Determine the rotational inertia of a ring on a turntable system in two ways sym

Question

Determine the rotational inertia of a ring on a turntable system in two ways symbolically; (1) via force and torque analysis, and (2) via mass and dimension measurement using table (10.2) (The pulley is massless)

Given: mass in hanging object, radius of innerside of ring, radius of outerside of ring, radius of turntable spindle (rs), linear acceleration with the ring on the turntable, and linear acceleration without the ring on the turntable.

Having a hard time with this one.

table 10.2

Determine the rotational inertia of a ring on a turntable system in two ways symbolically; (1) via force and torque analysis, and (2) via mass and dimension measurement using table (10.2) (The pulley is massless)

Explanation / Answer

1) Through force and torque analysis:

F = m*g

T = F*rs = m*g*rs

T =I(turntable)*alpha0 ,without the ring

T = (I(turntable)+I(ring))*alpha ,with the ring

alpha is the angular acceleration a = alpha*rs, a0=alpha0*rs

I(t)*alpah0 = (I(t)+I(r))*alpha

I(t) / (I(t)+I(r)) = alpha / alpha0

I(t)/ I(r) = alpha / (alpha0 -alpha)

i(t) / I(r) = a/ (a0-a)

I(r) = I(t) * (a0 -a)/a = I(t) *(a0/a -1)

Where the turntable moment of inertia is I(t) =M*R^2/2

if the mass of the turntable is M and radius of turntable is R.

or you know that

T =I(t)*a0/rs

I(t) = T*rs/a0 = m*g*rs*rs/a0 = m*g*rs^2/a0

Thus

I(r) = m*g*rs^2*(a0-a) / (a0*a)

2) Through computation

If you have z axis perpendicular to the plane of the ring, and x,y axis in the plane of the ring one can write

Iz = integral(r^2*dm) =integral(x^2+y^2)*dm = integral(x^2*dm) +integral(y^2*dm) =Ix+Iy

from symmetry you have Ix =Iy

and therefore

Iz =2Ix=2Iy

Now to determine Iz

Iz =integral (from R to r) (r^2*dm)

dm = sigma*dS =sigma*2*pi*r*dr , here sigma = M/S = M / [pi(R^2-r^2)], M is ring mass

Iz =integral(from R to r) (r^2*sigma*2*pi*r*dr) = (pi*sigma/2) (R^4 -r^4)= (M/2) (R^2+r^2)

Iz =(M/2) (R^2+r^2) r is outer radius, r is inner radious of ring

From tha table 10.2 it is the second option (I for cylinder or ring about central axis)

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