Question

A copper rod of diameter 1cm and length 8cm is drawn into a wire of length 18 m of uniform thickness. Find the thickness of the wire.

Answer 1

Hint: Since, the rod is drawn into wire, therefore, we can see that volume remains unchanged. As we know that the copper rod is in the form of a cylinder, so we are going to use the formula for finding the volume of the cylinder which is $ \pi {{r}^{2}}h $ where $ r $ is the radius of the base of the cylinder and $ h $ Is the height or length of the cylinder. Here, we will find the volume of both wires (initial copper rod and drawn out wire) and then compare them to find the radius of the wire and then the diameter of the wire.

Complete step by step answer:
$\Rightarrow$ As the copper rod is in the form of a cylinder, we are given \[diameter=1cm\].
We require a radius for finding the volume of the rod and hence converting the given diameter to radius by dividing the diameter by 2.
$\Rightarrow$ Therefore, \[radius=\frac{diameter}{2}=\frac{1}{2}cm\]
$\Rightarrow$ That is $ r=\frac{1}{2}cm $
Also, we are given the length of the rod as 8cm which is equal to the height of the cylinder. So, \[height=8cm\].
$\Rightarrow$ That is $ h=8cm $ .
$\Rightarrow$ As we know, volume of cylinder is given by $ \pi {{r}^{2}}h $ where $ r $ is radius of base of cylinder and $ h $ is height of cylinder. Therefore, volume of copper rod = $ \pi {{r}^{2}}h $ = \[\pi \text{ }x\text{ }{{(\frac{1}{2})}^{2}}(8)\]
\[\pi \text{ }x\text{ }(\frac{1}{4})(8)\] = \[\pi \text{ }x\text{ }(\frac{1}{4})(8)\]
  = $ 2\pi \text{ c}{{\text{m}}^{3}} $ ........equation (1)
$\Rightarrow$ Now, the rod is drawn to wire of $ length\text{ }18m $ . Therefore, length = height= 18m. As all the rest units are in cm, therefore changing 18 m to cm by multiplying 18 by 100, we get,
  $ 18\text{ }m=18x100\text{ cm = 1800 cm } $
  $ \Rightarrow $ Therefore, for cylinder(wire), $ \text{height= 1800 cm } $
Let ‘r’ ne the radius of base of cylinder. Since, volume of cylinder is given by $ \pi {{r}^{2}}h $ where $ \text{r} $ is radius of base of cylinder and $ \text{h} $ is height.
$\Rightarrow$ Therefore, volume of wire = $ \pi {{r}^{2}}h $
   = $ \pi \text{ }x\text{ }{{r}^{2}}\text{ }x\text{ }1800 $
   = $ 1800\pi {{r}^{2}}c{{m}^{3}} $ ………equation (2)
$\Rightarrow$ As the same rod is drawn into wire therefore volume of copper rod will be equal to the volume of wire.
  $ Volume\text{ }of\text{ }copper\text{ }rod\text{ }=\text{ }Volume\text{ }of\text{ }wire $
Substituting values from equation (1) and equation (2), we get,
  $ 2\pi \text{ = 1800}\pi {{\text{r}}^{2}} $
  $ {{\text{r}}^{2}}=\frac{2\pi }{\text{1800}\pi } $
  $ {{\text{r}}^{2}}=\frac{1}{900} $
Taking square root both sides, we get,
  $ \text{r}=\sqrt{\frac{1}{900}} $
  $ \Rightarrow $ \[\text{r}=\frac{1}{30}cm\]
$\Rightarrow$ Since, thickness of wire is determined by diameter of wire, therefore converting radius to diameter by multiplying by 2, we get,
  $ \Rightarrow $ \[diameter=radius\text{ * 2 =}\frac{1}{30}\text{ * 2 = }\frac{1}{15}cm\]
Hence, diameter or we can say, thickness of wire is \[\frac{1}{15}cm\].

Note:
Students should take care that in these types of questions both cylinders have the same volume only. The surface area or total surface area of both wires is different. They should also learn the formula for finding the volume of the cylinder. Most importantly, do not forget to change units to a single type only.