Moment of Inertia: Applications, Solved Examples

Moment of Inertia: Have students ever thought about why the door handles are attached far away from the hinged point? Have you ever experienced that it is easier to move the wrench if we hold it at the farthest position? What is the role of the moment of inertia in this phenomenon? Why is it easier to rotate the hammer if we rotate it about the head than if we rotate it about the tail? Is it the same as mass? The moment of inertia formula is important for students.

The moment of inertia is defined as the quantity reflected by the body resisting angular acceleration, which is the sum of the product of each particle’s mass and its square of the distance from the axis of rotation. In simpler terms, it is a number that determines the amount of torque required for a certain angular acceleration in a rotating axis. The angular mass or rotational inertia are other names for the moment of inertia. Continue reading the article to know more details like the moment of inertia formula and more.

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What is the Moment of Inertia Formula?

The moment of inertia means the moment of mass with respect to an axis. In practical scenarios, we deal with both translation and rotational motion. The role of the moment of inertia in rotational motion is analogous to the role played by the mass in translational motion.

The moment of inertia depends on the mass distribution and axis with respect to which we are calculating the moment of inertia; a body having a larger mass can have less moment of inertia than a body with a lower mass.

Rigid Body

In practical scenarios, most of the time, we do not deal with particles. Instead, we deal with continuous bodies which have some volume and occupy space. A Rigid body can be considered to be a collection of infinite numbers of particles. Rigid bodies are non-deformable; thus, the distance between any two constituent particles always remain the same; that is, if we mark any two points on the rigid body, then regardless of the orientation of the rigid body, the separation between the two points will not change.

Example: A block of iron.

Moment of Inertia Formula of Particles

The moment of inertia is defined for the system of particles, and it depends on the mass and the distance from the axis of rotation.

The moment of inertia of a simple particle is given by,

\({\rm{I}}\,{\rm{ = }}\,{\rm{M}}{{\rm{r}}^2}\)

Where,

\({\rm{M}}\) is the mass of the particle.

\({\rm{r}}\) is the distance of the rigid body with respect to the axis.

Moment of Inertia of Rigid Bodies

The rigid body is the collection of an infinite number of particles. Therefore, we integrate the differential element for the entire rigid body to get the value of the moment of inertia.

\({\rm{I}} = \int {{{\rm{r}}^{^2}}{\rm{dm}}} \)

Where,

\({\rm{r}}\) is the distance of the rigid body with respect to the axis.

\({{\rm{dm}}}\) is the differential mass element

Moment of Inertia of Some Common Figures

A point mass

\({\rm{m}}{{\rm{r}}^2}\)

A circular ring with respect to the axis about its centre.

\({\rm{m}}{{\rm{r}}^2}\)

A uniform disc with respect to the axis about its centre.

\(\frac{1}{2}{\rm{m}}{{\rm{r}}^2}\)

A Uniform log rod with respect to the axis about its centre.

\(\frac{1}{{12}}{\rm{m}}{{\rm{r}}^2}\)

A Uniform log rod with respect to the axis about its end.

\(\frac{1}{3}{\rm{m}}{{\rm{r}}^2}\)

A uniform sphere with respect to the axis about its centre.

\(\frac{2}{5}{\rm{m}}{{\rm{r}}^2}\)

A uniform rectangular plate with respect to the axis about its centre.

\(\frac{1}{{12}}{\rm{m(}}{{\rm{l}}^2} + {{\rm{b}}^2})\)

It is to be noted that the moment of inertia depends on the distribution of mass around the axes; thus, the figure with similar distribution has a similar expression for the moment of inertia.

Parallel Axis Theorem

The parallel axis theorem helps us to determine the moment of inertia easily about an axis passing through a point other than the centre of mass.

According to this theorem,

Moment of inertia about an axis which is at the distance of \(‘{\rm{d}}’\) from the centre of mass is given by,

\({{\rm{I}}_{\rm{O}}} = {{\rm{I}}_{{\rm{cm}}}} + \,{\rm{M}}{{\rm{d}}^2}\)

Where,

\({{\rm{I}}_{\rm{O}}}\) is the moment of inertia about the axis at the distance of \(‘{\rm{d}}’\) from the centre of mass.
\({{\rm{I}}_{{\rm{cm}}}}\) is the moment of inertial about the  centre of mass of the body.

