A disc `A` of radius `R` suspended by an elastic threat between two stationary plane `(` figure `)` performs torsional oscillations about its axis `OO^(')` . The moment of inertia of the disc relative to that axis is equal to `I`, the clearance between the disc and each of the planes is equal to `h`, with `h lt lt R`. Find the viscosity of the gas surrounding the disc `A` if the oscilaltion period of the disc equals `T` and the logarithmic damping decrement, `lambda`.

Explanation

From the law of viscosity, force per unit are `=eta(dv)/(dx)`
so when the disc executes torsional oscillations the resistive couple on it is
`=int _(0)^(R)eta. 2pi r . ( r varphi)/( h ). r. d r xx2= ( eta pi R^(4))/(h) dot(varphi)`
`(` factor 2 for the two sides of the disc, see the figure in the blook `)`
where `varphi` is torsion. The equation of motion is
`I ddot( varphi)+(eta pi R^(4))/( h ) dot(varphi)+ c varphi+0`
Comparing with `ddot( varphi) + 2 beta dot(varphi)+ omega_(0)^(2)=0` we get
`beta= eta pi R^(4)//2hI`
Now the logarithmic decrement `lambda` is given by `lambda = beta T , T=` time period
Thus `eta= 2 lambda I//pi R^(4)T`