A disk of radius R1 is carved out of a disk of radius R?. The surface mass densi

Question

A disk of radius R1 is carved out of a disk of radius R?. The surface mass density (kg/m2) of the structure is p. Find the location of center of mass Find the moment of inertia Ix, Iy, and Iz, If the structure in the figure is hung on a wall using a pin going through the center, what is the net gravitational torque about an axis going through the center? Which direction it will rotate? If the structure rotates about the center point, which point will reach the maximum linear velocity during this motion? Calculate this maximum velocity.

Explanation / Answer

We can assume that this disc is made up of two disc one of radius R2 and mass density '' and other of radius R1 and mass density '-' and centre at x = a .......... think about this

now mass of the larger disc: M = R22

and mass of smaller disc: m = -R12 .................. note that its negative(hypothetical)

now,

a)YCM = 0 ............. by symmetry

XCM = (M*0 +m*a)/(M+m) = -aR12/(R22-R12) ..............(1)

(lets assign 'r' to be equal to -XCM (=magnitude of XCM))

b)moment of inertia of larger disc: I2 = MR22/2

moment of inertia of smaller disc: I1 = mR12/2 + m a2 ......... using parallel axis theorem

so moment of inertia of whole system:

I = I2 + I1 = (/2)(R24 -R14 -2R12 a2)

c) torque = (M+m)g(XCM)(-k^)....... (k^ = unit vector in z direction)

or, torque = g(R22-R12)(-aR12/(R22 - R12))= -gaR12(k^)

therefore it will rotate anticlockwise

d) K.E. is going to be maximum when P.E. is going to be minimum as there sum is going to be constant by energy conservation.

i.e. KE is going to be maximum when CM is directly below the origin i.e.

YCM = -r ..... see statement below equation (1) for value of r

also as this is a rigid body point which is farthest from the axis of rotation is going to move with highest velocity (as remains constant for all points in object) so, the point that is going to move with fastest velocity is actually going to be the whole rim of the object as opposed to the intutional answer: leftmost point of the body.

conserving mechanical energy:

(M+m)gr =(0.5)I2

from this equation we get and the fastest velocity is given by R2

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