Question

The pitch of screw gauge is \(1 \mathrm{~mm}\) and there are 100 divisions on the circular scale. What measuring the diameter of a wire, the linear scale reads, \(1 \mathrm{~mm}\) and \(47^{\text {th }}\) division on the circular scale coincides with the reference line. The length of the wire is \(5.6 \mathrm{~cm}\). What will be the curved surface area (in \(\mathrm{cm}^{2}\) ) of the wire in appropriate number of significant figures. (A) \(2.6 \mathrm{~cm}^{2}\) (B) \(2.5848 \mathrm{~cm}^{2}\) (C) \(2.585 \mathrm{~cm}^{2}\) (D) \(2.5 \mathrm{~cm}^{2}\)

Short answer

The curved surface area of the wire is approximately 2.58 \(\mathrm{cm}^{2}\).

Step-by-step solution

Calculate Diameter from Screw Gauge Readings

To calculate the diameter, we can use the following formula: $$ Diameter = Pitch + (No. of divisions \times \frac{Pitch}{100}) $$ In this problem: - The pitch is given as 1 mm. - The linear scale reading is 1 mm. - The number of divisions on the circular scale is 47 divisions. Substitute these values to get the diameter: $$ Diameter = 1 + (47 \times \frac{1}{100}) $$ $$ Diameter = 1 + 0.47 $$ $$ Diameter = 1.47\,mm $$

Calculate Radius from Diameter

We can derive the radius from diameter using the following formula: $$ Radius = \frac{Diameter}{2} $$ Substitute the value of the diameter we calculated in Step 1 to get the radius: $$ Radius = \frac{1.47\, mm}{2}$$ $$ Radius = 0.735\, mm $$

Convert Length of Wire to mm

The length of the wire is given as 5.6 cm. To match the units, we need to convert it into mm. $$ Length = 5.6\, cm \times \frac{10\, mm}{1\, cm}$$ $$ Length = 56\, mm $$

Calculate Curved Surface Area of the Wire

Use the formula for the curved surface area of a cylinder, which is applicable to the curved surface area of a wire: $$ Curved\, Surface\, Area = 2\pi rL $$ Where, - r = radius - L = length of the wire Substitute the values of radius and length of the wire: $$ Curved\, Surface\, Area = 2\pi (0.735\,mm)(56\,mm) $$ $$ Curved\, Surface\, Area = 258.48\, mm^2 $$

Convert Curved Surface Area to \(\mathrm{cm}^{2}\) and Apply Significant Figures

Convert the curved surface area from \(\mathrm{mm}^{2}\) to \(\mathrm{cm}^{2}\): $$ Curved\, Surface\, Area = 258.48\, mm^2 \times \frac{1\, cm^2}{100\, mm^2}$$ $$ Curved\, Surface\, Area = 2.5848\, cm^2 $$ Now, apply the appropriate number of significant figures to the final answer. Since the given data has at most two significant figures, the answer should be rounded to two significant figures. $$ Curved\, Surface\, Area = 2.58\, cm^2$$ Thus, the curved surface area of the wire is approximately 2.58 \(\mathrm{cm}^{2}\), which corresponds to option (C).

Key concepts

Significant Figures

Understanding significant figures is essential in scientific measurements. They help ensure that calculations and results reflect the precision of the measurements involved. In our exercise, the length of the wire is given as 5.6 cm, which has two significant figures. This implies that our final result should echo this precision level.

When calculating the curved surface area, the raw result was 2.5848 cm². However, this number needs to be adjusted to two significant figures. We do this because our initial measurement (5.6 cm) only supports two significant figures. Therefore, the best representation of the curved surface area is 2.58 cm², ensuring the precision reflects our initial data.

Curved Surface Area of Cylinders

The curved surface area of a cylinder, like a wire, is crucial for understanding its properties. The formula for curved surface area is given by \(2\pi rL\), where \(r\) is the radius and \(L\) is the length. This formula captures the surface area that wraps around the circular cross-section.

In the provided problem, we calculated the radius from the diameter using the screw gauge readings and measured the length of the wire. By inserting these values into the formula, we calculated the curved surface area to be 258.48 mm² in its raw form before converting it into square centimeters. Understanding this concept allows us to find the surface exposure of cylindrical objects and is often used in physical and engineering contexts.

Measurement Units Conversion

Conversion between measurement units is often necessary to ensure uniformity in calculations. In this exercise, the wire's length was given in centimeters, whereas the diameter and the curved surface area required measurements in millimeters.

To convert the length from centimeters to millimeters, we used the conversion ratio \(1\, cm = 10\, mm\). Consequently, the length of 5.6 cm became 56 mm. Furthermore, the curved surface area was initially calculated in square millimeters, then converted to square centimeters using \(1\, cm^2 = 100\, mm^2\). Understanding these conversions is vital, as it helps maintain consistency and accuracy in scientific and mathematical work.