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Rotational Inertia
Storyboard

If an object is not acted upon, it will tend to maintain its current state, which corresponds to the angular velocity being constant.

The phenomenon is called inertia and gives rise to Newton's first principle in its version for rotation and generalizes the idea by defining that objects tend to keep the angular momentum constant which in the case of the moment of constant inertia is reduced to constant angular velocity.

The principle also leads to the fact that if the moment of inertia varies and the angular momentum is constant, the angular velocity will also be reversed.

>Model

ID:(1455, 0)


Rotational inertia
Definition

If we consider an object with a moment of inertia $I$ and angular velocity $\omega,"$ we can observe that there are two situations where changing its motion is more challenging:

• When its moment of inertia is very large (for example, trying to stop a carousel).
• When its angular velocity is very high (for example, trying to stop the shaft of a motor).

This is why a measure of motion is introduced that involves the body, which is the product of moment of inertia and angular velocity, known as the body's angular momentum.

In ballet, one can see how the dancer applies Newton's first principle for rotation in all their spins:

Bailarina Alina Cojocaru

ID:(10284, 0)


Angular momentum, right-hand rule
Image

The orientation of angular momentum can be determined using the right-hand rule: if you point your fingers in the direction of the radius and rotate in the direction of the moment,

ID:(11601, 0)


Rotational Inertia
Description

If an object is not acted upon, it will tend to maintain its current state, which corresponds to the angular velocity being constant. The phenomenon is called inertia and gives rise to Newton's first principle in its version for rotation and generalizes the idea by defining that objects tend to keep the angular momentum constant which in the case of the moment of constant inertia is reduced to constant angular velocity. The principle also leads to the fact that if the moment of inertia varies and the angular momentum is constant, the angular velocity will also be reversed.

Variables
Symbol
Text
Variable
Value
Units
Calculate
MKS Value
MKS Units
$L$
L
Angular Momentum
kg m^2/s
$\omega$
omega
Angular Speed
rad/s
$\Delta\omega$
Domega
Difference in Angular Speeds
rad/s
$L_0$
L_0
Initial Angular Momentum
kg m^2/s
$\omega_0$
omega_0
Initial Angular Speed
rad/s
$I_0$
I_0
Initial moment of inertia
kg m^2
$I$
I
Moment of Inertia
kg m^2
$\Delta I$
DI
Variation of the moment of inertia
kg m^2

Calculations

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Equation
Solved
Translated

Calculations

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Equations

Examples


(ID 15837)

If we consider an object with a moment of inertia $I$ and angular velocity $\omega,"$ we can observe that there are two situations where changing its motion is more challenging:

• When its moment of inertia is very large (for example, trying to stop a carousel).
• When its angular velocity is very high (for example, trying to stop the shaft of a motor).

This is why a measure of motion is introduced that involves the body, which is the product of moment of inertia and angular velocity, known as the body's angular momentum.

In ballet, one can see how the dancer applies Newton's first principle for rotation in all their spins:

Bailarina Alina Cojocaru

(ID 10284)


(ID 15834)

If the angular momentum is constant, then the angular Momentum ($L$) must be equal to the initial Angular Momentum ($L_0$), which implies that:

$ L = L_0 $

(ID 15841)

The moment ($p$) was defined as the product of the inertial Mass ($m_i$) and the speed ($v$), which is equal to:

$ p = m_i v $



The analogue of the speed ($v$) in the case of rotation is the instantaneous Angular Speed ($\omega$), therefore, the equivalent of the moment ($p$) should be a the angular Momentum ($L$) of the form:

$ L = I \omega $

.

the inertial Mass ($m_i$) is associated with the inertia in the translation of a body, so the moment of Inertia ($I$) corresponds to the inertia in the rotation of a body.

(ID 3251)

The moment ($p$) was defined as the product of the inertial Mass ($m_i$) and the speed ($v$), which is equal to:

$ p = m_i v $



The analogue of the speed ($v$) in the case of rotation is the instantaneous Angular Speed ($\omega$), therefore, the equivalent of the moment ($p$) should be a the angular Momentum ($L$) of the form:

$ L = I \omega $

.

the inertial Mass ($m_i$) is associated with the inertia in the translation of a body, so the moment of Inertia ($I$) corresponds to the inertia in the rotation of a body.

(ID 3251)

Acceleration is defined as the change in angular velocity per unit of time.

Therefore, the angular acceleration the difference in Angular Speeds ($\Delta\omega$) can be expressed in terms of the angular velocity the angular Speed ($\omega$) and time the initial Angular Speed ($\omega_0$) as follows:

$ \Delta\omega = \omega_2 - \omega_1 $

(ID 3681)

If the shape of the body changes during rotation, its moment of inertia will also change. Therefore, it makes sense to define the variation of the moment of inertia ($\Delta I$) by subtracting the value of the initial moment of inertia ($I_0$) from the moment of Inertia ($I$) as follows:

$ \Delta I = I - I_0 $

(ID 15842)

ID:(1455, 0)