Question

A copper wire $3$ mm in diameter is wound about a cylinder whose length is $1.2$ meter and diameter is $10$ cm, so as to cover the curved surface of the cylinder. Find the length and mass of the wire, assuming the density of copper wire to be $8.88\text{ }\dfrac{\text{g}}{\text{c}{{\text{m}}^{3}}}$. ​

Answer 1

Hint: First we will assume the number of wounds made by the copper wire and then we will apply the formula: \[\dfrac{\text{height of the cylinder}}{\text{diameter of wire}}\], and then to find the length of wire in one round equate it with the circumference of the cylinder and then multiply it with 400 to find the length of wire in $400$ rounds. Finally to find the mass of the wire, we will apply the formula: $\text{density}=\dfrac{\text{mass}}{\text{volume}}$ .

Complete step-by-step answer:
Now the following information is given in the question:
Diameter of the copper wire = $3$ mm
Length of the cylinder = $1.2$ m
Diameter of cylinder = $10$ cm
Density of the copper wire = $8.88\text{ }\dfrac{\text{g}}{\text{c}{{\text{m}}^{3}}}$.

Now, we have to find the length and the mass of the wire. Let us assume $n$ be the number of wounds made by the copper wire around the cylinder. In each wound, the wire covers a height of its diameter that is = $3$ mm on the cylinder.
Thus the number of wounds made by wire around the cylinder: \[n=\dfrac{\text{height of the cylinder}}{\text{diameter of wire}}\]
Now we know that the height of the cylinder = $1.2$ m
Therefore, $n=\dfrac{1.2\text{ m}}{3\text{ mm}}$
We will now convert both meter and millimetre into centimetre in order to have same unit, and we know that for converting metre to centimetre we will multiply the quantity by 100 and for millimetre to centimetre we will divide the quantity by 10 :
Therefore we will have:
 $\begin{align}
  & n=\dfrac{1.2\text{ m}}{3\text{ mm}}\Rightarrow n=\dfrac{1.2\times 100}{\left( \dfrac{3}{10} \right)}\Rightarrow n=\dfrac{120\text{ cm}}{0.3\text{ cm}} \\
 & n=400 \\
\end{align}$
Therefore, the number of wounds will be $400$.
Now the length of the wire required in one round will be equal to the circumference of the base of the cylinder.
As given the diameter of the cylinder is $10$ cm, therefore:
Circumference of the cylinder: $2\pi r=2\pi \left( \dfrac{d}{2} \right)=2\pi \left( 5 \right)=10\pi $
Now the length of the wire required in one round will also be equal to $10\pi $ ......... Equation 1.
Now the length of the wire required in $400$ rounds will be = $10\pi \times 400$
Now as $\pi =\dfrac{22}{7}$ , therefore : $10\pi \times 400=10\times \dfrac{22}{7}\times 400=\dfrac{88000}{7}=12571.42$cm
Therefore, the length of wire used will be: $12571.42\text{ cm}$
Now,
Since wire is in a cylindrical form, therefore the volume of wire used in each wound = $\pi {{r}^{2}}l$ ,
Where,
$\pi =\dfrac{22}{7}$ ,
$r$ = radius of the wire = $\dfrac{\left( \text{diameter} \right)}{2}=\dfrac{3}{2}=1.5\text{ mm}\Rightarrow \dfrac{1.5}{10}=0.15\text{ cm}$
$l$ = length of wire in one round = $10\pi $ (from equation 1)
Volume of wire used in each round = $\pi {{r}^{2}}l$= \[\pi \times {{\left( 0.15 \right)}^{2}}\times 10\pi \Rightarrow \dfrac{22}{7}\times {{\left( 0.15 \right)}^{2}}\times 10\times \dfrac{22}{7}=2.22\text{ c}{{\text{m}}^{2}}\]
Now the volume used in $400$ rounds = $400\times 2.22=888\text{ c}{{\text{m}}^{2}}$
We know that the formula for $\text{density}=\dfrac{\text{mass}}{\text{volume}}\Rightarrow \text{mass}=\left( \text{density }\times \text{ volume} \right)$
Given that the density of wire is $8.88\text{ }\dfrac{\text{g}}{\text{c}{{\text{m}}^{3}}}$, and we found out volume to be $888\text{ c}{{\text{m}}^{3}}$ ,
Therefore, the mass of copper wire used = $8.88\times 888=7855.44$ grams.
Hence, the length of wire used will be: $12571.42\text{ cm}$, And the mass of copper wire used = $7885.44\text{ g}$.

Note: Students can also take $\pi $ as $3.14$. Always convert the units into one particular unit as we saw initially we converted meter into centimeter and millimeter into centimeter. Also never forget to mention the units while writing the answer, as we wrote grams for mass and centimeter for length.