Mass moment of Inertia

 

This time, let's look at the 'Mass moment of inertia'.

 

Unlike the 'Geometrical moment of inertia',

Mass is used instead of Geometrical for the 'Mass moment of Inertia'.

 

Mass is about the inertia of an object, that is,

the property of trying to sustain a movement around mass.

And when 'moment' is attached to it, it shows the property of

continuing the rotational motion of the center of mass.

 

'Rat Fire Play' The larger the mass moment of inertia, the more difficult it is to turn, but the more difficult it is to stop. Reference Wikipedia

 

When an external force 'F' is generated that rotates an object

with a mass of 'm' based on the rotational axis as shown in Fig. 1 below.

 

Fig.1

 

 

According to Newton's law, acceleration occurs

in the tangential direction as shown in Equation 1 below.

 

Eq.1 External force

 

The moment for micro-mass due to external force is as shown in Equation 2 below.

 

Eq.2 Moment

 

The acceleration (a) is the product of the turning radius (r) and

the angular acceleration (α).

 

Eq.3

 

Substituting Equation.3 to Equation.1, Equation.4 is

 

Eq.4

 

Substituting into Equation.2 again, it is as shown in Equation.5 below.

 

Eq.5

 

 

Integrating Equation 5 to find the total moment.

 

Eq.6

 

Here, 'I' is called the mass 'Moment of inertia'.

 

Eq.7 Mass moment of inertia

 

As mentioned earlier,

 

The 'Mass moment of inertia' is inertia in the rotational motion of the center of mass.

In other words, it is a property that resists rotational motion.

---

 

The object in Fig. 1 above can be changed to

an equivalent model in the form of Fig. 2 below.

 

Fig.2

 

That is, if an object with the same mass is placed

at an appropriate distance (d), an equivalent model

with the mass moment of inertia as shown in Fig. 1 can be obtained.

 

Assuming the two models have the same mass moment of inertia,

 

Eq.8

 

The summary of 'd' is as in Eq. 9.

 

Fig.9 rotaional radius

 

In this case, 'd' is referred to as the rotaional radius.

 

When rotating an object based on an arbitrary position,

if only the mass is considered, the radius of rotation

can be obtained to have the same 'Mass moment of inertia'.

 

---

In addition, the 'parallel axis theorem' applies equally

to the 'Mass moment of inertia' as for the 'Geometrical moment of inertia'.

 

Fig.3

 

Eq.10 parallel axis theoerm

 

Also, the relationship between the 'Polar moment of inertia' and

the 'Right angle moment of inertia' is also the same

as that mentioned in the 'Geometrical moment of inertia'.

 

Eq.11

 

Thank you. ^^