Moment of Inertia: Definition, Factors, Equations in Shapes and Examples of Problems
Moment of Inertia: Definition, Factors, Equations in Shapes and Examples of Problems – What is meant by Moment of Inertia?, At this point About the knowledge.co.id will discuss it and of course about other things that also cover it. Let's look at the discussion together in the article below to better understand it.
Moment of Inertia: Definition, Factors, Equations in Shapes and Examples of Problems
The moment of inertia is a measure of an object's resistance to a change in the direction of rotation. Or simply put, the moment of inertia is the tendency of an object or objects to maintain their initial conditions to rotate either in a state of rest or rotating.
As formulated in Newton's 1st law which explains inertia or inertia which states that an object or object will tend to maintain its state unless there is an external force acting on the object the.
The greater the moment of inertia of an object, the more difficult it will be for the object to move. Conversely, if the moment of inertia of an object is small, it will cause the object to move more easily. The unit for the moment of inertia is kg.m2 or slug.ft2
The moment of inertia is one of the phenomena in physics, especially in the field of mechanics. So that we often encounter moments of inertia around us such as the behavior of an object's motion. For example, a ball that was originally stationary, then given a force so that it makes a rotational motion, after a while the ball will stop moving. Based on this phenomenon, it can be seen that an object, in this case a ball, will tend to maintain its initial state (at rest).
In Newton's first law it says "moving objects will tend to move and fixed objects tend to calm." Additionally, inertia is the tendency of objects to maintain a state (just being still or being able to move). Inertia is also called the inertia of an object.
Therefore, Newton's first law, also known as the law of inertia or the law of inertia. For example, objects that are difficult to move are said to have great inertia. Earth's rotation is always in a state that is called having inertial rotation.
The moment or moment of force is the product of the strength of the moment arm. So, the moment of inertia is a measure of the inclination or inertia of an object to rotate on its axis.
Moment of Inertia formula
Look at the picture below:
A particle with mass m is rotating about its axis with radius R. The moment of inertia of the particle point is expressed as the product of the mass of the particle times the distance of the particle to the axis of rotation or the radius. Thus, the moment of inertia is expressed by:
I = m. R2
Information:
I = Moment of Inertia (Kg m2)
m = Particle mass (Kg)
R = Radius of rotation (m)
The moment of inertia is the product of mass multiplied by the square of the distance from the mass to the axis. Systematically, the moment of inertia formula is formulated as follows:
I = Ʃm. R2
I = m1.R12+m2.R22+m3.R32+….+mn.Rn2
The Equation of the Moment of Inertia in Several Forms of Objects
-
Point Objects
For masses of objects or systems of masses in the form of points or points connected by ropes or rods whose mass is negligible, the following applies:
I = Ʃm. R2
Information:
I = Moment of Inertia (Kg m2)
m = Mass (Kg)
R = Distance kr pivot point (m)
-
Homogeneous Bar
Homogeneous rods are rods that have mass spread evenly so that the center of mass is in the middle. For homogeneous rods, it will be clear that there is an influence of the location of the axis of rotation on the moment of inertia.
- The shaft is at the Center
If the axis of rotation is at the center of mass then:
I = 1/12 ml2
Information :
I = moment of inertia (kg m2)
l = rod length (m)
m = mass (kg)
-
The shaft is at one end
-
If the axis of rotation is at one end of the bar, then:
I = 1/3m.l2
Information:
I = moment of inertia (kg m2)
l = rod length (m)
m = mass (kg)
-
Shift Shaft
-
If the rotary axis or shaft feels random or is not at the end or center, then the following applies:
Shift Shaft
I = 1/12 ml2 + m.(k.l)2
Information:
I = moment of inertia (kg m2)
l = rod length (m)
k.l = shift length (m)
m = mass (kg)
The length of the shift in question is how far the axis of rotation is shifted, for example, from the center it is shifted by 1/2l.
