Four small beads

PHYSICS · Multi-part · CBSE · Physics · System of Particles and Rotational Motion · System of Particles and Rotational Motion

Problem

Four small beads, each of mass

m

, are to be fixed on a thin light rod of negligible mass that can rotate about a vertical axis through its midpoint

O

. Consider the following two configurations: Configuration I: The beads are at distances

a

2a

3a

and

4a

from

O

on the same side of the axis. Configuration II: Two beads are kept at

a

and

3a

on one side of

O

, and the other two at

2a

and

4a

on the opposite side of

O

. Answer the questions below. (i) Show that the moment of inertia for each configuration can be written in the form

kma^{2}

, stating the value of

k

for both configurations. (ii) Prove that the two configurations possess the same moment of inertia about the axis through

O

. (iii) If one wishes to design a new configuration with the same total mass

4m

and the same set of allowable distances

\{a,2a,3a,4a\}

but a smaller moment of inertia than the above two, state a possible arrangement and justify your answer with calculation.