# A torque of $400\,Nm$ acting on a body of mass $40\,Kg$ produces an angular acceleration of $20\,rad\,{\sec ^{ - 2}}$. What is the moment of inertia of the body? And find the radius of gyration of the body.

Answer

Verified

280.2k+ views

**Hint:**When a body is in translational motion it moves with a linear acceleration while body rotates due to torque acting on it it produce an angular acceleration and the relation between torque and moment of inertia is related as $\vec \tau = I\vec \alpha $.

**Complete step by step answer:**

Let us assume that $I$ is the moment of inertia of the body and it’s given that:

The magnitude of angular acceleration of the body is $\alpha = 20\,rad\,{\sec ^{ - 2}}$.

The magnitude of torque acting on the body is $\tau = 400Nm$.

The mass of the given body is $m = 40\,Kg$.

Now, using the relation $\vec \tau = I\vec \alpha $ we get,

$400 = I \times 20$

$\therefore I = 20\,Kg{m^2}$

So, the moment of inertia of the body is $I = 20\,Kg{m^2}$.

Now, as we know that the general formula of Moment of inertia is written as,

$I = M{K^2}$

where $K$ denotes the radius of gyration

Now we have, the magnitude of moment of inertia is $I = 20\,Kg{m^2}$

Mass of the body is $m = 40\,Kg$

Putting these value in equation $I = M{K^2}$

We get,

$20 = 40 \times {K^2}$

$\Rightarrow K = \dfrac{1}{{\sqrt 2 }}$

$\therefore K = 0.707\,m$

So, the radius of gyration of the body is $K = 0.707m$

**Hence, the moment of inertia of the body is $I = 20\,Kg\,{m^2}$ and the radius of gyration of the body is $K = 0.707\,m$.**

**Note:**It should be remembered that, the radius of gyration of the body is the distance from the body to the axis of rotation which have same moment of inertia if whole mass of body considered to be act at that particular point and produce same moment of inertia with particular distance, this radius is called radius of gyration.

Recently Updated Pages

Basicity of sulphurous acid and sulphuric acid are

Define absolute refractive index of a medium

Which of the following would not be a valid reason class 11 biology CBSE

Why should electric field lines never cross each other class 12 physics CBSE

An electrostatic field line is a continuous curve That class 12 physics CBSE

What is meant by monosporic development of female class 11 biology CBSE

Trending doubts

What is 1 divided by 0 class 8 maths CBSE

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

What is the past tense of read class 10 english CBSE

What is pollution? How many types of pollution? Define it

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

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

How many crores make 10 million class 7 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

How fast is 60 miles per hour in kilometres per ho class 10 maths CBSE