Parallel Axis Theorem Formula Definition Equation Diagram Example

Parallel Axis Theorem: We call the parallel axis theorem as Huygens-Steiner theorem. It is named after two men, Christian Huygens and Jacob Steiner. As per the parallel axis theorem, the moment of inertia of a rigid body about the axis YY in the horizontal direction YY is equal to the moment of inertia about another axis Y’Y passing through the center of mass of the body, plus the total mass of the body and the distance between the two parallel axes and square of a perpendicular area.

Often in physics, objects rotate around an axis perpendicular to their center of mass. Some examples of this are a wheel rotating along a road around its central axis, a planet rotating through space on its central axis, and a fan rotating around its central axis. When analyzing such rotational motion about an object’s center of mass, one can easily see the moment of inertia of most objects of various shapes as they rotate about their center of mass.

Parallel Axis Theorem

As a reminder, an object’s moment of inertia describes how the object’s mass is distributed around a particular axis of rotation: the further the mass is distributed from the axis of rotation, the larger the object’s moment of inertia; The mass is distributed closer to the axis of rotation, the smaller the moment of inertia.

It is essential to know that the larger the moment of inertia of an object, the more difficult it is to accelerate the object rotationally. Sometimes, the situation may be more complicated. One may encounter situations where a rigid body rotates about a new, different axis parallel to the old axis instead of rotating about an axis through its Center of mass. It is essential to know that the larger the moment of inertia of an object, the more difficult it is to accelerate the object rotationally.

Sometimes, the situation may be more complicated. One may encounter situations where a rigid body rotates about a new, different axis parallel to the old axis instead of rotating about an axis through its center of mass.

Parallel Axis Theorem Proof

A simple example is spinning a hula-hoop around one’s finger, which requires the hula-hoop to rotate around its rim instead of its center of mass. What is parallel axis theory? The horizontal axis theory of inertia of an object allows us to describe more easily the rotational dynamics of such motion. The Huygens-Steiner theorem, also known as the Huygens-Steiner theorem or the Steiner theorem, named after Christian Huygens and Jacob Stein, allows a mass to “transform” through its center of mass without evaluation.  It is evaluated about any axis parallel to the axis through the center of mass.

Parallel Axis Theorem Example

The parallel axis theorem example problem was mentioned earlier in this lesson: If the hula-hoop weren’t rotating at its center, it would be rotating on the road, but instead, it would be rotating around its rim. If someone is twirling the hula-hoop around their finger at the edge of the hula-hoop: Is there a sweeping, sweeping motion around the finger?

To correctly analyze the rotational movement of the hula-hoop in this situation, the moment of inertia of the hula-hoop must be calculated perpendicular to its cross section with the hula-hoop and at its edge (not its center). A typical diameter of a hula-hoop is about 75.0 centimeters (cm), and its mass is 0.50 kilograms (kg), and in this situation, the hoop rotates around an axis perpendicular to its cross section and at its edge.

All the information needed to convert the hula-hoop’s moment of inertia from an axis through its center of mass to an axis around its rim is known: the mass of the hoop is known (m = 0.5 kg), is called the distance between the axis through the center of mass of the hoop and the axis at the edge of the hoop (the distance between the circle and its edge is simply the radius of the ring (R=d/2=R= 75.0 cm/2 = 37.5 cm = 0.375 m), and the moment of inertia of the ring rotating about its center of mass is known (ICm=mR2, where R is the radius of the circle). Thus, the hula-hoop revolves around its rim with a new moment of inertia.

Iedge=Icm+mr2=mR2+mr2=(0.5 kg)∗(0.375 m)2+(0.5 kg)∗(0.375 m)2=0.141 kgm2.

Another example is considering an unusually shaped object with a mass m= 2.0 kg. Its moment of inertia is known about its center of mass ICm=0.80Kgm2(Previous laboratory measurements and calculations probably determine this value). Still, it does not rotate around the center of the mass axis. Still, it revolves around a horizontal axis perpendicular to the distance R = 0.25 m from the center of the mass axis. The new moment of inertia (I) can be achieved by using the parallel axis theorem equation about this parallel axis:

I=Icm+mr2=0.80 kgm2+(2.0 kg)∗(0.25 m)2=0.925 kgm2.

Note again that this moment of inertia is greater than the moment of inertia about the object’s center of mass.

Parallel Axis Theorem Formula:

The parallel axis theorem can be expressed as follows:

I = I_c + Mh²

Where,

• I is the moment of inertia of the body
• I_c is the moment of inertia in the center
• M is the mass of the body
• h² is the square of the distance between the two axes 

Parallel axis theorem problems and solutions pdf

The common facts between the parallel and the perpendicular axes theorem arise because both are applied to determine the Moment of Inertia of a rotating body.

So in terms of purpose, they are the same. But they differ in words of the following:

• The parallel axes theorem determines the moment of inertia about any axis in the lamina’s plane. Whereas the perpendicular axes theorem determines the moment of inertia about an axis that is perpendicular (normal) to the plane of the lamina.

• In the parallel axes theorem, for establishing the moment of inertia about any axis in the plane of the lamina, you need data about parallel axes which would pass through the C.G. of the body. Whereas, for the perpendicular axes theorem, you would need data about two mutually perpendicular axes which would meet the axes at which moment of inertia is to be established at a common point such that the three axes can be considered similar to the three cartesian coordinate systems.

Conclusion

Parallel Axis Theorem determines the moment of inertia of a particular rigid body about any axis given the moment of inertia about a horizontal axis by the perpendicular distance between the center of mass of an object and the axes.

Statement:

The moment of inertia on the Z-axis can be represented as:

Iz=Icm + m(r^2)

Where

Icm is the moment of inertia of an object about its center of mass

m is the mass

r is the perpendicular distance between the two axes.