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If a disc has moment of inertia I about the axis which is tangential and in plane of the disc. then the moment of inertia about the axis which is tangential and perpendicular to its plane will be

If a disc has moment of inertia I about the axis which is tangential and in plane of the disc. then the moment of inertia about the axis which is tangential and perpendicular to its plane will be Correct Answer <span class="math-tex">\(\dfrac{6}{5} I\)</span>

Concept:

Moment of Inertia

  • Moment of inertia is analogous of mass in the rotational motion. 
  • A body with more mass will have more tendency to remain in its initial position. Similarly, a body with a higher moment of inertia will resist change in it is rotational motion. 
  • The moment of inertia of a single particle is given as I = MR2 (m is mass, r is the distance from the center of the axis. 
  • The moment of inertia of a system of particles is given as 

 

  • Radius of Gyration: The effective distance between the axis and the whole mass of a rigid body is called its radius of Gyration. 
  • The moment of inertia of a disc along center is and along perpendicular is 

Parallel Axis Theorem

  • The moment of inertia of a body about any axis is equal to the sum of the moment of inertia of the body about a parallel axis passing through its center of mass and the product of its mass and the square of the distance between the two parallel axes.

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  • By the Parallel Axis Theorem, the moment of inertia along the tangent of the above ring is 

Itangent = Idiameter + MR2

  • That means, the sum of moment of inertia of ring along its axis, and the MR2, m is mass of ring, R is the distance from center to the tangent. 

Perpendicular Axis Theorem

  • The moment of inertia of a planar body (lamina) about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes concurrent with perpendicular axis and lying in the plane of the body.

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It simply means the moment of inertia of the body in XY plane, along the perpendicular Z-axis is a sum of the moment of inertia along the X-axis and Y-axis. 

Iz = Ix + Iy 

Calculation:

Given the moment of inertia of the disc.

 

Now, the moment of inertia at the tangent of the disc will be given by parallel axis theorem as 

 

 -- (1)

Now, the moment of inertia perpendicular is given as

 -- (2)

Putting (1) in (2)

 

So, the correct option is 

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