Question

# The inner diameter of a cylindrical wooden pipe is 24 cm and its outer diameter is 28 cm. The length of the pipe is 35 cm. Find the mass of the pipe, if 1 $c{{m}^{3}}$ of wood has a mass of 0.6 g.

Hint:First find the volumes of inner and outer cylinders separately by using the formula $\pi {{r}^{2}}h$ and then use the formula, Total volume of wooden pipe = Volume of outer pipe – Volume of inner pipe to calculate the total volume of pipe. Then multiply the volume by 0.6 to get the total volume.

To find the mass of the cylinder we will first draw the diagram of the cylinder with given measures.

Now, we will write the given data with assuming the required notations,
Diameter of outer cylinder = D = 28 cm.
Therefore radius of outer cylinder $=R=\dfrac{D}{2}=\dfrac{28}{2}=14cm$ ………………………………….. (1)
Diameter of inner cylinder = d = 24 cm.
Therefore radius of inner cylinder $=r=\dfrac{d}{2}=\dfrac{24}{2}=12cm$ …………………………………………. (2)
Height of the cylinder = h =35 cm. ………………………………………………………………………………. (3)
Also, 1 $c{{m}^{3}}$ of wood has a mass of 0.6 g.
As we have given the mass per $c{{m}^{3}}$ of the wood therefore we will first find the total volume of the wooden cylinder so that we can easily calculate its total mass.
If we see the geometry of the figure then we can conclude that,
Total volume of wooden pipe = Volume of outer pipe – Volume of inner pipe ……………………… (4)
To find the total volume of wooden pipe we will first find out the inner and outer cylinder volumes. To find the individual volumes we should know the formula given below,
Formula:
Volume of cylinder = $\pi {{r}^{2}}h$
By using the formula given above we can write the formula for volume of outer cylinder as,
Volume of outer cylinder = $\pi {{R}^{2}}h$
If we put the value of equation (1) and equation (3) in the above equation we will get,
Volume of outer cylinder = $\pi {{\left( 14 \right)}^{2}}\left( 35 \right)$ ……………………………………………………………………………… (5)
Also the formula for the volume of inner cylinder can be written as,
Volume of inner cylinder = $\pi {{r}^{2}}h$
If we put the values of equation (2) and equation (3) in the above equation we will get,
Volume of inner cylinder = $\pi {{\left( 12 \right)}^{2}}\left( 35 \right)$ ………………………………………………………………………………. (6)
Now we will put the values of equation (5) and equation (6) in equation (4) to get the total volume of wooden pipe, Therefore,
Total volume of wooden pipe = $\pi {{\left( 14 \right)}^{2}}\left( 35 \right)-\pi {{\left( 12 \right)}^{2}}\left( 35 \right)$
Therefore, the total volume of wooden pipe = $35\pi \left[ {{\left( 14 \right)}^{2}}-{{\left( 12 \right)}^{2}} \right]$
Therefore, the total volume of wooden pipe = $35\pi \left[ 196-144 \right]$
Therefore, the total volume of wooden pipe = $35\pi \left( 52 \right)$
Therefore, total volume of wooden pipe = 5717.69 $c{{m}^{3}}$
Now as we have given in the problem,
1 $c{{m}^{3}}$ of wood has a mass of 0.6 g therefore,
5717.69 $c{{m}^{3}}$ of wood have mass = $5717.69\times 0.6$
Therefore, the total mass of the wooden pipe = 3430.61 g.
Therefore, the total mass of wooden pipe = $\dfrac{3430.61}{1000}$ Kg.
Therefore, the total mass of wooden pipe = 3.4306 Kg.
Therefore the total mass of the cylindrical wooden pipe is equal to 3.4306 Kg.

Note: You can directly use the formula $Mass=Density\times \left( Volume\text{ }of\text{ }outer\text{ }pipe\text{ }\text{ }Volume\text{ }of\text{ }inner\text{ }pipe \right)$ as 0.6 is basically the density of the material and it will give you quick answer in competitive exams. Just be aware of silly mistakes while using this formula.