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 face, at a height $\frac {3a}{4}$ above the base. The minimum value of $F$ for which the cube begins to tilt about the edge is (Assume that the cube does not slide)

For a rolling body, the velocity of $P_1$ and $P_2$ are ${\vec v_1}$ and ${\vec v_2}$ , respectively

Four particles of masses $1\,kg, 2 \,kg, 3 \,kg$ and $4\, kg$ are placed at the four vertices  $A, B, C$ and $D$ of a square of side $1\, m$. The coordinates of centre of  mass of the particles are

The centre of mass of two particles lies

A long horizontal rod has a bead which can slide along its length, and initially placed at a distance $L$ from one end $A$ of the rod. The rod is set in angular motion about $A$ with constant angular acceleration $\alpha$. If the coefficient of friction between the rod and the bead is $\mu$, and gravity is neglected, then the time after which the bead starts slipping is

Two bodies of mass $1\,kg$ and $3\,kg$ have position vectors $\hat i\,\, + \,\,2\hat j\,\, + \,\,\hat k$ and  $-\,3\hat i\,\, - \,\,2\hat j\,\, + \,\,\hat k$, respectively. The centre of mass of this system has a position vector