Two bodies of specific heats `s_1 and s_2` having same heat capacities are combinated to form a single composite body. Find the specific heat of the composite body.

Options

A. `S_1+S_2`
B. `(S_1+S_2)/(2)`
C. `(2S_1S_2)/(S_1+S_2)`
D. `1/(S_1)+1/(S_2)`

Answer

To find the specific heat of the composite body formed by two bodies with specific heats \( s_1 \) and \( s_2 \) that have the same heat capacities, we can follow these steps:

### Step 1: Understand the concept of heat capacity
Heat capacity (\( C \)) is defined as the amount of heat required to change the temperature of a substance. For a body, it can be expressed as:
\[ C = m \cdot s \]
where \( m \) is the mass of the body and \( s \) is its specific heat.

### Step 2: Write the heat capacities of the two bodies
Let the masses of the two bodies be \( m_1 \) and \( m_2 \). The heat capacities of the two bodies can be written as:
\[ C_1 = m_1 s_1 \quad \text{and} \quad C_2 = m_2 s_2 \]
Since it is given that both bodies have the same heat capacity, we can equate them:
\[ m_1 s_1 = m_2 s_2 \]

### Step 3: Express the mass ratio
From the equation \( m_1 s_1 = m_2 s_2 \), we can express the mass ratio:
\[ \frac{m_1}{m_2} = \frac{s_2}{s_1} \]

### Step 4: Calculate the total heat for a temperature change
Let \( Q_1 \) be the heat required to change the temperature of the first body and \( Q_2 \) for the second body. The total heat \( Q \) for a temperature change \( \Delta T \) can be expressed as:
\[ Q = Q_1 + Q_2 = m_1 s_1 \Delta T + m_2 s_2 \Delta T \]
Factoring out \( \Delta T \):
\[ Q = (m_1 s_1 + m_2 s_2) \Delta T \]

### Step 5: Find the specific heat of the composite body
The specific heat \( S \) of the composite body can be defined as:
\[ S = \frac{Q}{m_1 + m_2} \cdot \frac{1}{\Delta T} \]
Substituting for \( Q \):
\[ S = \frac{m_1 s_1 + m_2 s_2}{m_1 + m_2} \]

### Step 6: Substitute the mass ratio into the specific heat formula
Using the mass ratio \( \frac{m_1}{m_2} = \frac{s_2}{s_1} \), we can express \( m_1 \) in terms of \( m_2 \):
\[ m_1 = \frac{s_2}{s_1} m_2 \]
Substituting this into the specific heat formula:
\[ S = \frac{\left(\frac{s_2}{s_1} m_2\right) s_1 + m_2 s_2}{\frac{s_2}{s_1} m_2 + m_2} \]
This simplifies to:
\[ S = \frac{s_2 m_2 + s_1 m_2}{\frac{s_2 + s_1}{s_1} m_2} \]
\[ S = \frac{s_2 + s_1}{\frac{s_2 + s_1}{s_1}} = \frac{2 s_1 s_2}{s_1 + s_2} \]

### Final Answer
Thus, the specific heat of the composite body is given by:
\[ S = \frac{2 s_1 s_2}{s_1 + s_2} \]