Question

A metallic sphere cools from $50°\mathrm{C}$ to $40°\mathrm{C}$ in $300\mathrm{s}$. If atmospheric temperature around is $20°\mathrm{C}$, then the sphere’s temperature after the next $5\mathrm{minutes}$ will be close to :

• A. $35°\mathrm{C}$
• B. $31°\mathrm{C}$
• C. $33°\mathrm{C}$
• D. $28°\mathrm{C}$

Answer 1

The correct option is C

$33°\mathrm{C}$

Explanation for the correct option:

C) $\mathbf{33}\mathbf{°}\mathbf{C}$

• The temperature of a metallic sphere can be calculated by Newton's law of cooling according to which the rate of loss of heat from a body is directly proportional to the difference in temperature of the body and its surroundings.
• The law can be represented as follows: $\frac{T_1-T_2}{t_1-t_2}=k\left(\frac{T_1+T_2}{2}-T_{surr}\right)$ where $T_1$and $T_2$ are initial and final temperature, $t_1$ and $t_2$ are initial and final temperature, $k$ is rate constant and $T_{surr}$ is the temperature of surroundings of the sphere.
• Now, let's put the given values in the equation:$$\begin{gathered} \frac{50-40}{300}=k\left(\frac{50+40}{2}-20\right) \\ \frac{10}{300}=k\left(45-20\right)\left(1\right) \end{gathered}$$
• Another equation for determining the temperature after $5\mathrm{minutes}$is: $\frac{40-T_2}{300}=k\left(\frac{40+T_2}{2}-20\right)\left(2\right)$
• To determine sphere's temperature after $5\mathrm{minutes}$ dividing equation $\left(1\right)$ by $\left(2\right)$: $$\begin{gathered} \frac{10}{40-T_2}=\frac{25}{\left({\displaystyle \frac{40+T_2}{2}}-20\right)} \\ \frac{10}{40-T_2}=\frac{25}{T_2}\times 2 \\ {T}_2=\frac{100}{3} \\ {T}_2=33°C \end{gathered}$$

Explanation for the incorrect options:

A) $\mathbf{35}\mathbf{°}\mathbf{C}$

• As it can be observed from the above calculation that Newton's law of cooling is able to determine the temperature of the sphere as $33°\mathrm{C}$.
• Hence this value is incorrect.

B) $\mathbf{31}\mathbf{°}\mathbf{C}$

• As it can be observed from the above calculation that Newton's law of cooling is able to determine the temperature of the sphere as $33°\mathrm{C}$.
• Hence this value is incorrect.

D) $\mathbf{28}\mathbf{°}\mathbf{C}$

• As it can be observed from the above calculation that Newton's law of cooling is able to determine the temperature of the sphere as $33°\mathrm{C}$.
• Hence this value is incorrect.

Hence option C is correct, the sphere's temperature after $5\mathrm{minutes}$ is $33°\mathrm{C}$.