Question

9. An arc making 120° at the centre
of ring of mass m and radius r is
cut from the ring. The arc is made
to rotate about z-axis perpen-
dicular to its plane and passing
through the centre of the ring. The
moment of inertia of the arc about
the z-axis :
​

Answer 1

Explanation:

Let mass dm is varies for small angle dθ

If for 2π angle mass is M then for small θ small mass is

dm=

2π

M

dθ

We know that moment of inertia of ring is mr

2

For disc

moment of inertia =∫dmr

2

Substitute value of dm in above equation

=∫

2π

M

r

2

dθ

Integrating from 0 to 120

∘

=∫

0

6

4π

2π

M

r

2

dθ

=

2π

M

r

2

∫

0

6

4π

dθ

=

2π

M

r

2

[

6

4π

−0]

=

3

Mr

2

Moment of inertia is

3

Mr

2