-
Cylindrical Shaped Objects
- Solid Cylinder
For objects in the form of solid cylinders such as certain pulleys or wheels, the formula applies:
I = 1/2 m. R2
Information:
I = moment of inertia (kg m2)
R = cylinder radius (m)
m = mass (kg)
- Thin Hollow Cylinder
Thin cylindrical objects are hollow like thin rings, then the formula applies:
I = m. R2
Information:
I = moment of inertia (kg m2)
R = cylinder radius (m)
m = mass (kg)
- Thin Hollow Cylinder
A thick hollow cylinder is a cylinder that has an inner and outer radius. So the formula applies:
I = 1/2 m (R12 + R22)
I = moment of inertia (kg m2)
R1 = radius in cylinder (m)
R2 = outside radius of cylinder (m)
m = mass (kg)
-
Ball Shaped Objects
- Solid Ball
If the object is in the form of a solid ball, then the formula applies:
I = 2/5m. R2
Information:
I = moment of inertia (kg m2)
R = ball radius (m)
m = mass (kg)
- Hollow Ball
The formula that applies to hollow balls is:
I = 2/3m. R2
Information:
I = moment of inertia (kg m2)
R = ball radius (m)
m = mass (kg)
For more details, consider the following moment of inertia formula table:
Factors Affecting Moment of Inertia
There are several factors that affect the moment of inertia of an object, namely:
1. The distance between the object's axis of rotation
2. Position the swivel
3. Mass of objects
4. Shapes (Geometry
Example of a moment of inertia problem
Problem 1
It is known that a homogeneous rod has a mass of 0.6 kg and a length of 60 cm. If a lump of mud with a mass of 20 grams is thrown and sticks to one end of the rod, then determine the moment of inertia of the system through the center of the rod.
Discussion:
I = 1/12m.l2 + mR2
I = 1/12(0.6).(0.6)2 + 0,02(0,3)2
I = 0.018 + 0.0018
I = 0.0198
I = 1.98 x 10-2 kg m2
Question 2
If a solid cylinder with a mass of 2 kg and a radius of 0.1 m is rotated through the axis of the cylinder and a plume mud with a mass of 0.2 kg sticks to a distance of 0.05 meters from the edge of the cylinder, then calculate the moment of inertia system.
Discussion:
I = I cylinder + I mud
I = 1/2 mR2 + m.r2
I = 1/2 (2).(0,1)2 + 0,2. (0,05)2
I = 0.01 + 0.0005
I = 0.0105
I = 1.05 x 10-2 kg m2
Problem 3
A 100 gram ball is connected by a 20 cm long string as shown in the figure. The moment of inertia of the ball about the axis AB is...
Discussion:
The moment of inertia of a ball with mass m = 0.1 kg and a length of string r = 0.2 m is:
I = Mr2
I = (0.1) (0.2)2
I = 4 x 103 kg m2
Problem 4
A system below consists of 3 particles. If M1= 2 kg, m2= 1 kg and m3= 2 kg, determine the moment of inertia of the system if rotated according to:
a) P shaft
b) Q-axis
Discussion:
ip = m1r12 +m2r22+m3r32
IP = 2.02 + 1.12 + 2.22
ip = 9 kg m2
IQ = m1r12 +m2r22+m3r32
IQ = 2.12 + 1.02 + 2.12
IQ = 4 kg m2
Problem 5
A solid rod has a mass of 2 kg and a length of 2 meters. Determine the moment of inertia of the rod if the axis of rotation is in the center of the rod!
Discussion:
The moment of inertia of a solid rod, the axis of rotation is located at the center of the rod
I = 1/12 ml
I = (1/12) (2) (2)2
I = 0.67 kg m2
Problem 6
Determine the moment of inertia of a solid (solid) disc with a mass of 10 kg and a radius of 0.1 meter, if the axis of rotation is at the center of the disc, as shown in the figure!
Discussion:
A solid disc has a moment of inertia
I = 1/2 mr2
I = (1/2) (10) (0.1)2
I = 0.05 kg m2
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