Question

This problem describes an experimental method for determining the moment of inertia of an irregular shaped object such as the payload for a satellite. A counter weight of mass m, is suspended by a cord wound around a spool of radius r, forming part of a turntable supporting the object. The turntable can rotate w/o friction. When the counterweight is released from rest it descends through a distance h, acquiring a speed v. Show that the moment of inertia I of the rotating apparatus is mr2 (2gh/v2 – 1).

Answer 1

Answer:

Explanation:

 The turntable that is rotating apparatus whose moment of inertia ( I )  is to be calculated is used as a spool of radius r. m mass is going down with acceleration a and rotating the apparatus with the help of cord with tension T  wound around it

For motion of weight m

mg - T = ma

and for the motion of spool or apparatus

Tr = I a/r

T = Ia/r²

mg - Ia/r²  = ma

mg   =  Ia/r² + ma

a (  I/r² + m ) = mg

When  the counterweight m  is released from rest it descends through a distance h, acquiring a speed v

v² = 2ah

a = v² / 2h

v² / 2h  (  I/r² + m ) = mg

(  I/r² + m ) = 2mgh / v²

 I/r²  = 2mgh / v²-  m

I = r² (2mgh / v²-  m )

= mr² ( 2gh/ v² -1 )

Proved .