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Inertia

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Inertia is originated from the Latin word, iners, means idle or lazy. The fundamental principles of inertia helps to describe the theories of classical mechanics.And it gives a correlation between the motion of an object and the applied force.

Generally, the term inertia is referred as the amount of resistance to change in velocity. The concept of inertia is coined by Newton through his laws of motion. His first law of motion is mentioned about the inertia of an object.There are different types of inertia such as thermal inertia, rotational inertia etc.which we will discuss the above sections in detail.


Inertia Definition

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Inertia means that the resistance or impedance of a body to change its velocity. So, inertia is defined as the property of matter which continue its state of rest or uniform motion in a straight line until an unbalanced force is applied to this system. And it is denoted by I. Newton's first law of motion tells about the inertia of an object. Hence, it is also known as the law of inertia.

Inertia Coin Tumbler Experiment

Inertia Equation

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Inertia of a rotating body is known as the moment of inertia. It is defined as the resistance of a rotating body to change its angular velocity about an axis of rotation. In classical mechanics, moment of inertia is also known as rotational inertia, mass moment of inertia etc. The mathematical representation of the moment of inertia is given by the equation:
I = MR2
Where I is the moment of inertia of a rotating body
M is the mass of the body in Kg
R is the distance between the body and axis of rotation in m

Inertia Units

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Moment of inertia is generally represented by the letter I and the SI unit is given by Kgm2. The cgs unit of the moment of inertia is given by gcm2.

Law of Inertia

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The three Newton's laws of motion are the building blocks of the classical mechanics. These laws gives an account of the relation between the object and the forces acting upon it, and the corresponding motions. Newton's first law is basically known as law of inertia. It states that an object will remain at rest or in uniform motion in a straight line unless acted on by an external, unbalanced force.

Uniform motion in a straight line means that the velocity is constant. An object at rest has a constant velocity of zero. An external force is an applied force - that is one applied on or to the object or system.
Inertia Law

Thermal Inertia

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Thermal Inertia

Thermal inertia is the tendency of a material to resist changes in temperature. It is a scalar quantity. It depends upon the volumetric properties of the materials. This dependence on volumetric properties  that makes I as a useful parameters in remote sensing applications. It is a useful element for geological studies also. The mathematical expression for thermal inertia is given by: 
                  $I = (K \rho c)^{\frac{1}{2}}$

Where I is the thermal inertia, k is thermal conductivity, ρ is the bulk density of the material, and c is the specific heat capacity per unit mass.

The unit of thermal inertia is given by J/m2S½K                                 

Inertia and Mass

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As we know, inertia is a natural tendency of an object to remain in a state of rest or in uniform motion in a straight line. There is a linear relation between the mass and inertia. Mass is the measure of inertia. The greater the mass of an object, the greater its inertia and vice versa. So, mass is directly proportional to inertia.
The tendency of an object to resist changes in its state of motion varies with mass. Mass is that quantity that is solely depend upon the inertia of an object. The more inertia that an object has, the more mass that it has. A more massive object has a greater tendency to resist changes in its state of motion.

Mass Figure

Gravity and Inertia

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As we known, gravity is the natural force of attraction exerted by a celestial body, such as Earth, tending to draw them toward the center of the body. So, gravity is defined as the force of attraction between any two bodies of different masses, which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

G = $\frac{Mm}{R^{2}}$

Gravity and Inertia
So we can see that gravity is the force of attraction between two different masses, and inertia is linearly related to mass of an object and represents the resistance to velocity change. This reveals the relationship between the gravity and inertia.

Torque and Inertia

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Torque
Torque or moment of force is defined as the tendency of a force to rotate an object through an axis. Normally, a force is a push or a pull but a torque can cause a twist to an object. Torque is defined mathematically as the cross product of perpendicular distance and force, which produce rotation. Torque is generally represented as τ. The SI unit of torque is N.m (Newton meter). The equation is given by,

τ = r × F

Where
τ is the torque vector
r is the displacement vector
F is the force vector


Both torque and moment of inertia is depend upon the axis of rotation. The moment of inertia depends on the mass of an object as well as how that mass is distributed to the rotational axis. The spinning movement of an object of mass m is placed very near to the axis of rotation is easier than that of the mass is placed far from the rotational axis.

