1. **Stating the problem:** We are given an expression for velocity and asked to find the kinetic energy of the rigid bar (කෂ්ඨයේ ආතීතය). The velocity of the slider Q along the inclined plane is given by $$v = \sqrt{\frac{2mg \sin \alpha}{M + (5 - 4 \cos \alpha) m}}$$ 2. **Formula for kinetic energy:** The kinetic energy (KE) of a rigid body is given by $$KE = \frac{1}{2} M v^2$$ where $M$ is the mass of the body and $v$ is its velocity. 3. **Applying the formula:** Substitute the given velocity expression into the kinetic energy formula: $$KE = \frac{1}{2} M \left(\sqrt{\frac{2mg \sin \alpha}{M + (5 - 4 \cos \alpha) m}}\right)^2$$ 4. **Simplify the expression:** Squaring the square root removes the root: $$KE = \frac{1}{2} M \cdot \frac{2mg \sin \alpha}{M + (5 - 4 \cos \alpha) m}$$ 5. **Further simplification:** $$KE = \frac{M \cdot 2mg \sin \alpha}{2 \left(M + (5 - 4 \cos \alpha) m\right)} = \frac{M mg \sin \alpha}{M + (5 - 4 \cos \alpha) m}$$ 6. **Final answer:** The kinetic energy of the rigid bar is $$\boxed{KE = \frac{M mg \sin \alpha}{M + (5 - 4 \cos \alpha) m}}$$ This expression shows how the kinetic energy depends on the masses $M$, $m$, gravitational acceleration $g$, and the angle $\alpha$.