Animated Solution for Physics - Properties of Solids and Liquids: A metallic sphere cools from $50^{\circ }\text{C}$ to $40^{\circ }\text{C}$ in $300\text{ s}$. If atmospheric temperature around is $20^{\circ }\text{C}$, then the sphere's temperature after the next $5\text{ min}$ will be close to

Imagine a hot metallic sphere sitting in a room. The room is at a comfortable $20^{\circ }\text{C}$, but the sphere is initially at $50^{\circ }\text{C}$. Because nature loves equilibrium, the sphere starts losing heat to the surrounding air.

According to Newton's Law of Cooling, the rate at which this temperature drops isn't constant. It is directly proportional to the temperature difference between the sphere and the room. When the sphere is very hot, it cools rapidly. As it gets closer to room temperature, the cooling slows down.

For small temperature drops, we can avoid complex exponential calculus by using the average form of Newton's Law of Cooling:

Here, $T_1$ and $T_2$ are the initial and final temperatures of the interval, $\Delta t$ is the time taken, $T_s$ is the surrounding temperature, and $k$ is the cooling constant specific to this sphere.

Finding the Cooling Constant

Let's look at the first $5\text{ minutes}$ (which is $300\text{ seconds}$). The sphere cools from $50^{\circ }\text{C}$ to $40^{\circ }\text{C}$.

Let's plug these values into our master equation:

Simplifying the left side, the temperature dropped by $10^{\circ }\text{C}$ over $5\text{ minutes}$, giving a rate of $2^{\circ }\text{C}$ per minute. On the right side, the average temperature of the sphere during this time was $45^{\circ }\text{C}$. Subtracting the room temperature ($20^{\circ }\text{C}$) gives us $25^{\circ }\text{C}$.

Predicting the Future

Now, the question asks for the temperature after the next $5\text{ minutes}$. The sphere is now starting at $40^{\circ }\text{C}$ and will cool down to some unknown temperature $T$.

We set up the exact same equation for this new interval, using the $k$ we just found:

Let's simplify the right side. Finding a common denominator inside the bracket:

This beautiful cancellation makes our algebra much cleaner! The equation becomes:

The $2$ in the numerator and denominator cancel out:

Cross-multiplying by $25$:

Looking at our options, the closest value is $33^{\circ }\text{C}$.

This perfectly illustrates the exponential nature of cooling: in the first $5\text{ minutes}$, it dropped by $10^{\circ }\text{C}$, but in the next $5\text{ minutes}$, it only dropped by about $6.67^{\circ }\text{C}$!