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).

Key concepts

Vernier Caliper

Vernier calipers are precise measuring tools used to measure dimensions like length and diameter. They have two scales: the main scale and the vernier scale. The main scale provides the baseline measurement, while the vernier scale allows for more precise readings by showing the increments between the main scale divisions. To read a vernier caliper, note the main scale reading first. Then, identify which vernier scale division aligns exactly with a division on the main scale. This aligned division determines the additional fraction of a main scale division to be added.
This exercise demonstrates the importance of the vernier caliper in achieving measurements with higher accuracy. The least count of a vernier caliper is a crucial concept here, calculated as the smallest difference between one main and one vernier scale division. In this case, it is 0.001 cm, or 0.01 mm. Calculating precise measurements with a vernier caliper ensures that tiny differences in scale readings are noticed and accounted for.

Significant Figures

Significant figures are important in expressing measurements accurately in science and engineering. They include all the certain digits in a number plus one final digit, which is uncertain or estimated. The number of significant figures conveys the precision of the measurement. When performing calculations, it's crucial to maintain the correct number of significant figures to reflect the precision of your initial measurements.
In this exercise, the edge length measured by the vernier caliper and the mass both have four significant figures. Therefore, the resultant calculations involving these measurements, such as density, also require four significant figures to be accurate. This concept ensures accuracy and avoids overstating the precision beyond the initial measurement capabilities.

Cube Volume

The volume of a cube is determined using the cube volume formula, which is simply the length of one side raised to the power of three. A perfect cube has equal sides, so if the edge of a cube is measured as 1.001 cm, the volume is calculated using \[\text{Volume} = \text{Edge Length}^3 = (1.001\, \text{cm})^3 = 1.003\, \text{cm}^3\]Calculating the volume is a straightforward step once you have an accurate measurement for the cube's edge, which in this case was measured using a vernier caliper.
Understanding the relationship between edge length and volume in cubes is essential in geometry and physics, as it connects linear dimensions to spatial properties, essential in contexts involving materials and density computations.

Measurement Accuracy

Measurement accuracy refers to how close a measured value is to the true value of the quantity being measured. Achieving high measurement accuracy in experiments and calculations ensures reliability and validity of the results. Tools like the vernier caliper enhance accuracy by providing finer readings than other measuring devices like a simple ruler.
This exercise highlights how to achieve accurate measurements of a cube's edge, thereby allowing for precise calculations in finding volume and density. Improving measurement accuracy also relates to using correct significant figures, which maintain the integrity of measurements through arithmetic processes. A strict adherence to accurate methods like using a vernier caliper will result in trustworthy data useful for engineering and scientific purposes.