Askiitians Tutor Team
Last Activity: 7 Days ago
To tackle this question, we need to consider the principles of heat transfer, specifically Newton's Law of Cooling. This law states that the rate of heat loss of a body is proportional to the difference in temperature between the body and its surroundings, provided this temperature difference is small. In this scenario, we have two metallic spheres, S1 and S2, with S1 being three times the mass of S2. Let's break down the problem step by step.
Understanding the Variables
We have two spheres:
- S1: Mass = 3m
- S2: Mass = m
Both spheres are made of the same material and have identical surface finishes, which means they will have the same specific heat capacity and surface area for heat transfer. Let's denote the initial temperature of both spheres as T_initial and the ambient temperature as T_ambient.
Applying Newton's Law of Cooling
According to Newton's Law of Cooling, the rate of cooling (dT/dt) can be expressed as:
dT/dt = -k(T - T_ambient)
Where:
- dT/dt: Rate of temperature change
- k: Cooling constant (depends on the material and surface area)
- T: Temperature of the object
- T_ambient: Ambient temperature
Rate of Cooling for Each Sphere
For both spheres, the cooling rate can be expressed as:
- For S1: dT1/dt = -k1(T_initial - T_ambient)
- For S2: dT2/dt = -k2(T_initial - T_ambient)
Since both spheres are identical in material and surface finish, we can assume that the cooling constants k1 and k2 are the same. Therefore, we can simplify our equations to:
- dT1/dt = -k(T_initial - T_ambient)
- dT2/dt = -k(T_initial - T_ambient)
Considering Mass and Heat Capacity
The rate of cooling is also influenced by the mass of the objects. The larger the mass, the more heat energy it can store, which affects how quickly it cools down. The specific heat capacity (c) is defined as the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius.
For S1, the heat capacity is 3mc, and for S2, it is mc. This means that S1 can store three times more heat energy than S2. Therefore, when both spheres are subjected to the same cooling conditions, S1 will cool down more slowly than S2.
Calculating the Ratio of Cooling Rates
To find the ratio of the initial rates of cooling, we can express it as:
Ratio = (Rate of cooling of S1) / (Rate of cooling of S2)
Since the rate of cooling is inversely proportional to the mass (or heat capacity), we can express this ratio as:
Ratio = (1 / (3mc)) / (1 / (mc)) = 1/3
Final Answer
The ratio of the initial rate of cooling of S1 to that of S2 is therefore 1/3. This means that S1 cools at a rate that is one-third that of S2, due to its greater mass and heat capacity. Thus, the correct answer is option a. 1/3.
