Moment of Inertia of a Disc

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Inertia

The resistance of any physical object to any change in its velocity is known as Inertia. The changes to the object's speed or direction of motion are included in inertia. When no forces act upon them then an aspect of this property is the tendency of objects to keep moving in a straight line at a constant speed.


The meaning of Inertia has derived from the Latin word, iners, meaning idle, sluggish. One of the primary manifestations of mass is Inertia, which is a quantitative property of physical systems. Inertia is defined by Isaac Newton as his first law in his Philosophiæ Naturalis Principia Mathematica, which states:


The innate force of matter or vis insita, is a power of resisting by which every body, as much as in it lies, endeavors to preserve its present state, whether it be of moving uniformly or rest forward in a straight line.


In constant velocity, an object will continue moving at its current velocity position until some other external force causes its speed or direction to change.


On the Earth’s surface, inertia is often masked by effects of friction and gravity and air resistance, both of which tend to decrease the speed of moving objects (commonly to the point of rest). The philosopher Aristotle was misled to believe that objects would move only as long as a force was applied to them.


One of the fundamental principles in classical physics is the principle of inertia that is still used today to describe the motion of objects and how they are affected by the applied forces on them.


Moment of Inertia

The moment of inertia is otherwise known as the moment of the mass of inertia, a quantity that determines the torque needed for a desired angular acceleration about a rotational axis is an angular mass or rotational inertia of a rigid body is similar to how mass determines the force needed for the desired acceleration. It depends on the axis chosen and the body's mass distribution, with larger moments requiring more torque to change the body's rate of rotation.


As an extensive (additive) property: for a point of mass, the moment of inertia is simply the mass times the square of the perpendicular distance to the axis of rotation. The sum of the moments of inertia of its component subsystems (all taken about the same axis) is the moment of inertia of a rigid composite system. The second moment of mass with respect to distance from an axis is its simplest definition.


The moment of inertia about an axis perpendicular to the plane, a scalar value, matters for bodies constrained to rotate in a plane. The moments for bodies which are free to rotate in three dimensions, can be understood by asymmetric 3 × 3 matrices, with a set of we can say mutually perpendicular principal axes for this matrix is diagonal and torques which is a kind of force itself is around the axes act independently of each other.


Moment of Inertia of a Disk

The moment of inertia which is also denoted by the letter “i”, measures the extent to which resistance of an object is rotational acceleration about a particular axis, and is the rotational analog to mass. ML2([mass] × [length]2) is the unit of the dimension of Mass moments of inertia. The second moment of the area should not be confused with, which is used in beam calculations. rotational inertia, and sometimes as the angular mass is often also known as the mass moment of inertia.


Symmetric mass distribution is mainly considered by this article, that unless otherwise specified with constant density throughout the object, and the axis of rotation is taken to be through the center of mass.


Derivation

(image will be uploaded soon)


The mass is distributed all over the x and y plane on a thin disk. Then, we move on to establishing the relation for surface mass density (σ) where it is defined as or said to be the mass per unit surface area. The surface mass density will also be constant, since the disk is uniform, therefore;

σ = m / A

Or

σA = m

So,

dm = σ(dA)


For simplification of the area where it can be assumed that the area to be made of a collection of rings that are mostly thin in nature. the mass increment (dm) of radius r which are at equal distance from the axis is said to be thin rings. Expressing the small area (dA) of every ring by the length (2πr) times the small width of the rings (dr.) It is given as:

\[A = \pi r^{2}\],

\[dA = d(\pi r^{2}) = \pi dr^{2} = 2rdr\]


To get the full area of the disk, we add all the rings from a radius of range 0 to R. The value that is used in the integration of dr is the range of the radius that is given. After putting all these together we get:

\[I = O \int R r^{2} \sigma (\pi r)dr\]

\[I = 2 \pi \sigma O \int R r^{3}dr\]

\[I = 2 \pi \sigma \frac{r^{4}}{4} dR\]

\[I = 2 \pi \sigma (\frac{r^{4}}{4 - 0})\]

\[I = 2 \pi (\frac{m}{4}) (\frac{R^{4}}{4})\]

\[I = 2 \pi (\frac{m}{\pi r^{2}})(\frac{R^{4}}{4})\]

\[I = \frac{1}{2} m R^{2}\]

FAQ (Frequently Asked Questions)

Q1. What are Inertia’s Moment Factors?

Ans: The moment of inertia depends on both the mass of the body and its shape or geometry, as defined by the distance to the axis of rotation.

Q2. What is a Moment of Inertia? Explain with an Example.

Ans: The example of Moment of inertia is if the amount of inertia or resistance to charge is fairly slight in a wheel with an axis in the middle. The small amount of torque on the wheel in the right direction will change its velocity if all the mass is eventually distributed around a pivot point.

Q3. Describe the Moment of Inertia of a Disc.

Ans: Presuming that the moment of inertia of a disc about an axis which is perpendicular to it and through its center to be known it is mr2/2, where m is defined as the mass of the disc, and r is the radius of the disc. Assuming The disc is a planar body.

Q4. When the Moment of Inertia Becomes Important?

Ans: In almost all the physics problems the inertia is important, especially rotational inertia, that involves a mass in the rotation motion. For the calculation of an angular momentum, it is used and allows us to explain how rotational momentum changes when the distribution of mass changes.