A uniform cube of side a and mass m rests on a rough horizontal table

By

Problem: 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 face, at a height 3a/4 above the base. The minimum value of F for which the cube begins to tip about the edge is________ (Assume that the cube does not slide.)  (IIT JEE 1984)

Solution: Let the cube topples about the point O.

A uniform cube of side a and mass m

The forces acting on the cube are applied force $F$, weight $mg$, frictional force $f$ and normal reaction $N$. The centre of mass of the cube lies on its geometrical centre because the cube is uniform. Thus, perpendicular distance of the weight $mg$ from the point O is $a/2$. The applied force $F$ is minimum when the cube is about to topple. At this instant, only point O of the cube is in contact with the ground. Thus, normal reaction $N$ passes through the point O. The torques of $N$ and $f$ about the point O are zero because these forces pass through this point. When the cube is about to topple, net torque on the cube about the point O is zero i.e., \begin{align} F(3a/4)-mg(a/2)=0, \end{align} which gives, \begin{align} F=2mg/3.\nonumber \end{align}

More Solved Problems on Equilibrium of Rigid Bodies

See Our Book

  1. 300 Solved Problems on Rotational Mechanics by Jitender Singh and Shraddhesh Chaturvedi
  2. IIT JEE Physics by Jitender Singh and Shraddhesh Chaturvedi