Question

A copper wire, $3mm$ in diameter, is wound about a cylinder whose length is $12cm$, and diameter $10cm$, 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 $8.88gperc{m}^3$.

Answer 1

Given:Diameter of the wire$=3mm=\frac{3}{10}cm$

$\therefore$One round of the wire covers $\frac{3}{10}cm$ of the height of the cylinder.

Now,given,Length of cylinder$=12cm$

Now, wire will cover entire length of cylinder

Let's find number of rounds

$\frac{3}{10}cm$ is covered in $=1$ round.

$1cm$ is covered in $=1\times \frac{10}{3}$ rounds

$12cm$ is covered in $=12\times \frac{10}{3}=40$ rounds

Length of the wire required to complete one round

$=$Circumference of the cylinder

$=2\pi r=2\pi \times \frac{10}{2}=10\pi cm$

Length of the wire required to cover whole cylinder

$=$Length required to cover $1$ round$\times$Number of rounds

$=10\pi \times 40=400\times 3.14=1256cm$

Now,we need to find mass of the wire,assuming the density of copper to be $8.88gper{cm}^3$

So,first we find volume of wire

Given:Diameter of the wire$=\frac{3}{10}cm$

$\therefore$Radius of the wire$=\frac{\frac{3}{10}}{2}=\frac{3}{20}cm$

Volume of the wire$=\pi r^2\times length=\pi {\left(\frac{3}{20}\right)}^2\times 1256=\frac{9\times 1256\times 3.14}{400}=88.73{cm}^3$

Now, Density of the wire$=\frac{Massofthewire}{Volumeofthewire}$

$Density\times volume=Mass$

Putting density of the wire$=8.88gper{cm}^3$ and Volume$=88.73{cm}^3$

Mass$=8.88\times 88.73=787.83gm$

$\therefore$Length of the wire$=1256cm$ and Mass of the wire$=788gm$(approx)