Inertia vs Momentum

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In classical view, momentum or linear momentum is a product of mass and velocity. It is represented by P, and it is a vector quantity, possessing direction as well as magnitude. The mathematical expression is given by,

P = mv
Where P is the linear momentum
m is the mass of an object
v is the velocity of the object

Linear momentum is a conserved quantity, that is, for a closed system which is not affected by external force,then the total momentum is constant. In classical mechanics, using this momentum conservation, Newton formulate the laws of motion. In the case of inertia and momentum, both are directly proportional to the mass of an object. So, we can say that these quantities have linear relationship with the mass of an object.

Linear Momentum

Product of Inertia

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Product of inertia is related to two rectangular axes, the sum of the products formed by multiplying the mass of each element of a figure by the product of the coordinates corresponding to those axes.

Products of inertia of a rigid body are defined as,
Ixy = ∫ (xy) dm,
Ixz = ∫ (xz) dm,
Iyz = ∫ (yz) dm

Product of Inertia
The figure illustrate the above equations of products of inertia.

Rotational Inertia

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Rotational inertia is also known as the moment of inertia, and it is defined as the tendency of a rotating body to rotate until acted upon by an external force or it is a quantity which express a rotating body's tendency to resist  the angular acceleration.

Rotational Inertia

Rotational Inertia Formula

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Rotational inertia is generally represented as I. The equation for rotational inertia is given by,

I = MR2
where I is the rotational inertia
M is the mass of a rotating body
R is the distance between the body and axis of rotation

The SI unit of rotational inertia is Kg.m2

Inertia Examples

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Some of the examples of inertia are given below:
  1. Body movement of a person who is sitting inside a car when it makes a turn.
  2. A ball rolling down from a hill continue its motion until any external force stops it.
  3. If a car is moving forward it will continue to move forward unless the brakes interfere with its movement.
  4. When a car is suddenly accelerated, drivers and passengers may feel as their bodies are moving backward.
  5. If a bus suddenly stops and you aren't holding onto a support, you will be pushed to the front of the bus.

Inertia Problems

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The problems related to the inertia are given below:

Solved Examples

Question 1: Calculate the moment of inertia of a propeller with three blades with mass m, length l at 1200 relative to each other?

Solution:
 
We know that moment of inertia of a single rod rotating around its end is $\frac{1}{3}$ ml2.
If there are three blades rotating around the same axis and in the same plane, the moment of inertia is just three times than $\frac{1}{3}$ ml2.
So, I = ml2

 

Question 2: The moment of inertia of a straight wire and that of a circular frame are L1 and L2 respectively. If the composition of the wires are same and both lengths are equal, calculate the ratio of the moment if inertia of these two, that is I1 and I2 .

Solution:
 
The moment of Inertia of the straight wire is given by,
I1$\frac{ML^{2}}{12}$
 The moment of inertia of the circular frame is given by,
 I2 = MR2
From the question, it is given that the length and composition of the wires are same.so,
L = 2 $\pi$ R
There fore, R = $\frac{L}{2 \pi}$
Substitute the value of R in I2. We get
l2 = $\frac{ML^{2}}{4\pi ^{2}}$
So, $\frac{l_{1}}{l_{2}}$ = $\frac{ML^{2}}{12}$ $\times$ $\frac{4 \pi ^{2}}{ML^{2}}$
$\frac{l_{1}}{l_{2}}$ = $\frac{\pi ^{2}}{3}$

 

Question 3: Calculate the radius of the ring, where the radius of the sphere of same moment of inertia is R.

Solution:
 
Let assume that the radius of sphere is Rs and radius of ring is Rr. And the moment of inertia of the sphere and ring be Is, Ir respectively.
The moment of inertia of the solid sphere is given by,
Is = $\frac{2}{5}$ $MR_{s}^{2}$
Moment of inertia of ring is,
Ir = MRr2
According to the question, Is = Ir
So,
MRr2= $\frac{2}{5}$ $MR_{s}^{2}$
Rr2= $\frac{2}{5}$ $R_{s}^{2}$
Rr = $\sqrt{\frac{2}{5}}R_{s}$