A rigid body in the X-Y plane consists of two point masses (1 kg each) attached to the ends of two massless rods, each of 1 cm length, as shown in the figure. It rotates at 30 RPM counter-clockwise about the Z-axis passing through point O. A point mass of √2 kg, attached to one end of a third massless rod, is used for balancing the body by attaching the free end of the rod to point O. The length of the third rod is ________ cm.

The correct option is 1.

To dynamically or statically balance a rotating rigid body in a single plane (the X-Y plane) about a fixed point O, the net centrifugal force must be zero. This condition is met when the vector sum of the product of each mass and its corresponding position vector from the center of rotation O equals zero:
$$\sum m_i{\mathbf{r}}_i=0$$

From the provided diagram, we identify the parameters of the original system:
• Mass 1: $m_1=1\text{ kg}$ at a radius of $r_1=1\text{ cm}$
• Mass 2: $m_2=1\text{ kg}$ at a radius of $r_2=1\text{ cm}$
• The angle between the two massless rods is ${90}^{\circ }$, meaning their position vectors are mutually perpendicular.

The magnitude of the resultant unbalance moment $R$ of the two masses is calculated using the vector addition of orthogonal components:
$$R=\sqrt{{\left(m_1{r}_1\right)}^2+{\left(m_2{r}_2\right)}^2}$$

Substituting the given values:
$$R=\sqrt{{\left(1\times 1\right)}^2+{\left(1\times 1\right)}^2}=\sqrt{1+1}=\sqrt{2}\text{ kg⋅cm}$$

To achieve complete balance, a third mass $m_b=\sqrt{2}\text{ kg}$ must be attached at a distance $r_b$ in the direction diametrically opposite to the resultant vector, satisfying:
$$m_b{r}_b=R$$

Solving for $r_b$:
$$\sqrt{2}\times r_b=\sqrt{2}$$

Dividing both sides by $\sqrt{2}$ gives:
$$r_b=1\text{ cm}$$

Thus, the required length of the third rod is 1 cm.