contestada

a solid disk and a solid sphere are released simultaneously at the top of an inclined plane. they roll down without slipping. the two objects have the same mass m, and the same size radius r. the moments of inertia about an axis going through the center of each object are:

Respuesta :

The two objects have the same mass m, and the same size radius r, the moments of inertia about an axis going through the center of each object are   [tex]\frac{1}{2} MR^{2}[/tex]  and  [tex]\frac{2}{5} MR^{2}[/tex] .

How is moment of inertia of sphere calculated ?

We will calculate the moment of inertia of a solid sphere by integrating multiple inertias of the disc.

I = 1/2MR²

[tex]dI = \frac{1}{2} dMR^{2}[/tex]

[tex]dM =[/tex] ρ [tex]dV[/tex]

V = 4/3 πR³

[tex]dV[/tex] = πR²dx

[tex]dM =[/tex] ⍴πR²dx

Putting [tex]dM[/tex] in dI

dI =1/2⍴πR4dx

y² = R² – r²

dI = 1/2⍴π(R² – r²)2dx

Integrating both sides, from 0 to R

I = 8/15⍴πR5

Replacing ⍴ = M/V

⍴ = M/4/3 πR³

Substituting ⍴ in I,

Final Formula: – I = 2/5MR²

What is moment of inertia and how can it be calculated ?

  1. The moment of inertia of an object is a computed measure for a stiff body that is rotating around a fixed axis.
  2. It measures how difficult it would be to modify the rotational speed of an object.
  3. That measurement is derived based on the distribution of mass within the object and the position of the axis.
  4. This means that the same object might have quite varied moment of inertia values depending on where and how the axis of rotation is located.

Can learn more about moment of inertia from https://brainly.com/question/28196763

#SPJ4