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Torsion is responsible for causing rotational energy inside a body. In this context, the formula of moment of inertia or polar moment of inertia are two aspects that are brought up occasionally. Both of these terms are quantities that showcase the tendency of the body to resist any change after a certain amount of torque is applied.
The polar moment of inertia focuses on calculating the ability of the body to stay strong against the torsional force that the torque applies. On the other hand, a moment of interest primarily focuses on angular acceleration resistance. Read to know the difference in better detail.
Moment of Inertia- Brief Overview
In studying this phenomenon, the moment of inertia is also called angular mass or mass moment of inertia. It refers to the body’s resistance when it is subjected to angular acceleration specifically. The unit of moment of inertia and the equation depend on the object’s mass and its placement related to the rotating axis.
Moment of inertia primarily showcases how the mass of an object is distributed from the main rotating axis. The formula of moment of inertia reads like this.
𝙸 = ∫ r².dm
For the measurement, you can use the following equation
𝙸 = ∑Mi.ri2
Here, ‘r’ stands for the distance between the rotational axis and the mass, while ‘M’ represents each object’s mass.
Polar Moment of Inertia- Brief Overview
The meaning of polar moment of inertia relates to doing a cross-section of the main object on the rotational axis. This process makes it possible to avoid the deformation that the torsional loading causes.
The measurement of the value of the polar moment of inertia involves calculating the degree of the axis that stands perpendicular to the cross-section plane. This axis moves through the cross-sections centre.
The main formula that the polar moment of inertia holds is related to the z-axis and is written as follows.
J = ∫ r².dA
The equation for the calculation involves the following formula.
J= Ix + Iy
Here, ‘Ix’ stands for the value of the moment of inertia related to the x-axis, while the ‘Iy’ refers to the value about the y-axis.
Representation of Polar Moment of Inertia vs Moment of Inertia
The following is a basic representation of the main points of difference between Moment of Inertia and Polar Moment of Inertia in tabular format.
Polar Moment of Inertia | Mass Moment of Inertia | |
What it Means | This is a calculation of the ability of one object to resist high levels of torsion when applied to the object under one specific axis. | This is a type of quantity that checks and represents the tendency of an object to withstand angular acceleration. |
Units | The main unit of measurement here is m⁴. | The main unit of measurement here is Kg.m². |
Mathematical Formula | The formula of polar moment of inertia is represented as J = ∫r².dA. | The formula of moment of inertia is shown as 𝙸 = ∫r².dm. |
Differences Between Polar Moment of Inertia and Mass Moment of Inertia
There are notable differences between the two concepts of Moment of Inertia and Polar Moment of Inertia. Here are the main points of concern that differ for each option.
Point of resistance
Both the formula of moment of inertia and polar moment of inertia relates to the resistance that the object offers. On the other hand, the moment of inertia or angular mass relates mainly to angular acceleration. So, as mentioned, this quantification relates to how the object resists the modifications in angular acceleration.
On the other hand, the polar moment of inertia focuses mainly on how the object resists torsional deformation. The equation calculates and showcases the degree of rotational energy exerted by torsion and the object resistance power.
Points of focus for the calculation
To measure the polar moment of inertia, it is important to measure the main region of the cross-section that occurs in the object. In contrast, the unit of moment of inertia is based on the location of the object’s mass on the rotational axis and the full object mass amount.
Equation for the calculation
The mathematical equations for calculating the two quantities are different as well. For example, the formula of the moment of inertia stands as 𝙸 = ∫r².dm, while the formula of polar moment of inertia calculation is J = ∫r².dA.
Axis distribution
The main thing that is represented as the mass moment of inertia is the mass distribution quantity from the rotational axis. On the other hand, the polar moment of inertia showcases the area distribution value from the axis used for the reference, e.g., the z-axis.
Dimensional formula
The dimensional formula of moment of inertia stands as [L²M¹T⁰], while the same formula for the polar moment of inertia appears as [L⁴M⁰T⁰].
Measurement units
The polar moment of inertia is measured with m⁴. For the moment of inertia’s values, the main unit of measurement is Kg.m².
Requirements
Calculating the moment of inertia is important to conduct the design of objects like gears, flywheels, gears, and rotary machines. On the other hand, the polar moment of inertia is used to calculate and handle rotating component designs for objects like bolts, coupling, and shafts.
Conclusion
Both moment of inertia and polar moment of inertia are crucial points to consider related to understanding an object’s reaction to torsion. The former focuses on mainly angular acceleration resistance, while the latter measures the torsional deformation resistance.
The formula of moment of inertia and polar moment of inertia differ, including other points like the measurement, calculation, and requirements. Therefore, it is important to understand both separately to use the right equation for the correct type of measurement.
