Question

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 $\sqrt{2$ 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.}

• A. 1
• B. $\sqrt{2}$
• C. $1/\sqrt{2}$
• D. $1/(2\sqrt{2})$

Hint:

Dynamic balancing of rotating systems requires the vector sum of all $m r$ terms to be zero. Always treat unbalance as a vector quantity.

Answer 1

The Correct Option is A

Since the system rotates about O, the rotating masses must be dynamically balanced. Dynamic balance requires the vector sum of the rotating mass moments to be zero: \[ \sum m_i r_i = 0. \]
Step 1: Identify the mass positions.
Two masses of 1 kg each are located at:
- One at 1 cm along the positive X-axis
- One at 1 cm rotated by 90° (i.e., along the positive Y-axis)
Thus their position vectors are: \[ \mathbf{r}_1 = 1\,\hat{i}, \qquad \mathbf{r}_2 = 1\,\hat{j}. \]
Step 2: Compute their combined unbalance vector.
\[ \mathbf{U} = 1(1\,\hat{i}) + 1(1\,\hat{j}) = \hat{i} + \hat{j}. \]
This vector has magnitude: \[ |\mathbf{U}| = \sqrt{1^2 + 1^2} = \sqrt{2}. \]
Step 3: Balance using a third mass of $\sqrt{2$ kg.}
For perfect balance, \[ m_3 r_3 = |\mathbf{U}|. \] Given: \[ m_3 = \sqrt{2}. \]
Thus, \[ \sqrt{2}\, r_3 = \sqrt{2} \Rightarrow r_3 = 1 \text{ cm}. \]
Therefore, the length of the third rod must be 1 cm.
Final Answer: 1 cm