Question

A thin wire of length L and uniform linear mass density p is bent into a circular coil Moment of inertia of the coil about tangential axis in its plane is_______​

Answer 1

Explanation:

Given: A thin wire of length L and uniform linear mass density ρ is bent into a circular coil.

To find the moment of inertia of the coil about the tangential axis in its plane

Solution:

The moment of inertia of the thin wire coil about its diameter

I=

2

1

MR

2

Here M=V×ρ (mass=volume × density)

and we know L=2πR⟹R=

2π

L

(as thin wire of length L is bent into a circular coil)

I=

2

1

MR

2

⟹I=

2

1

Lρ×(

2π

L

)

2

⟹I=

8π

2

L

3

ρ

Now using parallel axis theorem, we get

I

xx

′

=I+MR

2

⟹I

xx

′

=

8π

2

L

3

ρ

+Lρ(

2π

L

)

2

⟹I

xx

′

=

8π

2

L

3

ρ+2L

3

ρ

⟹I

xx

′

=

8π

2

3L

3

ρ

is the moment of inertia of the coil about the tangential axis in its plane8π

2

3L

3

ρ