Question

A thin rod of mass 3kg and length 3m is bent in the form of an equilateral triangle the moment of inertia of the triangle about an axis passing through it's center of mass and perpendicular to its plane would be ?

Answer 1

Answer:

Predict. And Write what happen if Artificial fibre not existed in our Life

Answer 2

Answer:

                    =0.045kgm²

Explanation:

Let us take the general case of a bar of mass m and length L.

When you convert it into an equilateral triangle, the centre of mass lies at the centroid, and each arm has mass m/3 and length L/3.

The plan is to calculate the MOI of one arm, and multiply it by 3, simple asthat.

We will use the integral technique here. Let us take a small length dx at length x from the centre of an arm. Let's take distance of centre of the arm from the centroid to be h.

Now,

the distance l between dx and centroid O is√x²+h²

Mass of that element= m/Ldx

Thus MOI of the small element is-  m/L l²dx

And for the full arm-

∫L/6−L/6 m/L(x²+h²)dₓ

After solving which you get,

I=mL²/162 after putting h=L/6√3

Since MOI of full rod is 3I

Thus, final answer is mL²/54

Plugging in the values of mass and length, we get-  0.045kgm²