C

Step 1: Identify the parameters.
The inclined plane has angular dimensions of 3-4-5, meaning the height (h) is 3x, the base (b) is 4x, and the hypotenuse is 5x (where x is a scaling factor). The angle of inclination, θ, can be calculated as:
$$ an heta = \frac{h}{b} = \frac{3}{4}$$
Therefore, using trigonometric functions:
$$\sin \theta = \frac{h}{5x} = \frac{3}{5}$$, $$\cos \theta = \frac{b}{5x} = \frac{4}{5}$$
Step 2: Calculate the forces acting on the block.
The gravitational force acting on the block is $$mg$$, which can be broken into two components: parallel (down the incline) and perpendicular (normal to the incline).
- Perpendicular component: $$F_{N} = mg \cos \theta$$
- Parallel component (down the incline): $$F_{g} = mg \sin \theta$$
Step 3: Consider the frictional force that opposes the motion. The maximum static friction force is given by:
$$F_{f} = \mu_{s} \cdot N = \mu_{s} \cdot mg \cos \theta$$
Step 4: Set up the equations for forces. For the block to not slide down, the frictional force must be greater than or equal to the gravitational component pulling it down the incline, hence:
$$F_{f} \geq F_{g} \implies \mu_{s} (mg \cos \theta) \geq mg \sin \theta$$
This simplifies to:
$$\mu_{s} \geq \tan \theta$$
Step 5: Substitute values. Given that $$\mu_{s} = \frac{1}{4}$$, find the angle:
$$\mu_{s} = \frac{1}{4} \geq \tan \theta$$. The maximum threshold satisfies the condition for static friction.
Step 6: Calculate the minimum centrifugal force required to balance gravity for circular motion. The centripetal force (provided by static friction) is equal to:
$$F_{c} = m \omega^2 r$$ where $$r = 0.4 m$$
Thus:
$$\mu_{s} mg \cos \theta = m \omega^2 r$$
Canceling out $$m$$ and solving for $$\omega$$ gives:
$$\omega = \sqrt{\frac{\mu_{s} g \cos \theta}{r}}$$
Use $$g = 9.8 \, m/s^2$$, $$\theta = \tan^{-1}(\frac{3}{4})$$ leading to $$\cos \theta = \frac{4}{5}$$.
Substitute:
$$\omega = \sqrt{\frac{(1/4)(9.8)(4/5)}{0.4}} = \sqrt{\frac{9.84}{8}} = \sqrt{1.23} \approx 1.11 \, rad/s$$
Therefore, the answer is approximately:
$$\omega \, \approx 1.11 rad/s$$ which aligns closely with option C.