Question

The edge of a cube is measured using a vernier caliper \((9\) divisions of the main scale is equal to 10 divisions of vernier scale and 1 main scale division is \(1 \mathrm{~mm}\) ). The main scale division reading is 10 and 1 division of vernier scale was found to be coinciding with the main scale. The mass of the cube is \(2.736 \mathrm{~g}\). What will be the density in $\left\\{\mathrm{g} /\left(\mathrm{cm}^{3}\right)\right\\}$ upto correct significant figures? (A) $2.66 \times 10^{-3}\left\\{\mathrm{~g} /\left(\mathrm{cm}^{3}\right)\right\\}$ (B) $2.66 \times 10^{3}\left\\{\mathrm{~g} /\left(\mathrm{cm}^{3}\right)\right\\}$ (C) \(2.66\left\\{\mathrm{~g} /\left(\mathrm{cm}^{3}\right)\right\\}\) (D) $2.66 \times 10^{-6}\left\\{\mathrm{~g} /\left(\mathrm{cm}^{3}\right)\right\\}$

Short answer

The density of the cube is \(2.729\left\\{\mathrm{~g} /\left(\mathrm{cm}^{3}\right)\right\\}\).

Step-by-step solution

Understand the vernier caliper readings

The main scale has 10 divisions and 1 main scale division is 1 mm. The vernier scale has 9 divisions equal to 10 divisions of the main scale. This means that 1 vernier scale division is equal to \(\frac{9}{10}\) main scale divisions. The main scale reading is 10, and 1 division of the vernier scale coincides with the main scale.

Calculate the actual edge length

We can find the actual length of the edge using the formula: Actual Length = Main Scale Reading + (Vernier Scale Reading × Least Count) Least Count is the difference between one main scale division and one vernier scale division. It can be calculated by the formula: Least Count = \(\frac{\text{Value of 1 Main Scale Division}}{\text{Number of divisions in which the vernier scale is divided}}\) Here, the Least Count = \(\frac{1 \mathrm{~mm} - 0.9 \mathrm{~mm}}{10}\) = \(0.01 \mathrm{~mm}\) = \(0.001 \mathrm{~cm}\) Now, let's find the Actual Length: Actual Length = Main Scale Reading + (Vernier Scale Reading × Least Count) = \(10 \mathrm{~mm} + (1 \times 0.01 \mathrm{~mm})\) Actual Length = \(10.01 \mathrm{~mm}\) = \(1.001 \mathrm{~cm}\)

Calculate the volume of the cube

Now that we have the edge length, we can calculate the volume of the cube using the formula: Volume = Edge Length³ Volume = \((1.001 \mathrm{~cm})³\) Volume = \(1.003 \mathrm{~cm}^3\)

Calculate the density of the cube

We are given the mass of the cube as \(2.736 \mathrm{~g}\). Now we can calculate the density of the cube using the formula: Density = \(\frac{\text{Mass}}{\text{Volume}}\) Density = \(\frac{2.736 \mathrm{~g}}{1.003 \mathrm{~cm}^3}\) Density = \(2.729 \mathrm{~g/cm}^3\) Now, let's consider the correct significant figures. The edge length has four significant figures and the mass has four significant figures, so our final answer should have four significant figures as well. Therefore, the density of the cube is \(2.729\left\\{\mathrm{~g} /\left(\mathrm{cm}^{3}\right)\right\\}\), which matches option (C).