A sphere, a cube and a thin circular plate are all made of the same material, have the same mass and are initially heated to a temperature of $200^{\circ} \mathrm{C}$. Arrange then in the ascending order of rate of cooling. (A) sphere, plate, cube (B) sphere, cube, plate (C) cube, sphere, plate (D) plate, cube, sphere

Short Answer

Expert verified
The ascending order of rate of cooling for a sphere, a cube, and a thin circular plate made of the same material and having the same mass is (B) sphere, cube, plate. This is based on comparing their surface area to volume ratios, which determines the rate of cooling.

Step by step solution

01

Calculate Surface Area to Volume Ratios

Let's find the surface area to volume ratio for each object.
02

Sphere

For a sphere, the surface area \(A_S\) and volume \(V_S\) are given by the formulas: \[A_S = 4\pi r^2 \quad \text{and} \quad V_S = \frac{4}{3}\pi r^3\] The surface area to volume ratio of a sphere is: \[\frac{A_S}{V_S} = \frac{4\pi r^2}{\frac{4}{3}\pi r^3} = \frac{3}{r}\]
03

Cube

For a cube, the surface area \(A_C\) and volume \(V_C\) are given by the formulas: \[A_C = 6s^2 \quad \text{and} \quad V_C = s^3\] The surface area to volume ratio of a cube is: \[\frac{A_C}{V_C} = \frac{6s^2}{s^3} = \frac{6}{s}\]
04

Thin Circular Plate

For a thin circular plate, the area \(A_P\) and volume \(V_P\) are given by the formulas: \[A_P = 2\pi R^2 \quad \text{and} \quad V_P = \pi R^2t\] where 't' is the thickness of the plate. The surface area to volume ratio of a thin circular plate is: \[\frac{A_P}{V_P} = \frac{2\pi R^2}{\pi R^2t} = \frac{2}{t}\] Note that since all objects have the same mass, the sphere, the cube, and the plate have the same material, meaning that we can compare these ratios directly without taking into account their respective densities.
05

Compare the Ratios

Now that we have the surface area to volume ratios for each object, let's compare them: For the sphere: \(\frac{3}{r}\) For the cube: \(\frac{6}{s}\) For the thin circular plate: \(\frac{2}{t}\) Since the mass of all three objects is the same and they are made of the same material, intuitively, the sphere would have the smallest surface area, followed by the cube and then the circular plate. Therefore, the ascending order of rate of cooling would be sphere, cube, and plate. Hence, the correct answer is: (B) sphere, cube, plate

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