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).