Question

A copper wire of diameter is evenly wrapped on a cylinder of length 15cm and diameter 49cm to cover its whole surface. Find the length and volume of wire. If the specific gravity of copper is $9gc{{m}^{-3}}$, then find the weight of the wire.

Answer 1

Hint: The number of rotations of the copper wire on the cylinder can be calculated by dividing the length of the cylinder with the width (that is equal to the diameter) of the copper wire. Once we have the number of rotations, we can easily calculate the length of wire required by multiplying the number of rotations with the circumference of any of the two circular faces. Volume and weight of the wire can be then further calculated.

Complete step-by-step answer:
Let the number of rotation that the copper wire makes over the cylinder be equal to ‘N’. Then this can be calculated by dividing the length of the cylinder with the width (that is equal to the diameter) of the copper wire. But before that, let us define some terms as follows:
Diameter of copper wire (d) $=6mm=0.6cm$
Radius of copper wire (r) $=0.3cm$
Density of the copper wire $\left( \rho \right)=9gc{{m}^{-3}}$
Length of cylinder (l) $=15cm$
Diameter of cylinder (D) $=49cm$

Now, the number of rotations can be calculated as follows:
$\begin{align}
  & \Rightarrow N=\dfrac{l}{d} \\
 & \Rightarrow N=\dfrac{15}{0.6} \\
 & \Rightarrow N=25 \\
\end{align}$
Therefore, the length of wire used can be written as the product of number of rotations and circumference of any of the two circular faces. This length (say, L) can be calculated as follows:
$\begin{align}
  & \Rightarrow L=\pi D\times N \\
 & \Rightarrow L=\dfrac{22}{7}\times 49\times 25cm \\
 & \Rightarrow L=3850cm \\
\end{align}$

Now, the volume of this copper wire (say, L) will be equal to:
$\begin{align}
  & \Rightarrow V=\pi {{r}^{2}}\times L \\
 & \Rightarrow V=\dfrac{22}{7}\times {{\left( 0.3 \right)}^{2}}\times 3850c{{m}^{3}} \\
 & \Rightarrow V=108.9c{{m}^{3}} \\
\end{align}$
And at last, the weight of the copper wire (say, W) can be calculated by finding the product of its volume and density. This is done as follows:
$\begin{align}
  & \Rightarrow W=V\times \rho \\
 & \Rightarrow W=108.9\times 9g \\
 & \Rightarrow W=980.1g \\
\end{align}$
Hence, the length, volume and weight of the copper wire are $3850cm,\text{ }108.1c{{m}^{3}}\text{ and }980.1g$ respectively.

Note: Here, the copper wire was assumed to be wound very tightly on the cylinder. This is because the formula used by us to calculate the length is the formula for circumference of a circle and since the wire is not breaking anywhere in the middle, all the rotation almost forms a circle. The tighter these wounds are, the more accurate our answer is.