Question

Calculate MI of a cylinder of length 1.5m radius 0.05m density $8\times {{10}^{3}}kg/{{m}^{3}}$ about the axis of the cylinder to its center?

Answer 1

Hint: To find moment of inertia of a cylinder about its central axis we will use the equation of moment of inertia for the cylinder in which mass of the cylinder can be found by using density of the cylinder.

Formula used:
$I=\dfrac{1}{2}m{{r}^{2}}$
$\rho =\dfrac{m}{v}$

Complete step by step solution:
We know that formula of moment of inertia about axis of cylinder is given by,
$I=\dfrac{1}{2}m{{r}^{2}}.....\left( 1 \right)$
Where,
I = moment of inertia
m = total mass of the cylinder
r = radius of the cylinder

Here radius of the cylinder is given as,
r = 0.05m

Now we have to find mass of cylinder
To find the mass of the cylinder we will use the following equation.
$\rho =\dfrac{m}{v}$
Where, ρ = density of the cylinder
v = volume of the cylinder
Here,
$m=\rho \times v$

Now the density of cylinder is given as
$\rho =8\times {{10}^{3}}kg/{{m}^{3}}$

And volume of cylinder can be given by the following expressions
$\begin{align}
  & \Rightarrow v=\text{area }\!\!\times\!\!\text{ height of the cylinder} \\
 & \Rightarrow v=A\times h \\
 & \therefore v=\pi {{r}^{2}}h \\
\end{align}$

Now mass,
$m=\rho \pi {{r}^{2}}h...\left( 2 \right)$
Now put value of equation (2) in equation (1)
$I=\dfrac{1}{2}\times \rho \pi {{r}^{2}}h\times {{r}^{2}}$

Now let’s put all the values in above equation
$\begin{align}
  & \Rightarrow I=\dfrac{1}{2}\times 8\times {{10}^{3}}\times 3.14\times {{\left( 0.05 \right)}^{2}}\times 1.5\times {{\left( 0.05 \right)}^{2}} \\
 & \therefore I=0.1178kg/{{m}^{2}} \\
\end{align}$

So the moment of inertia of the cylinder is $0.1178kg/{{m}^{2}}$.

Additional information:
Definition of moment of inertia,
Moment of inertia is defined as the quantity expressed by the body resisting angular acceleration which is the sum of the product of the mass of every particle with its square of a distance from the axis of rotation.
It is depends upon following factors,
$\bullet$ Shape and size of the body
$\bullet$ Axis of rotation
$\bullet$ The density of the material

Note:
When we are calculating mass in which we have to take the height of cylinder h in the form of length of the cylinder so don’t get confused between height and length of the cylinder both are the same phenomena in terms of cylinder dimensions.