1. Given data

• Initial intensity from source: $I_0 = 256\ \text{W/m}^2$
• Three polaroids are placed successively.
• Axis of $P_1$ is the reference.
• Axis of $P_2$ makes $60^\circ$ with $P_1$.
• Axis of $P_3$ makes $90^\circ$ with $P_1$.

So the angle between successive polaroids is:

• Between $P_1$ and $P_2$: $60^\circ$
• Between $P_2$ and $P_3$: $90^\circ - 60^\circ = 30^\circ$

2. Intensity after first polaroid $P_1$

Since the source is ordinary unpolarized light, after passing through the first polaroid, intensity becomes half: $I_1 = \frac{I_0}{2} = \frac{256}{2} = 128\ \text{W/m}^2$

3. Intensity after second polaroid $P_2$

Apply Malus' law: $I_2 = I_1 \cos^2 60^\circ$ Since $\cos 60^\circ = \frac{1}{2}$ we get $I_2 = 128 \times \left(\frac{1}{2}\right)^2 = 128 \times \frac{1}{4} = 32\ \text{W/m}^2$

4. Intensity after third polaroid $P_3$

The angle between $P_2$ and $P_3$ is $30^\circ$, so again by Malus' law, $I_3 = I_2 \cos^2 30^\circ$ Now, $\cos 30^\circ = \frac{\sqrt{3}}{2}$ so $\cos^2 30^\circ = \frac{3}{4}$ Therefore, $I_3 = 32 \times \frac{3}{4} = 24\ \text{W/m}^2$

5. Final answer

The intensity at point $O$ is $\boxed{24\ \text{W/m}^2}$