Moment of Inertia of a Solid Cylinder: Formula and Derivation

The moment of inertia of a rigid body is a quantity that determines the torque required for a desired angular acceleration about a rotational axis.

• It is also known as the mass moment of inertia, angular mass, second moment of mass, or rotational inertia.
• It is determined by the mass distribution of the body and the axis chosen, with larger moments requiring more torque to change the body's rate of rotation by a given amount.
• The moment of inertia of a point mass is simply the mass multiplied by the square of the perpendicular distance to the axis of rotation. 
• The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all measured about the same axis).
• The SI unit of moment of inertia is kg m²

Key Terms: Moment of inertia, Radius of gyration, Mass, Axis of rotation, Angular momentum, Density, Rigid body, Solid cylinder

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Moment of Inertia

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Moment of inertia is a quantitive measure of the rotational inertia of a body i.e. the opposition that the body exhibits to having its speed of rotation about an axis altered by the application of torque. 

• The axis of the rotation of the body may be internal, external, or may or may not be fixed.
• The moment of inertia is however always specified with respect to that axis.
• For the calculation of angular momentum, torque, and kinetic energy of a rotating body concept of moment of inertia is very helpful.
• As the mass of the body is the measure of its in linear motion, the moment of inertia about a given axis of rotation resists a change in its rotational motion.

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Moment of Inertia Formula

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The moment of inertia of a body about a given axis of rotation is defined as the sum of the product of masses of the constituting particles and the square of their distances from the axis of rotation.

Let n particles of a rigid body of masses m₁, m₂, m₃, …….., m_n located at distances r1, r₂, r₃ ,......r_n respectively from the axis of rotation, then the moment of inertia of the whole body is given by

I = m₁r₁² + m₂r₂² + m₃r₃² + ……. + m_nr_n²

\(I = \sum_{i=1}^{i=n} m_ir_i^2\)

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Derivation of Moment of Inertia for a Solid Cylinder

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Consider a solid cylinder of mass M and length L. Let R be the radius of the cylinder.

Moment of Inertia of a Solid Cylinder

The volume of the cylinder, V = πR²L

The moment of inertia for a solid cylinder can be calculated in two ways

1. About its own axis
2. About an axis passing through its center and perpendicular to its own axis.

Moment of Inertia for a solid cylinder about its own axis

Mass per unit volume of the cylinder = \(\frac{M}{\pi R^2L}\)

Let this solid cylinder be made up of a large number of co-axial hollow cylinders.

Consider such a cylinder of radius x, thickness dx, and length L.

The volume of the co-axial cylinder, V'=(x+dx)²L

⇒ V' = 2πxdxL

Mass of the co-axial cylinder, M’ = Mass per unit volume of the cylinder x volume of the co-axial cylinder

⇒ M' = \(\frac{M}{\pi R^2L}\) x 2πxdxL

⇒ M' = \(\frac{2M}{R^2}xdx\)

The moment of inertia of the co-axial cylinder about its axis OO’ is given by

I’ = Mass of the co-axial cylinder x (Distance from the axis of rotation)²

⇒ I' = \(\frac{2M}{R^2}xdx\) x x²

⇒ I' = \(\frac{2M}{R^2} x^3dx\)

The moment of inertia of the whole cylinder is given by

\(I = \int _{x=0}^{x=R} \frac{2M}{R^2}x^3dx\)

⇒ I =\( \frac{2M}{R^2} [ \frac{x^4}{4}]_0^R = \frac{M}{2R^2} R^4\)

⇒ I = \(\frac{1}{2}\)MR²

Hence the moment of inertia of a solid cylinder about its own axis is, 

I = \(\frac{1}{2}\) MR²

Moment of Inertia of a solid cylinder about an axis passing through its center and perpendicular to its own axis

Let the solid cylinder of mass M and length L be made up of a large number of discs.

Consider such a disc of width dx.

Mass per unit length of the cylinder = \(\frac{M}{L}\)

Mass of the small disc = \(\frac{M}{L}\) dx

We know, that the moment of inertia of a disc about its diameter (A’B’) is

I_D = (1/4) x mass of the disc x (radius of the disc)²

I_D = \(\frac{M^4}{L}\)R²dx

Using the parallel axes theorem, the moment of inertia of the disc about the AB axis is given by

dI = I_D + (mass of the disc x distance between AB and A’B’)²

⇒ \(dI = \frac{M^4}{L} R^2dx + \frac{M}{L} x^2dx\)

The moment of inertia of the whole cylinder about the AB axis is given by

⇒ \(I = \int _{x=-L/2}^{x=L/2} [\frac{M}{4L} R^2dx + \frac{M}{L} x^2dx]\)

⇒ \(I = \frac{MR^2}{4L} \int _{x=-L/2}^{x=L/2} dx + \frac{M}{L} \int _{x=-L/2}^{x=L/2} x^2dx\)

After solving the above equation, we get

⇒ \(I = \frac{M}{4} [R^2 + \frac{L^2}{3}]\)

Hence the moment of Inertia of a solid cylinder about an axis passing through its center and perpendicular to its own axis is

\(I = \frac{M}{4} [R^2 + \frac{L^2}{3}]\)

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Factors Affecting Moment of Inertia

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The following are the factors on which the moment of inertia of a body depends

• Mass of the body: A body of heavier mass has a larger value of the moment of inertia. Hence, a higher amount of torque is required to change the rotation of the body.
• Shape and Size of the Body: The rotational axis of a body affects the moment of inertia. The rotating axis of a body is determined by its size and shape.
• Axis of Rotation (Distribution of Mass Relative to the Axis): When the mass on one side is heavier, the rotational axis of the body is closer to it, and the moment of inertia on that side requires a greater torque to change the motion.
• Position and Orientation of the Axis of Rotation concerning the Body: Changing the point at which the axis of rotation passes through the body, that is, changing its location, would likewise affect the body's moment of inertia.

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Radius of Gyration

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The distance of a point in a body from the axis of rotation, at which if the whole of the mass of the body were supposed to be concentrated, its moment of inertia about the axis of rotation will be the same as that determined by the actual distribution of the body is called Radius of Gyration.

Suppose the whole mass M of a body is concentrated at a point whose distance from the axis of rotation is k, then the moment of inertia of the body is given by

I = Mk²

Where k is known as the radius of gyration. Its SI unit is a meter (m).

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Things to Remember

• Moment of inertia is a quantitative measure of the rotational inertia of a body.
• The moment of inertia plays the same role in rotational motion as mass does in linear motion.
• The SI unit of moment of inertia is kg m²
• For a body of mass M having distance R from the axis of rotation, the moment of inertia is given as I = MR²
• The moment of inertia of a solid cylinder about its own axis is, I = \(\frac{1}{2}\) MR²
• The moment of Inertia of a solid cylinder about an axis passing through its center and perpendicular to its own axis is \(I = \frac{M}{4} [R^2 + \frac{L^2}{3}]\)

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