\({\rm{M}}\) is the mass of the rigid body.

\(d\) is the distance of the axis from the centre of mass.

It is important to note that the axes must parallel each other. If the axes are not parallel to each other, then the parallel axes theorem is not applicable. Hence the name parallel axis theorem.

From the parallel axis theorem, we can say that the moment of inertia is minimum about an axis passing through the centre of mass.

Perpendicular Axis Theorem

Perpendicular axis theorem makes it easier to determine the moment of inertia of a planar object. This theorem is only valid for planar objects.

According to this theorem, the moment of inertia about the axis perpendicular to the plane of the body is equal to the sum of the moment of inertia about any two axes perpendicular to each other in the plane of the body.

\({{\rm{I}}_{\rm{z}}} = {{\rm{I}}_{\rm{x}}} + {{\rm{I}}_{\rm{y}}}\)

Where,

\({{\rm{I}}_{\rm{z}}}\) is the moment of inertia about the axis perpendicular to the plane of the body.

\({{\rm{I}}_{\rm{x}}}\;{\rm{and }}\;{{\rm{I}}_{\rm{y}}}\) are the two axes perpendicular to each other in the plane of the body.

Radius of Gyration

The radius of gyration of a body about any axis is defined as the effective distance from this axis where the whole mass of the body is assumed to be concentrated so that the moment of inertia is the same as the initial.

The moment of inertia in  terms of the radius of gyration is given by,

\({\rm{I}}\, = {\rm{M}}{{\rm{k}}^2}\)

Where,

\({\rm{M}}\) is the mass of the body, and \({\rm{k}}\) is the radius of gyration.

Thus, the radius of gyration will depend upon the mass distribution and axis of rotation.

Application of Moment of Inertia

The moment of inertia in rotation dynamics and statics is analogous to the mass in linear statics and dynamics.

Moment of inertia is used to write the torque equations in analysing the dynamics of a rigid body.

\({\rm{\tau }}\, = {\rm{I\alpha }}\)

Where,

\({\rm{\tau }}\,\) is the net torque in the body.

\({\rm{\alpha }}\) is the angular acceleration of the body.

\({\rm{I}}\) is the moment of inertia of the body.

Sugarcane crusher employs the use of wheels that have more moment of inertia so as to increase angular momentum and easily crush the sugarcane.

Solved Examples on Moment of Inertia

Q.1. A thin wire of length \({\rm{l}}\) and mass \({\rm{m}}\) is bent in the form of a semicircle. Its moment of inertia about an axis joining its ends will be:

For semicircle, we know that,
\({\rm{\pi r}}\, = \,{\rm{l}}\)
Where,
\({\rm{r}}\) is the radius of the semicircle.
\(\therefore {\rm{r}} = \frac{1}{{\rm{\pi }}}\)
Using perpendicular axis theorem, we have,
\({{\rm{I}}_{{\rm{CM}}}}\, = \,{\rm{2}}{{\rm{I}}_{\rm{d}}}\)
As the mass distribution of the circular ring or a semicircle or a circular quadrant is same about the axes.
\({{\rm{I}}_{{\rm{CM}}}}\, = \,{\rm{2}}{{\rm{I}}_{\rm{d}}}\)
where \({{\rm{I}}_{{\rm{CM}}}}\,\) is moment of inertia about its centre of mass and perpendicular to its plane and \({{\rm{I}}_{\rm{d}}}\) is moment of inertia about its diameter.

Further solving, we get,
\({\rm{m}}{{\rm{r}}^2} = 2{{\rm{I}}_{\rm{d}}} \Rightarrow {{\rm{I}}_{\rm{d}}} = \frac{1}{2}{\rm{m}}{{\rm{r}}^2}\)
Substituting the value of \(‘\,{\rm{r}}’\), we get,
\({{\rm{I}}_{\rm{d}}}\, = \,\frac{{{\rm{m}}{{\rm{r}}^2}}}{2} = \frac{{\rm{m}}}{2}{\left( {\frac{{\rm{l}}}{{\rm{\pi }}}} \right)^2} = \frac{{{\rm{m}}{{\rm{l}}^2}}}{{2{{\rm{\pi }}^2}}}\)

Q.2. The moment of inertia of a disc about its own axis is \({\rm{I}}\).Its moment of inertia about a tangential axis in the plane will be.

