Answer

Verified

373.5k+ views

**Hint:**We have to assume the moment of inertia at first about an axis perpendicular to it and through its center. Now take three concurrent axes, and by the theory of perpendicular axes form an equation.

**Complete step by step answer:**

In the above diagram, we see a circular disc and ab is the diameter of the disc and we are asked to find the moment of inertia by one of its diameter.

We know that the instant of inertia of the disc about an axis perpendicular thereto and thru its center to be known; it's $\dfrac{M{{R}^{2}}}{2}$, where M is that the mass of the disc and R is its radius.

The disc is taken into account to be a planar body. Hence the idea of perpendicular axes is applicable thereto .We take three concurrent axes through the center of the disc, O because the x,y,z axes;x and y-axes dwell the plane of the disc and z is perpendicular thereto. By the theory of perpendicular axes,

\[{{I}_{z}}={{I}_{x}}+{{I}_{y}}\]

And

Now, the axes x and y are along two diameters of the disc, and by design, the moment of inertia of the disc is the same at any diameter. Hence

${{I}_{x}}={{I}_{y}}$

${{I}_{z}}=2{{I}_{x}}$

But,

${{I}_{z}}=\dfrac{M{{R}^{2}}}{2}$

On solving,

${{I}_{x}}=\dfrac{{{I}_{z}}}{2}$

${{I}_{x}}=\dfrac{M{{R}^{2}}}{4}$

Therefore the instant of inertia of a disc about any of its diameter is MR$^{2}$ /4.

**Note:**The axes x and y are along two diameters of the disc. Assume the moment of inertia at the right place. The moment of inertia of the disk about any of its diameter will be the same. In order to find the moment of inertia about any axis, we can use the parallel axis theorem or the perpendicular axis theorem.

Recently Updated Pages

Draw a labelled diagram of DC motor class 10 physics CBSE

A rod flies with constant velocity past a mark which class 10 physics CBSE

Why are spaceships provided with heat shields class 10 physics CBSE

What is reflection Write the laws of reflection class 10 physics CBSE

What is the magnetic energy density in terms of standard class 10 physics CBSE

Write any two differences between a binocular and a class 10 physics CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Name 10 Living and Non living things class 9 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE