Two metallic spheres S1 and S2 are made of the same material and have identical surface finish. The mass of S1 is three times that to S2. Both the spheres are heated to the same high temperature and placed in the same room having lower temperature but area thermally insulated from each other. The ratio of the initial rate of cooling of S1 to that S2 is ? a. 1/3 b. (1/3)1/3 c. (3)-1/2 d. 31/2

To determine the ratio of the initial rate of cooling of the two metallic spheres, S1 and S2, we can apply Newton's Law of Cooling. This law states that the rate of heat loss of a body is directly proportional to the difference in temperature between the body and its surroundings, provided this difference is small. The formula can be expressed as:

Understanding the Variables

Let’s define the variables involved:

• m1: mass of sphere S1
• m2: mass of sphere S2
• T1: initial temperature of S1
• T2: initial temperature of S2
• T_room: temperature of the room

Given that the mass of S1 is three times that of S2, we can express this relationship as:

• m1 = 3m2

Applying Newton's Law of Cooling

The rate of cooling can be expressed as:

• Rate of cooling ∝ (T_initial - T_room) / m

For sphere S1, the rate of cooling (R1) can be expressed as:

• R1 ∝ (T1 - T_room) / m1

For sphere S2, the rate of cooling (R2) can be expressed as:

• R2 ∝ (T2 - T_room) / m2

Finding the Ratio of Cooling Rates

Now, we can find the ratio of the initial rates of cooling:

• R1 / R2 = [(T1 - T_room) / m1] / [(T2 - T_room) / m2]

Substituting m1 = 3m2 into the equation gives us:

• R1 / R2 = [(T1 - T_room) / (3m2)] / [(T2 - T_room) / m2]

This simplifies to:

• R1 / R2 = (T1 - T_room) / (3(T2 - T_room))

Considering the Temperature Difference

Assuming both spheres are heated to the same initial temperature (T1 = T2), we can denote this common temperature as T_initial. Thus, the equation becomes:

• R1 / R2 = (T_initial - T_room) / (3(T_initial - T_room))

Now, simplifying this further, we find:

• R1 / R2 = 1 / 3

Final Result

Therefore, the ratio of the initial rate of cooling of S1 to that of S2 is:

• 1/3

So, the correct answer is option a. 1/3.