Two spheres of the same metal have radii in the_ratio 1 : 2 Their heat capacities are in what ratio

Options

A. `1:2`
B. ` 1:4 `
C. ` 2:1`
D. ` 1:8 `

Explanation

To find the ratio of the heat capacities of two spheres made of the same metal with radii in the ratio 1:2, we can follow these steps:

### Step 1: Understand the relationship between heat capacity, mass, and specific heat.
The heat capacity \( C \) of an object is given by the formula:
\[ C = m \cdot c \]
where \( m \) is the mass of the object and \( c \) is the specific heat capacity of the material.

### Step 2: Determine the mass of each sphere.
The mass \( m \) of a sphere can be expressed in terms of its volume and density:
\[ m = \rho \cdot V \]
For a sphere, the volume \( V \) is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, the mass of the first sphere (with radius \( r_1 \)) is:
\[ m_1 = \rho \cdot \frac{4}{3} \pi r_1^3 \]
And for the second sphere (with radius \( r_2 \)):
\[ m_2 = \rho \cdot \frac{4}{3} \pi r_2^3 \]

### Step 3: Substitute the radii into the mass equations.
Given that the radii are in the ratio \( r_1 : r_2 = 1 : 2 \), we can express \( r_2 \) as \( 2r_1 \). Therefore:
\[ m_1 = \rho \cdot \frac{4}{3} \pi r_1^3 \]
\[ m_2 = \rho \cdot \frac{4}{3} \pi (2r_1)^3 = \rho \cdot \frac{4}{3} \pi \cdot 8r_1^3 = 8 \rho \cdot \frac{4}{3} \pi r_1^3 \]

### Step 4: Find the ratio of the masses.
Now, we can find the ratio of the masses:
\[ \frac{m_1}{m_2} = \frac{\rho \cdot \frac{4}{3} \pi r_1^3}{8 \rho \cdot \frac{4}{3} \pi r_1^3} = \frac{1}{8} \]

### Step 5: Determine the ratio of heat capacities.
Since both spheres are made of the same material, their specific heat capacities \( c \) are equal. Therefore, the ratio of their heat capacities can be expressed as:
\[ \frac{C_1}{C_2} = \frac{m_1 \cdot c}{m_2 \cdot c} = \frac{m_1}{m_2} \]
Thus:
\[ \frac{C_1}{C_2} = \frac{1}{8} \]

### Conclusion:
The heat capacities of the two spheres are in the ratio \( 1 : 8 \).