Moment of Inertia of a Disc

Imagine a merry-go-round spinning around its central axis. Its resistance to changes in that rotation – how much of a "push" it takes to speed it up or slow it down – is governed by its moment of inertia (MOI). It quantifies an object's "rotational inertia" around a specific axis.

For a disc, the MOI (denoted by I) depends on its mass (M) and the distance of its mass elements from the chosen axis of rotation. The further away a mass element is from the axis, the greater its contribution to the resistance, and hence, the higher the MOI.

Inertia is a way to understand how movement, time, and energy are connected. When things are in motion, they can also be still, either naturally or by resisting the movement. Resistance occurs when electric charges face opposition in their flow. Now, let's briefly explore the moment of inertia and moment of inertia of a disc to learn more.

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 word "inertia" comes from the Latin word "iners," which means idle or sluggish. Isaac Newton defined inertia in his first law, stating that objects resist changes to their state of motion.

In simple terms, if an object is moving at a steady speed, it will continue to do so unless a force makes it speed up, slow down, or change direction. On Earth, inertia is affected by factors like friction, gravity, and air resistance, which can make objects appear to slow down or come to a stop. The principle of inertia is a key concept in physics, explaining how objects move and respond to forces.

Moment of Inertia

In simpler terms, the moment of inertia is a measure of how difficult it is to make an object rotate around a certain axis. It depends on the distribution of mass in the object and the axis chosen. Larger moments of inertia mean it takes more force (torque) to change how fast the object is rotating.

For a single point of mass, the moment of inertia is just the mass multiplied by the square of the distance to the rotation axis. When dealing with a system of objects rotating together, you can add up the moments of inertia of each part to find the total moment of inertia.

In simpler situations, like objects rotating in a plane, the moment of inertia is a scalar value. But for objects free to rotate in three dimensions, we use matrices and consider axes around which torques act independently. This is important for understanding the motion of objects in physics, particularly in exams like JEE Main.

Factors Affecting Moment of Inertia

The basic factors affecting the moment of inertia are the following:

• Density - The density of the material affects the moment of inertia. 

• Size of the material

• The shape of the material

• Axis rotation - A line of points as to which a rigid body rotates is known as axis of rotation. 

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. 

\[ML^2\] (mass×length2) 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: 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 is to be made of a collection of rings that are mostly thin in nature. The mass increment (dm) of radius r which are at an 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:

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

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

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

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

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

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

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

While memorising formulas is crucial, understanding their derivation strengthens your grasp. We can derive the MOI of a disc using integration, considering it as a collection of thin rings. This process not only clarifies the concept but also empowers you to tackle similar problems involving other shapes.

Moment of Inertia of a Circular Disc for Different Situations

Solid Disk:

• Imagine spinning the disc like a coin on a table. The rotation happens around the center of the disc.

• Formula: $\dfrac{1}{2}MR^2$

Axis at Rim:

• Picture the disc spinning, but this time it's rotating around the outer edge.

• When the axis passes through the rim of the disc, the MOI increases significantly due to the larger distances involved.

• Formula: $\dfrac{3}{2}MR^2$

Disk with a Hole:

• If there's a hole in the middle of the disc and it spins around its center.

• If the disc has a central hole, the MOI depends on the radii of both the disc and the hole. We need to consider the mass distribution differently in this case.

• Formula: $\dfrac{1}{2}M(a^2 + b^2)$

Area of the Moment of Inertia

It is also known as the second area moment. It displays the dispersion of points in a random axis. Its property is of a two-dimensional plane. 

Polar Moment of Inertia

The polar moment of inertia measures an objects’ extent to resist when torque is applied on a specific axis. Torsion is the twisting of the body with a force apple on it. 

The polar moment puts forth the resistance shown by a cylindrical object towards the torsion when applied in a cross-sectional area that is vertical to the central axis of the object. The magnitude of the polar moment of inertia is directly proportional to the torsional resistance of the thing. 

There are three types of cross-sectional moments.

• Hollow-cylinder shaft

• Solid-cylinder shaft

• Thin-walled shaft

The limitations of the Polar moment of inertia include its inability to measure shafts and beams which are not cylindrical cross-sectional.

More about the moment of inertia and its; various concepts are available in the Vedantu learning platform. 

Differentiation between Moment of Inertia and Polar Moment of Inertia

Mass Moment of Inertia

Also known as rotational inertia, the mass moment of inertia is the volume that measures the objects’ resistance when a change occurs in angular momentum. 

Factors affecting mass moment of Inertia are:

• Angular acceleration

• Kinesthetic energy-storing

When a torsional force is applied to the body, its mass moment of inertia will be inversely proportional to angular acceleration.

Some important things about the mass moment of inertia are:

The body mass and its location depend on the mass moment of inertia.

The larger the mess moment of inertia becomes, the farther the rotational axis from its’ mass is. 

The MOI can be mathematically represented as:

\[I=mr^{2}\]

Applications of Moment of Inertia:

The MOI of a disc plays a vital role in several applications, including:

• Rotational Dynamics: Calculating the torque required to accelerate or decelerate the disc, understanding its rotational kinetic energy, and analyzing its angular momentum.

• Equilibrium and Stability: Determining the stability of discs in various configurations, like a spinning gyroscope.

• Mechanical Design: Optimizing the design of rotating machinery involving discs, such as wheels, gears, and turbines.

Key Points to Remember:

1. The formula, I = ½ * M * R², applies only for a disc rotating about its central axis.

2. The MOI increases with both mass and radius squared.

3. The concept of parallel and perpendicular axes can be applied to find the MOI about other axes (e.g., diameters).

4. Practice diverse problems involving discs and various axes of rotation to solidify your understanding.