Question

A copper wire of 3 mm in diameter is wound about a cylinder whose length is 1.2 m and diameter is 10 cm so as to cover the curved surface area of the cylinder. Find the length and mass of wire, assuming the density of copper wire to be 8.88 gm per cm.

Answer 1

Hint: Calculate the curved surface area using the formula for curved surface area of the cylinder.
Curved Surface area of cylinder = 2π rh
where r is the radius of the cylinder, h is the height (or length) of the cylinder.
Then find the area covered by a unit cm length of copper wire.
Then divide the curved surface area of the cylinder by the area covered by a unit cm length of wire.

Complete step-by-step answer:

We are given the dimension of cylinder so that the curved surface area of cylinder can be found out using the formula for curved surface area of cylinder which is as shown below
Curved Surface of cylinder = 2π rh
where r is the radius of the cylinder and h is the height (or length) of the cylinder.
Let R and L be the radius and length of the given cylinder. Then it is given that the diameter of the cylinder is 10 cm. So its radius is $\dfrac{10}{2}$ = 5 cm.
R = 5 cm
L = 1.2 m = 1.2 x 100 = 120 cm
So the curved surface area of cylinder $=2\pi RL$
                                                                     $\begin{align}
  & =2\times \dfrac{22}{7}\times 5\times 120 \\
 & =\dfrac{44\times 600}{7} \\
 & =3771.43\,c{{m}^{2}} \\
\end{align}$

From the diagram above we easily see that 1 cm length of copper wire covers an area which is equal to the area of a rectangle of length 1 cm and breadth equal to the diameter of copper wire which is 3 cm.
So area covered by 1 cm length of copper wire = 1 cm x 3 mm
                                                                                    = 10 mm x 3 mm
                                                                                    = 30 $mm^2$
Length of copper wire needed to cover the entire cylinder can be determined by dividing the surface area of the cylinder by the area covered by 1 cm length of wire.
Let the length of copper wire needed be s. Then
$\begin{align}
  & s=\dfrac{3771.43\,c{{m}^{2}}}{30\,m{{m}^{2}}} \\
 & \Rightarrow s=\dfrac{3771.43\times {{10}^{2}}\,m{{m}^{2}}}{30\,m{{m}^{2}}} \\
 & \Rightarrow s=12571.43 \\
\end{align}$
So the length of copper wire needed to cover the cylinder is 125.714 m.

Note: There is one high chance of making mistakes related to units of various measurements given. Take care of conversion of units from m to cm, cm to mm when doing multiplication to get the area otherwise the answer will not come correct.