Question

a square of mass m is made of thin wire with sides of length a. Calculate the rotational inertia about z-axis​

Answer 1

Given:

A square of mass m is made of thin wire with sides of length a.

To find:

Rotational Inertia (Moment of Inertia) about the Z axis.

Calculation:

Let us consider that a square plate of mass m is placed along the XY plane , such that the geometric axis is along Z axis.

Now , for any rectangle of sides x and y, the moment of inertia can be considered as the net rotational inertia due to expansion of 2 wires of length x and y along the respective axes.

Now, refer to the diagram:

$\therefore \: MI = (MI)_{x} + (MI)_{y}$

$\implies\: MI = \dfrac{m {x}^{2} }{12} + \dfrac{m {y}^{2} }{12}$

For a square , x = y = a :

$\implies\: MI = \dfrac{m {a}^{2} }{12} + \dfrac{m {a}^{2} }{12}$

$\implies\: MI = 2 \times \dfrac{m {a}^{2} }{12}$

$\implies\: MI = \dfrac{m {a}^{2} }{6}$

So, rotational inertia along Z axis is ma²/6.