The moment of inertia of disc about its own axis is given as ,
\({\rm{I}}\,{\rm{ = }}\,\frac{1}{2}{\rm{M}}{{\rm{R}}^2}\)
Where,
\({\rm{M}}\) is mass of disc, and \({\rm{R}}\) is its radius.
Using perpendicular axis theorem, we get,
\( \Rightarrow {\rm{I’}}\, = \,{\rm{I}}’ = \frac{1}{2}{\rm{M}}{{\rm{R}}^2}\)
\( \Rightarrow {\rm{I}}’ = \frac{1}{4}{\rm{M}}{{\rm{R}}^2}\)
Where,
\({\rm{I}}’\) is the moment of inertia about an axis parallel to the plane and passing through its centre.
Moment of inertia of disc about a tangent in a plane is given by using parallel axis theorem.
\({\rm{I}}”\, = \,{\rm{I}}’ + {\rm{M}}{{\rm{R}}^2}\)
\( \Rightarrow \frac{1}{4}{\rm{M}}{{\rm{R}}^2} + {\rm{M}}{{\rm{R}}^2} = \frac{5}{4}{\rm{M}}{{\rm{R}}^2}\)
On substituting the value of \({\rm{M}}{{\rm{R}}^2}\) we get
\( \Rightarrow {{\rm{I}}^{”}} = \frac{5}{2}{\rm{I}}\)

Summary

The role played by the mass in translational motion is analogous to the role of the moment of inertia in rotational motion. The moment of inertia depends on the mass distribution and axis with respect to which we are calculating the moment of inertia. A body having a larger mass can have fewer moments of inertia than a body with a lower mass. Moment of inertia about an axis which is at the distance of ‘d’ from the centre of mass is given by,

\({{\rm{I}}_{\rm{O}}} = {{\rm{I}}_{{\rm{cm}}}}\, + \,{\rm{M}}{{\rm{d}}^2}\)

According to the Perpendicular axis theorem, the moment of inertia about the axis perpendicular to the plane of the body is equal to the sum of moment of inertia about any two axes perpendicular to each other in the plane of the body.

\({{\rm{I}}_{\rm{z}}} = {{\rm{I}}_{\rm{x}}}\, + \,{{\rm{I}}_{\rm{y}}}\)

FAQs on Moment of Inertia

Q.1: What is meant by the moment of a quantity?
Ans: The moment of a physical quantity can be defined as the effectiveness of the particular quantity with respect to a point or axis. For example, the effect of force on a point or axis is known as the moment of force or torque. The effect or the effectiveness of the mass or inertia about a point or axis is known as the moment of inertia.

Q.2: Can a body with less mass have a greater moment of inertia than a body with more mass?
Ans: The moment of inertia depends on the mass distribution and axis with respect to which we are calculating the moment of inertia; therefore, it is possible to have a larger mass that can have less moment of inertia than a body with a lower mass.

Q.3: Why does the expression for moment of inertia for a point mass and a circular ring about an axis passing through the centre perpendicular to the ring be the same?
Ans: The moment of inertia depends on the mass distribution and axis with respect to which we are calculating the moment of inertia, and the circular ring and a particle have the same mass distribution, that is, the mass is concentrated at some distance; thus, they have the same expression for moment of inertia.

Q.4: What is the radius of gyration?
Ans: The radius of gyration of a body about any axis is defined as the effective distance from this axis where the whole mass of the body can be assumed to be concentrated so that the moment of inertia is the same as the initial.

Q.5: Can a body with mass and some volume have zero moments of inertia?
Ans: No, a body has the minimum moment of inertia if the axis passes through the centre of mass, thus for rigid bodies, the moment of inertia cannot be zero, but for particles, if the axis passes through it, then the moment of inertia will be zero about that axis.

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