mrow> </msub> <mo>?</mo> <mi>n</mi> </math> (The time is to be counted from the instant t = 0 ). The value of n is

Newton's law of cooling, - d T d t = k T - T 0 ⇒ - ∫ d T T - T 0 = k ∫ d t ⇒ - l n ? T - T 0 T T + T 0 / 2 = k t ? Heat ∝ T - T 0 ⇒ - l n ? T - T 0 / 2 T - T 0 = k t ⇒ - l n ? 1 2 = k t ⇒ t = l n ? 2 k = l n ? 4 2 k ∴ n = 4

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Physics NCERT Exemplar Solutions Class 11th Chapter Fifteen Physics Class 11th Physics Thermodynamics

A hot body is being cooled in air according to Newton's law of cooling, the rate of fall of temperature being K times the difference of its temperature with respect to that of surroundings. The time after which the rate of heat loss of the body will be half of the maximum rate of heat loss is $\frac{1}{2 k} {l o g}_{e} ? n$ (The time is to be counted from the instant t = 0 ). The value of n is

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Raj Pandey | Contributor-Level 9

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Newton's law of cooling, $\begin{array}{r} - \frac{d T}{d t} = k (T - T_{0}) \Rightarrow - \int \frac{d T}{T - T_{0}} = k \int d t \\ \Rightarrow {[- l n ? |T - T_{0}|]}_{T}^{(T + T_{0}) / 2} = k t (? Heat \propto (T - T_{0})) \\ \Rightarrow - l n ? |\frac{(T - T_{0}) / 2}{T - T_{0}}| = k t \Rightarrow - l n ? |\frac{1}{2}| = k t \Rightarrow t = \frac{l n ? 2}{k} = \frac{l n ? 4}{2 k} ∴ n = 4 \end{array}$

Newton's law of cooling, <math> <mtable columnalign="right"> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow> <mi> </mi> <mo>-</mo> <mfrac> <mrow> <mrow> <mi>d</mi> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mrow> </mfrac> <mo>=</mo> <mi>k</mi> <mfenced separators="|"> <mrow> <mrow> <mi>T</mi> <mo>-</mo> <msub> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msub> </mrow> </mrow> </mfenced> <mo>⇒</mo> <mo>-</mo> <mo>∫</mo> <mrow> <mrow> <mi> </mi> </mrow> </mrow> <mfrac> <mrow> <mrow> <mi>d</mi> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mi>T</mi> <mo>-</mo> <msub> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msub> </mrow> </mrow> </mfrac> <mo>=</mo> <mi>k</mi> <mo>∫</mo> <mrow> <mrow> <mi> </mi> </mrow> </mrow> <mi>d</mi> <mi>t</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow> <mi> </mi> <mo>⇒</mo> <mrow> <mi mathvariant="normal"> </mi> </mrow> <msubsup> <mrow> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <mrow> <mo>-</mo> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">n</mi> <mo>?</mo> <mfenced open="|" close="|" separators="|"> <mrow> <mrow> <mi>T</mi> <mo>-</mo> <msub> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msub> </mrow> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mfenced separators="|"> <mrow> <mrow> <mi>T</mi> <mo>+</mo> <msub> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msub> </mrow> </mrow> </mfenced> <mo>/</mo> <mn>2</mn> </mrow> </mrow> </msubsup> <mo>=</mo> <mi>k</mi> <mi>t</mi> <mrow> <mi mathvariant="normal"> </mi> </mrow> <mfenced separators="|"> <mrow> <mrow> <mo>?</mo> <mtext> </mtext> <mtext>Heat</mtext> <mtext> </mtext> <mo>∝</mo> <mfenced separators="|"> <mrow> <mrow> <mi>T</mi> <mo>-</mo> <msub> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msub> </mrow> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow> <mi> </mi> <mo>⇒</mo> <mrow> <mi mathvariant="normal"> </mi> </mrow> <mo>-</mo> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">n</mi> <mo>?</mo> <mfenced open="|" close="|" separators="|"> <mrow> <mrow> <mfrac> <mrow> <mrow> <mfenced separators="|"> <mrow> <mrow> <mi>T</mi> <mo>-</mo> <msub> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msub> </mrow> </mrow> </mfenced> <mo>/</mo> <mn>2</mn> </mrow> </mrow> <mrow> <mrow> <mi>T</mi> <mo>-</mo> <msub> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msub> </mrow> </mrow> </mfrac> </mrow> </mrow> </mfenced> <mo>=</mo> <mi>k</mi> <mi>t</mi> <mrow> <mi mathvariant="normal"> </mi> </mrow> <mo>⇒</mo> <mrow> <mi mathvariant="normal"> </mi> </mrow> <mo>-</mo> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">n</mi> <mo>?</mo> <mfenced open="|" close="|" separators="|"> <mrow> <mrow> <mfrac> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> </mrow> </mrow> </mfenced> <mo>=</mo> <mi>k</mi> <mi>t</mi> <mrow> <mi mathvariant="normal"> </mi> </mrow> <mo>⇒</mo> <mrow> <mi mathvariant="normal"> </mi> </mrow> <mi>t</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">n</mi> <mo>?</mo> <mn>2</mn> </mrow> </mrow> <mrow> <mrow> <mi>k</mi> </mrow> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mrow> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">n</mi> <mo>?</mo> <mn>4</mn> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </mrow> </mfrac> <mrow> <mi mathvariant="normal"> </mi> </mrow> <mo>∴</mo> <mrow> <mi mathvariant="normal"> </mi> </mrow> <mi>n</mi> <mo>=</mo> <mn>4</mn> </mrow> </mtd> </mtr> </mtable> </math>

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