Moment of Inertia
A Moment of Inertia is a physical measure that is quantifies the torque needed for an object to change its rotation.
- AKA: Mass Moment of Inertia, Angular Mass, Rotational Inertia.
- Context:
- It can be defined for an object rotating around the z-axis as: $I=\displaystyle \sum m_i\left(x_i^2+y_i^2\right)=\int \left(x^2+y^2\right) dm$
where $\left(x_i^2+y_i^2\right)$ and $\left(x^2+y^2\right)$ are the square of its distance from the z-axis; $m_i$ and $dm$ are mass elements; and $I$ is the object's Inertia.
- It can be defined for an object rotating around the z-axis as:
- Example(s):
- The moment of inertia of a thin rod, length $L$ and mass $M$ is: $\dfrac{ML^2}{12}$
- The moment of inertia of a thin concentric circular ring, mass $M$, radii $r_1$ and $r_2$ is: $\dfrac{M \left(r_1^2+r_2^2\right)}{2}$
- The moment of inertia of a sphere, mass $M$ and radius $r$ is: $\dfrac{2MR^2}{5}$
- Counter-Example(s):
- See: Inertia, Rigid Body, Torque, Angular Acceleration, Mass, Force, Acceleration.
References
2020
- (Wikipedia, 2020) ⇒ https://en.wikipedia.org/wiki/Moment_of_inertia Retrieved:2020-11-29.
- The moment of inertia, otherwise known as the mass moment of inertia, angular mass or rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis; similar to how mass determines the force needed for a desired acceleration. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rate of rotation.
It is an extensive (additive) property: for a point mass the moment of inertia is simply the mass times the square of the perpendicular distance to the axis of rotation. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis). Its simplest definition is the second moment of mass with respect to distance from an axis.
For bodies constrained to rotate in a plane, only their moment of inertia about an axis perpendicular to the plane, a scalar value, matters. For bodies free to rotate in three dimensions, their moments can be described by a symmetric 3 × 3 matrix, with a set of mutually perpendicular principal axes for which this matrix is diagonal and torques around the axes act independently of each other.
- The moment of inertia, otherwise known as the mass moment of inertia, angular mass or rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis; similar to how mass determines the force needed for a desired acceleration. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rate of rotation.
1965
- (Feynman, 1965) ⇒ Richard Feynman (1965) "Chapter 9: Center of Mass; Moment of Inertia". In: Feynman Lectures On Physics. Volume I.
- QUOTE: To summarize, the moment of inertia of an object about a given axis, which we shall call the z-axis, has the following properties:
- 1. The moment of inertia is $I_z\displaystyle \sum m_i\left(x_i^2+y_i^2\right)=\int \left(x^2+y^2\right) dm$
- 1. The moment of inertia is
- 2. If the object is made of a number of parts, each of whose moment of inertia is known, the total moment of inertia is the sum of the moments of inertia of the pieces.
- 3. The moment of inertia about any given axis is equal to the moment of inertia about a parallel axis through the CM plus the total mass times the square of the distance from the axis to the CM.
- 4. If the object is a plane figure, the moment of inertia about an axis perpendicular to the plane is equal to the sum of the moments of inertia about any two mutually perpendicular axes lying in the plane and intersecting at the perpendicular axis.
2020
- (Wikipedia, 2020) ⇒ https://en.wikipedia.org/wiki/Moment_of_inertia Retrieved:2020-11-29.
- {{Infobox physical quantity
| name = Moment of inertia
| image = Маховик.jpg
| caption = Flywheels have large moments of inertia to smooth out rotational motion. This example is in a Russian museum.
| unit = kg m2
| otherunits = lbf·ft·s2
| symbols = I
| baseunits =
| dimension = M L2
| derivations = [math]\displaystyle{ I = \frac{L}{\omega} }[/math] }}
The moment of inertia, otherwise known as the mass moment of inertia, angular mass or rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis; similar to how mass determines the force needed for a desired acceleration. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rate of rotation.
It is an extensive (additive) property: for a point mass the moment of inertia is simply the mass times the square of the perpendicular distance to the axis of rotation. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis). Its simplest definition is the second moment of mass with respect to distance from an axis.
For bodies constrained to rotate in a plane, only their moment of inertia about an axis perpendicular to the plane, a scalar value, matters. For bodies free to rotate in three dimensions, their moments can be described by a symmetric 3 × 3 matrix, with a set of mutually perpendicular principal axes for which this matrix is diagonal and torques around the axes act independently of each other.
- {{Infobox physical quantity