The ratio of the acceleration for a solid sphere (mass $$m$$ and radius $$R$$) rolling down an incline of angle '$$\theta$$' without slipping down the incline without rolling is.

A particle of mass $m$ is projected with a velocity $$v$$ making an angle of $$45^{\circ}$$ with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height $$h$$ is.

A uniform cube of side $a$ and mass $m$ rests on a rough horizontal table. A horizontal force $F$ is applied normal to one of the faces at a point that is directly above the centre of the faces at a point that is directly above the centre of the face at a height 3a/4 above the base. The minimum value of F for which the cube begins to topple an edge is (assume that cube does not slide)

Consider a rod of mass $$M$$ and length $$L$$ pivoted at its centre free to rotate in a vertical plane. The rod is at rest in the vertical position. A bullet of mass $$M$$ moving horizontal at a speed $$v$$ strikes and gets embedded in one end of the rod. The angular velocity of the rod just after the collision will be