how to find the moment of inertia about an axis of several particles given the positions and masses of the particles

Moment of inertia is the measure of the rotational inertia of the body. In simple words, the opposition exhibited by the body to the change in the speed of rotation of the body about the axis due to the applied torque (turning force) acting on the body.
The formula for the moment of inertia is the “sum of the product of mass” of each particle with the “square of its distance from the axis of the rotation”. The formula of Moment of Inertia is expressed as I = Σ miri2.
Moment of inertia resists rotational motion and hence called moment of inertia and not moment of force.
The moment of inertia (MOI) is one of the many mass properties that describe an object's stability and the forces needed to alter its motion.

The definition of moment of inertia allows to find the results for the moment of inertia with respect to each axis are:

1-Iₓ = m r²

2-Iy=9m r²

3-Iz=10m r².

The moment of inertia is a scalar quantity is obtained by the expression.

Sol-

I =∫ r² dm

Where I is the moment of inertia, m the mass of the body and r the distance from the axis of rotation.

In the case of point particles, the expression reduces to

I = m r²

In this case we calculate with respect to each coordinate axis.

a) With respect to the x axis the distance is x = r, the moment of inertia is Iₓ = m r²

with respect to the y-axis, indicate that the distance is y = 3 r.

We calculate

I_y = m (3r) ²

I_y = 9 m r²

c) Regarding the z-axis.

We look for the distance.

z =√x^2+y^2

z = √ 1^2+3^2

We calculate the moment of inertia Iz = m (10r²)

Iz = 10 m r²

In conclusion using the definition of moment of inertia we can look for the results of the moment of inertia for each axis are:

a) Iₓ = m r²

b) I_y = 9 m r²

c) I_z = 10 m r²

To learn more about inertia refer to:

https://brainly.com/question/3406242

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