Question

A copper wire of 4 mm diameter is evenly wound about a cylinder whose length is 24 cm and diameter 20 cm so as to cover the whole surface. Find the length and weight of the wire assuming the density to be 8.68 $gm/cm^3$.

Answer 1

Hint: Here, we will firstly calculate the numbers of turns/ number of rounds of wire. We know the length of wire in one turn is equal to the circumference of the circle. And multiplying it with the number of turns will give a total length of wire.

Complete Step-by-Step solution:
Given,
Length of the cylinder = 24 cm
Diameter of copper wire = 4 mm = 0.4 cm
Then,
the number of rounds of wire to cover the length of cylinder
$
   = \dfrac{{24}}{{0.4}} \\
   = 60 \\
$
Now, diameter of cylinder = 20 cm
length of wire in one round = circumference of base of the cylinder = $ = \pi d$
$
   = \dfrac{{22}}{7} \times 20 \\
   = \dfrac{{440}}{7}cm \\
$
Length of wire for covering the whole surface of cylinder = length of wire in 60 rounds
$ = 60 \times \dfrac{{440}}{7} = 3771.428$
Radius of copper wire = 0.2 cm
Therefore, volume of wire = $\pi $r2h
$ = \dfrac{{22}}{7} \times {(.2)^2} \times 3771.428$
= 474.122 cu. cm
Weight of wire = volume × density = 474.122 × 8.68 gm = 4115.38 gm = 4.11538 kg $ \approx $4.12 kg.

Note: For calculating the volume of wire, we have to use the formula of the cylinder. As wire is also in cylindrical shape but its radius is small whereas height is large. Also, we will use the formula for mass/weight of the wire as mentioned below:\[Density = \dfrac{{Mass}}{{Volume}}\]. Here, we need to take care of units of measurement as data is given in mm and cm.