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MCQMediumJEE 2023Heat Transfer (Conduction, Convection, Radiation)

JEE Physics 2023 Question with Solution

A body cools in 77 minutes from 60∘C60^\circ \text{C} to 40∘C40^\circ \text{C}. The temperature of the surroundings is 10∘C10^\circ \text{C}. The temperature of the body after the next 77 minutes is:

  • A

    30∘C30^\circ \text{C}

  • B

    34∘C34^\circ \text{C}

  • C

    32∘C32^\circ \text{C}

  • D

    28∘C28^\circ \text{C}

Answer

Correct answer:D

Step-by-step solution

Standard Method

Given: The body cools from 60∘C60^\circ \text{C} to 40∘C40^\circ \text{C} in the first 77 minutes, and the surroundings are at 10∘C10^\circ \text{C}.

Find: The temperature after the next 77 minutes.

Using Newton’s Law of Cooling, the rate of cooling is proportional to the excess temperature above surroundings. For the interval method shown in the solution, use average temperature in each interval:

T−TsΔt=k(T−Ts)\frac{T - T_s}{\Delta t} = k(T - T_s)

where TsT_s is the surrounding temperature.

For the first interval:

60−406=k(60+402−10)\frac{60 - 40}{6} = k\left(\frac{60 + 40}{2} - 10\right)

So,

k=206×50k = \frac{20}{6 \times 50}

For the second interval, let the final temperature be TT:

40−T6=k(40+T2−10)\frac{40 - T}{6} = k\left(\frac{40 + T}{2} - 10\right)

Substituting the value of kk and solving, we get:

T=28∘CT = 28^\circ \text{C}

Therefore, the temperature of the body after the next 77 minutes is 28∘C28^\circ \text{C}. The correct option is D.

Common mistakes

  • Using the initial temperature instead of the average temperature over each interval is incorrect here because the provided solution applies the interval form with mean temperature. Use 60+402\frac{60+40}{2} for the first interval and 40+T2\frac{40+T}{2} for the second interval.

  • Ignoring the surrounding temperature of 10∘C10^\circ \text{C} leads to a wrong proportionality. In Newton’s law of cooling, always work with excess temperature, that is, body temperature minus surrounding temperature.

  • Directly assuming the temperature drop in the next 77 minutes is again 20∘C20^\circ \text{C} is wrong because cooling is not linear in time. The rate decreases as the body temperature approaches the surrounding temperature.

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