A copper wire, 3 mm in diameter, is wound about a cylinder whose length is 12 cm, and diameter 10 cm, so as to cover the curved surface of the cylinder. Find the length and mass of the wire, assuming the density of copper to be 8.88 g per cm3
Solution:
A figure is drawn to visualize the cylinder.
As we know that the curved surface of the cylinder is to be covered with the copper wire, then one round of copper wire will cover the same height of the cylinder as of the width (diameter) of the wire.
Therefore, Length of the wire × diameter the of wire = CSA of the cylinder
Let us find the CSA of the cylinder by using the formula;
CSA of the cylinder = 2πrh, where r and h are the radius and height of the cylinder respectively.
To find the mass of the wire:
Mass = Density × Volume
As the density of the copper is given, we need to find the volume of the wire first.
As the wire is cylindrical in shape
Therefore, Volume of the wire = volume of the cylinder
Let us find the volume of the cylindrical wire by using the formula:
Volume of the cylinder = πr₁2h₁, where r₁ and h₁ are the radius and the length of the cylindrical wire
Height of cylinder, h = 12 cm
Diameter of cylinder, d = 10 cm
Radius of cylinder, r = 10 cm/2 = 5 cm
Diameter (width) of wire d₁ = 3 mm = 3 / 10 cm = 0.3 cm
Radius of wire, r₁ = 0.3 cm/2 = 0.15 cm
Let the length of the wire be h₁
Since curved surface of the cylinder to be covered with the wire
Length of the wire × diameter the of wire = CSA of the cylinder
d₁ × h₁ = 2πrh
h₁ = 2πrh/d₁
= (2 × 22 × 5 cm × 12 cm) / 7 × 0.3 cm
= 1257.14 cm
Volume of the wire = volume of the cylinder
= πr₁2h₁
= (22/7) × 0.15 cm × 0.15 cm × 1257.14 cm
= 88.898 cm3
Density of copper = 8.8 g /cm3
Mass of the copper Wire = Density × volume
= 8.88g / cm3 × 88.898 cm3
= 789.41 g
Thus, the length of the wire is 12.57 m and the mass is 790 g (approx.)
☛ Check: Class 10 Maths NCERT Solutions Chapter 13
Video Solution:
A copper wire, 3 mm in diameter, is wound about a cylinder whose length is 12 cm, and diameter 10 cm, so as to cover the curved surface of the cylinder. Find the length and mass of the wire, assuming the density of copper to be 8.88 g per cm3
NCERT Solutions for Class 10 Maths Chapter 13 Exercise 13.5 Question 1
Summary:
A copper wire, 3 mm in diameter, is wound about a cylinder whose length is 12 cm, and diameter 10 cm, so as to cover the curved surface of the cylinder. Assuming the density of copper to be 8.88 g per cm3 , the length and mass of the wire are 12.57 m and 790 g respectively.
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