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

Step-by-step 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 .