Problem 40 - Arihant AIEEE Physics

Question

The ratio of thermal capacities of two spheres \(A\) and \(B\), if their diameters are in the ratio \(1: 2\), densities in the ratio \(2: 1\), and the specific heat in the ratio of \(1: 3\), will be: (a) \(1: 6\) (b) \(1: 12\) (c) \(1: 3\) (d) \(1: 4\)

Step-by-Step Solution

Answer

The ratio of thermal capacities is \(1: 12\). (Option b)

Step 1: Define Thermal Capacity

Thermal capacity, or heat capacity, of an object is given by the formula: \( C = mc \), where \( m \) is the mass of the object and \( c \) is the specific heat of the material. For spheres \( A \) and \( B \), we need to find the thermal capacities \( C_A \) and \( C_B \) and then find their ratio \( \frac{C_A}{C_B} \).

Step 2: Find the Mass of Each Sphere

The mass \( m \) of a sphere is given by the formula \( m = \text{density} \times \text{volume} \). The volume \( V \) of a sphere is \( V = \frac{4}{3}\pi r^3 \). Since the diameters of spheres \( A \) and \( B \) are in the ratio \( 1: 2 \), their radii \( r_A \) and \( r_B \) are also in the ratio of \( 1: 2 \). So, \( V_A = \frac{4}{3}\pi (r_A)^3 \) and \( V_B = \frac{4}{3}\pi (2r_A)^3 = \frac{4}{3}\pi 8(r_A)^3 \).

Step 3: Calculate Mass of Each Sphere

Using the volume expressions from Step 2 and given density ratios:- For sphere \( A \): \( m_A = 2 \times \frac{4}{3}\pi (r_A)^3 \).- For sphere \( B \): \( m_B = 1 \times \frac{4}{3}\pi 8(r_A)^3 \).Thus, \( m_A = \frac{8}{3}\pi(r_A)^3 \) and \( m_B = \frac{32}{3}\pi(r_A)^3 \).

Step 4: Calculate Thermal Capacity Ratio

Using heat capacity formula \( C = mc \):- For sphere \( A \): \( C_A = m_A c_A = \frac{8}{3}\pi(r_A)^3 \times c_A \).- For sphere \( B \): \( C_B = m_B c_B = \frac{32}{3}\pi(r_A)^3 \times c_B \).The specific heat ratio is \( c_A : c_B = 1:3 \).

Step 5: Solve for the Ratio \( \frac{C_A}{C_B} \)

Substitute the expressions from Step 4 into the ratio:\[ \frac{C_A}{C_B} = \frac{ \frac{8}{3}\pi(r_A)^3 \times c_A }{ \frac{32}{3}\pi(r_A)^3 \times c_B } = \frac{8 \times c_A}{32 \times c_B} \]Substitute the specific heat ratio \( c_A : c_B = 1:3 \):\[ \frac{C_A}{C_B} = \frac{8 \times 1}{32 \times 3} = \frac{8}{96} = \frac{1}{12} \].

Key Concepts

Density

Density is a key concept in understanding various physical properties of objects. It essentially measures how much mass is contained within a specific volume. The formula for density \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \) helps you determine how compact or sparse an object is. Higher density means more mass is packed in a given volume, whereas lower density means less mass is packed in the same space.

Density is crucial for calculating the mass of a sphere, as it directly influences how heavy the sphere is. In our example, two spheres have densities in the ratio of 2:1. This means sphere \( A \) is denser than sphere \( B \). Because the density is directly proportional to mass, sphere \( A \) will have twice the mass for the same volume as sphere \( B \).

Understanding density also requires knowing about different materials and their densities. For example, metals generally have high densities compared to liquids like water. So, when working through calculations, be sure to determine the right density values for the material you are dealing with.

Specific Heat

Specific heat is an essential property of a material, indicating how much heat is needed to change the temperature of a substance. Its formula \( q = mc\Delta T \) tells us that the amount of heat \( q \) required to change the temperature is proportional to the mass \( m \), the specific heat \( c \), and the temperature change \( \Delta T \). This means the higher the specific heat, the more energy you need to change an object's temperature.

In the solution, spheres \( A \) and \( B \) have specific heat ratios of 1:3. This indicates that sphere \( B \) requires more energy to achieve the same temperature change compared to sphere \( A \). Specific heat is vital for calculating thermal capacity, as it reflects both the material's nature and the required energy. Materials like water have a relatively high specific heat, implying they can store more heat energy without quickly changing temperature.

When comparing substances, remember that a higher specific heat means more heat energy is stored for every degree change in temperature. This concept is tremendously useful in practical applications like designing thermal systems.

Volume of a Sphere

The volume of a sphere is an important geometrical and physical property that allows us to calculate other attributes like mass and thermal capacity. The formula for the volume of a sphere is \( V = \frac{4}{3}\pi r^3 \), where \( r \) is the radius. This formula shows how a slight change in radius has a significant impact on the sphere's volume, due to the cube of the radius.

In the given exercise, the radii of spheres \( A \) and \( B \) are in the ratio 1:2. This indicates that the volume of sphere \( B \) will be eight times larger than that of sphere \( A \) because volume increases with the cube of the radius. Therefore, if you double the radius, the volume increases by a factor of eight \( 2^3 \).

Understanding volume and its relation to radius is essential for calculations involving spherical objects, ranging from simple school exercises to real-world applications such as calculating the volume of planets or bubbles. In thermal physics, this relationship helps in determining the thermal capacity by understanding how much space the substance occupies.