The thermal capacities of two bodies A and B are in the ratio 1:4. if the rate of heat loss are equal for these two bodies then the rate of fall of the temperature will be in the ratio

Options

A. `1:1`
B. `1:4`
C. `4:1`
D. `(4)^(1//4):1`

Answer

To solve the problem, we need to analyze the relationship between the thermal capacities of two bodies and their rates of temperature change. Here’s a step-by-step solution:

### Step 1: Understand the given information
The thermal capacities of two bodies A and B are in the ratio 1:4. This means if we denote the thermal capacity of body A as \( C_A \) and that of body B as \( C_B \), we can write:
\[ C_A : C_B = 1 : 4 \]
This implies:
\[ C_A = k \quad \text{and} \quad C_B = 4k \]
for some constant \( k \).

### Step 2: Relate heat loss to temperature change
According to the principle of heat transfer, the rate of heat loss \( \frac{dQ}{dt} \) for each body can be expressed as:
\[ \frac{dQ}{dt} = -C \frac{dT}{dt} \]
where \( C \) is the thermal capacity and \( \frac{dT}{dt} \) is the rate of fall of temperature.

### Step 3: Set up the equations for both bodies
For body A:
\[ \frac{dQ_A}{dt} = -C_A \frac{dT_A}{dt} \]
For body B:
\[ \frac{dQ_B}{dt} = -C_B \frac{dT_B}{dt} \]
Since the rate of heat loss is equal for both bodies, we can set these equations equal to each other:
\[ -C_A \frac{dT_A}{dt} = -C_B \frac{dT_B}{dt} \]

### Step 4: Substitute the thermal capacities
Substituting the values of \( C_A \) and \( C_B \):
\[ -k \frac{dT_A}{dt} = -4k \frac{dT_B}{dt} \]
We can cancel \( -k \) from both sides (assuming \( k \neq 0 \)):
\[ \frac{dT_A}{dt} = 4 \frac{dT_B}{dt} \]

### Step 5: Find the ratio of the rates of fall of temperature
Rearranging the equation gives:
\[ \frac{dT_A}{dT_B} = 4 \]
This means that the rate of fall of temperature for body A is 4 times that of body B. Therefore, the ratio of the rates of fall of temperature (which we denote as \( R_A : R_B \)) is:
\[ R_A : R_B = 4 : 1 \]

### Final Answer
The rate of fall of the temperature will be in the ratio \( 4 : 1 \).

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