How Do You Calculate the Rotational Inertia of a Hollow Cylinder?

[Oomair]

Homework Statement

A pipe made of aluminum with a density of 2.7 g/cm3 is a right cylinder 43 cm long whose outer diameter is 4.8 cm and whose inner diameter is 2.4 cm. What is the rotational inertia about the central axis of the pipe? Note that the rotational inertia of the thick cylinder can be expressed as the rotational inertia for a solid cylinder of radius R2 minus the rotational inertia of the solid cylinder of radius R1.

Homework Equations

D= m/v I= mr^2 for a solid cylinder volume of a cylinder = piR^2H

The Attempt at a Solution

what i basically did is find the mass of the two cylinders

D=m/v outer cylinder = 2.7g/cm^2 = m/v and inner cylinder = 2.7g/cm^2 = m/v

mass of the outer cylinder comes out to be 2100.89g and inner cylinder comes out to be 525.22g

then i put them into the moment of inertia equation I= mr^2

((2100.89g)(2.4cm)^2) - ((525.2g)(1.2cm)^2) = 11344.81 g*cm^2, then i convert it into kg*m^2 and i get .001134 kg*m^2, anything wrong I am doing here?

 

[alphysicist]

Hi Oomair,

The moment of inertia of a solid cylinder about its central axis is:

$$I=\frac{1}{2}MR^2$$

 

[Moetasim]

Dear student,

Your approach to finding the moment of inertia is correct. However, there are a few errors in your calculations.

First, when finding the mass of the outer and inner cylinders, you should use the volume formula for a cylinder, which is V = πr^2h. So the mass of the outer cylinder should be (2.7 g/cm^3)(π(2.4 cm)^2(43 cm)) = 2031.24 g, and the mass of the inner cylinder should be (2.7 g/cm^3)(π(1.2 cm)^2(43 cm)) = 507.81 g.

Next, when plugging these values into the moment of inertia equation, you should use the radius of the entire cylinder, not just the inner or outer radius. So the moment of inertia should be ((2031.24 g)(2.4 cm)^2) - ((507.81 g)(2.4 cm)^2) = 17072.86 g*cm^2. Converting to kg*m^2, we get 0.001707 kg*m^2.

So your final answer for the rotational inertia of the pipe should be 0.001707 kg*m^2.

Hope this helps clarify your approach! Keep up the good work.