The aim of this question is to learn the **volume of a sphere** and the **density of different materials**.

If the radius **r** is known, the **volume** **V** of a sphere is given by:

\[ V \ = \ \dfrac{ 4 }{ 3 } \ \pi r^3 \ … \ … \ … \ (1) \]

Also, for a given material the **density** $ d $ is defined as:

\[ d \ = \ \dfrac{ m }{ V } \ … \ … \ … \ (2) \]

Where **m** is the **mass of the body**. We will manipulate the above two equations to solve the given problem.

## Expert Answer

**Substituting equation (1) in equation (2)**:

\[ d \ = \ \dfrac{ m }{ \bigg ( \ \frac{ 4 }{ 3 } \ \pi r^3 \ \bigg ) } \]

\[ \Rightarrow d \ = \ \dfrac{ 4 m }{ 3 \pi r^3 } \]

**For lead** (say material no. 1 ), the above equation becomes:

\[ d_1 \ = \ \dfrac{ 4 m_1 }{ 3 \pi r_1^3 } \ … \ … \ … \ (3) \]

**For Aluminum** (say material no. 2 ), the above equation becomes:

\[ d_2 \ = \ \dfrac{ 4 m_2 }{ 3 \pi r_2^3 } \ … \ … \ … \ (4) \]

**Dividing and simplifying equation (3) by equation (4)**:

\[ \dfrac{ d_1 }{ d_2 } \ = \ \dfrac{ m_1 r_2^3 }{ m_2 r_1^3 } \]

Given that:

\[ m_1 = m_2 \]

The above equation further reduces to:

\[ \dfrac{ d_1 }{ d_2 } \ = \ \bigg ( \dfrac{ r_2 }{ r_1 } \bigg )^3 \ … \ … \ … \ (5) \]

\[ \Rightarrow \dfrac{ r_2 }{ r_1 } \ = \ \bigg ( \dfrac{ d_1 }{ d_2 } \bigg )^{ 1/3 } \]

**From density tables:**

\[ d_1 \ = \ 11.29 \ g/cm^3 \text{ and } d_2 \ = \ 2.7 \ g/cm^3 \]

Substituting these in equation no. (5):

\[ \dfrac{ r_2 }{ r_1 } \ = \ \bigg ( \dfrac{ 11.29 }{ 2.7 } \bigg )^{ 1/3 } \]

\[ \dfrac{ r_2 }{ r_1 } \ = \ \bigg ( 4.1814 \bigg )^{ 1/3 } \]

\[ \Rightarrow \dfrac{ r_2 }{ r_1 } \ = \ 1.61 \]

## Numerical Result

\[ \dfrac{ r_2 }{ r_1 } \ = \ 1.61 \]

## Example

Find the **ratio of the radiuses** of two uniform spheres. One is made up of **copper** and the other one is made of **Zinc**.

Let copper and zinc be materials no. 1 and 2, respectively. Then **from density tables**:

\[ d_1 \ = \ 8.96 \ g/cm^3 \text{ and } d_2 \ = \ 7.133 \ g/cm^3 \]

**Substituting these in equation no. (5)**:

\[ \dfrac{ r_2 }{ r_1 } \ = \ \bigg ( \dfrac{ 8.96 }{ 7.133 } \bigg )^{ 1/3 } \]

\[ \dfrac{ r_2 }{ r_1 } \ = \ \bigg ( 1.256 \bigg )^{ 1/3 } \]

\[ \Rightarrow \dfrac{ r_2 }{ r_1 } \ = \ 1.0789 \]