Question

A sphere, a cube and a thin circular plate all of same material and same mass initially heated to same high temperature are allowed to cool down under similar conditions. Then the

• A. plate will cool the fastest and cube the slowest
• B. sphere will cool the fastest and cube the slowest
• C. plate will cool the fastest and sphere the slowest
• D. cube will cool the fastest and plate the slowest

Answer 1

The Correct Option is C

The rate at which an object cools is determined by the surface area and volume.
According to Newton's law of cooling, the rate of heat loss is directly proportional to the surface area and inversely proportional to the volume. 
The sphere has the least surface area for a given volume, which makes it lose heat more slowly.
The plate has the greatest surface area relative to its volume, allowing it to cool down the fastest.
The cube has a moderate surface area and will cool at a rate between the sphere and plate.

Thus, the plate will cool the fastest and the sphere the slowest.

Answer 2

The rate of cooling of an object depends on its surface area and volume. The cooling of a body is governed by Newton's Law of Cooling, which states that the rate of heat loss is proportional to the surface area available for heat dissipation.

- Sphere: For a sphere, the surface area is \(A_{\text{sphere}} = 4 \pi r^2\) and the volume is \(V_{\text{sphere}} = \frac{4}{3} \pi r^3\). The ratio of the surface area to the volume is \( \frac{A_{\text{sphere}}}{V_{\text{sphere}}} = \frac{3}{r} \).

- Cube: For a cube, the surface area is \(A_{\text{cube}} = 6a^2\) and the volume is \(V_{\text{cube}} = a^3\). The ratio of the surface area to the volume is \( \frac{A_{\text{cube}}}{V_{\text{cube}}} = \frac{6}{a} \).

- Plate (Thin Circular Plate): For a thin circular plate, the surface area is \(A_{\text{plate}} = \pi r^2\) and the volume is \(V_{\text{plate}} = \text{thickness} \times \pi r^2\). The ratio of the surface area to the volume is much larger compared to the sphere and the cube because of its thin nature.

Thus, the thin plate has the highest surface area to volume ratio, meaning it will cool the fastest because it can lose heat more efficiently. On the other hand, the sphere has the lowest surface area to volume ratio, so it will cool the slowest.

Therefore, the correct answer is (C) plate will cool the fastest and sphere the slowest.