Question

The diameter of a sphere is measured using a vernier caliper whose 9 divisions of main scale are equal to 10 divisions of vernier scale. The shortest division on the main scale is equal to l mm. The main scale reading is 2 cm and second division of vernier scale coincides with a division on main scale. If mass of the sphere is 8.635 g, the density of the sphere is:

• A. \(2.5 \, \text{g/cm}^3\)
• B. \(1.7 \, \text{g/cm}^3\)
• C. \(2.2 \, \text{g/cm}^3\)
• D. \(2.0 \, \text{g/cm}^3\)

Answer 1

The Correct Option is D

To ascertain the sphere's density, we must first compute its volume using the provided measurements. The process involves the following stages:

1. Determine the least count of the vernier caliper.
2. Obtain the total reading from the vernier caliper.
3. Calculate the sphere's volume utilizing the determined diameter.
4. Finally, compute the density using the equation \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \).

Step 1: Calculate the Least Count of the Vernier Caliper

Provided information:

• \(9\) main scale divisions correspond to \(10\) vernier scale divisions.
• The length of one main scale division is \(l \, \text{mm} = 0.1 \, \text{cm}\) (given that 1 cm = 10 mm).

The least count (LC) of the vernier caliper is calculated as:

\(\text{LC} = \frac{\text{Value of one main scale division}}{\text{Number of divisions on vernier scale}} = \frac{l}{9} \, \text{cm} = \frac{0.1}{9} \, \text{cm} \approx 0.0111 \, \text{cm}\)

Step 2: Determine the Total Reading

Given values:

• Main scale reading is \(2 \, \text{cm}\).
• The second division on the vernier scale aligns with a main scale division.

The vernier scale reading is calculated as \(2 \times \text{Least Count} = 2 \times 0.0111 \, \text{cm} = 0.0222 \, \text{cm}\).

The total reading, representing the sphere's diameter, is:

\(\text{Total reading} = \text{Main Scale Reading} + \text{Vernier Scale Reading} = 2 + 0.0222 \approx 2.0222 \, \text{cm}\)

Step 3: Calculate the Volume of the Sphere

The formula for the volume of a sphere is:

\(V = \frac{4}{3} \pi r^3\)

where \( r \) denotes the sphere's radius.

With a diameter of \(2.0222 \, \text{cm}\), the radius \( r = \frac{2.0222}{2} \, \text{cm} = 1.0111 \, \text{cm}\).

Substituting the radius into the volume formula yields:

\(V = \frac{4}{3} \pi (1.0111)^3 \approx \frac{4}{3} \times 3.1416 \times 1.033 \approx 4.32 \, \text{cm}^3\)

Step 4: Calculate the Density

Using the density formula:

\(\text{Density} = \frac{\text{Mass}}{\text{Volume}} = \frac{8.635 \, \text{g}}{4.32 \, \text{cm}^3} \approx 2.0 \, \text{g/cm}^3\)

Consequently, the sphere's density is \( 2.0 \, \text{g/cm}^3 \).