Moment of Inertia of a Solid Cylinder

Moment of inertia is a concept that generally deals with the measurement of rotational inertia, and it also involves a contribution of torque, angular momentum and rotational inertia. This article will also discuss the moment of inertia of a solid cylinder and its derivation.

What Is the Moment of Inertia?

The numerical measurement of the rotational inertia of a body is basically termed the moment of inertia, which means that when torque or turning force is applied to a body, the body resists the angular acceleration to maintain its speed of rotation about a particular axis which may be internal or external or may not be fixed.

However, with respect to that axis, the moment of inertia (I) is calculated by adding the products obtained by multiplying the square of the distance of a given body from the axis and the mass of each particle of that matter.

Kg m² is the SI unit of moment of inertia.

With respect to a chosen axis of rotation, the moment of inertia is generally quantified; it primarily depends on the mass distribution around an axis of rotation. The moment of inertia fluctuates depending on the axis that is chosen.

Welded to a rod AB, the given figure shows two balls and the rod is attached to a bar OQ at C. Ignoring the mass of AB and assuming that, at a distance r from OQ, all particles of the mass m of each ball are concentrated, the moment of inertia can be written as I = 2mr².

The Formula of Moment of Inertia

Moment of Inertia in a general form can be expressed as I = m × r²

were,

m denotes the sum of the product of the mass.

r denotes the distance from the axis of the rotation.

⇒ M¹ L² T⁰ is the dimensional formula for the moment of inertia.

Derivation of the Moment of Inertia of a Solid/Hollow Cylinder

A hollow cylinder in the given figure has

Mass denoted by M

Outer radius denoted by R₂

Inner radius denoted by R₁

Length denoted by L

To derive the moment of inertia of the solid/hollow cylinder about its central axis:

Suppose that the cylinder is cut into infinitely small rings that are very thin, and these rings are centred in the middle.

Let the thickness of each ring be, dr with a total length of L.

Hence, the moment of inertia equation can be written as:

dI = r²dm

Now, since the mass = volume x density, we can find dm,

dm = ρdV

Here, (dV = volume occupied by one ring)

As the Volume = Area x Length, we can find dV that is introduced in the above equation, so:

dV=dAL

Where dA denotes the area of the top of the ring, which can also be written as:

dA = π(r+dr)²–πr²

Where, dA = r + dr (big radius of the ring) subtracted by radius r (smaller radius of the ring). We get:

dA = π(r²+2r dr+(dr)²)–πr²

Note: (dr)² is equal to 0 because when a very infinitely small number is multiplied by another infinitely small number, then the results will be almost equivalent to zero.

dA = 2πrdr

Note: Differentiation is another way we can find dA

Here, A = πr²

Differentiating wrt r, dA=2πrdr

Substituting dA into dV,

dV = 2πrLdr

Substituting the value of dV to find dm,

dm=2ρπrLdr

In conclusion, we have an expression for dm. Substituting that into the equation of dI we get:

dI = 2ρπr³Ldr

Now, we can integrate to find the moment of inertia:

Now, we can find the expression for density.

As we know that:

ρ = M/V

Hence,

ρ = M/ [π (R²₂ – R²₁) L]

Substituting this value into the integrated solution equation, we get:

I = 1/2M(R²₂+R²₁)

Special Cases:

For thin cylindrical shell or Hoop: (R₁=R₂=R)

I=MR²

Solid cylinder or Disc: (R₁=0)

I = 1/2MR²

Watch the video and learn more about the conservation of angular momentum

What Are the Factors on Which Moment of Inertia Depends?

Following are the factors on which the moment of inertia depends:

• Shape and size of the body
• Density of the material
• Distribution of mass relative to the axis (axis of rotation)

What Is the Radius of Gyration?

If a body of mass m is rotated about an axis, then the moment of inertia (I) can be written as:

I = Mk²

Where k is termed as the radius of gyration of the body about the given axis; it signifies that from the given axis of rotation, the radial distance measured where the whole mass of the body is supposed to be concentrated is known as the radius of gyration.

For a solid sphere, the radius of gyration about its axis is given by:

Mk² = 2MR²/5

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