A hollow sphere of radius R, a hollow cube of side R and a thin circular plate of radius R, made up of the same material, are all heated to 20°C above room temperature. When left to cool in the room, which of them will reach the room temperature first?

Correct Option

The rate of cooling of an object is governed by Newton's Law of Cooling, which states that the rate of heat loss (dQ/dt) is directly proportional to the surface area (A) of the object and the temperature difference (ΔT) between the object and its surroundings. Simultaneously, the rate of temperature change (dT/dt) is inversely proportional to the object's heat capacity (mass 'm' multiplied by specific heat capacity 'c').

Thus, the rate of cooling (dT/dt) is proportional to the ratio of the object's surface area to its mass (dT/dt ∝ A/m). For objects made of the same material and heated to the same initial temperature, the object with the highest surface area to mass ratio will cool fastest.

Among a hollow sphere of radius R, a hollow cube of side R, and a thin circular plate of radius R, the hollow sphere, in this comparative context, possesses the highest surface area to mass ratio. This allows it to dissipate heat more efficiently to the surroundings.

Therefore, the hollow sphere will lose heat most rapidly and reach room temperature first.

Incorrect Options

• (A) Circular plate: The thin circular plate generally has a lower surface area to mass ratio compared to the hollow sphere under the given conditions. Consequently, its rate of heat loss per unit mass is lower, leading to a slower cooling process.
• (B) Cube: The hollow cube has a lower surface area to mass ratio than the hollow sphere. This results in a less efficient rate of heat dissipation compared to the sphere, causing it to cool more slowly.
• (D) All of them will reach the room temperature at the same time: Since the objects possess different geometric shapes, their respective surface area to mass ratios vary. This variation directly influences their rates of cooling, meaning they will not reach room temperature simultaneously.