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

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 $\frac{3 a}{4}$ 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) (a) $\frac{m g}{4}$ (b) $\frac{2 m g}{3}$ (c) $\frac{3 m g}{4}$ (d) $m g$

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 $\frac{3 a}{4}$ 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)
(a) $\frac{m g}{4}$
(b) $\frac{2 m g}{3}$
(c) $\frac{3 m g}{4}$
(d) $m g$

Key Concepts

Static Equilibrium

Static equilibrium refers to the state where the sum of all forces and the sum of all torques (moments) acting on a body are zero. In problems involving tipping, such as a cube about to rotate about an edge, analyzing static equilibrium involves ensuring that the moments produced by external forces (like gravity and applied force) balance exactly at the critical tipping point.

Torque or Moment

Torque is the rotational equivalent of force and is calculated as the product of a force and its perpendicular distance from a chosen pivot point. In the context of tipping, comparing the torques generated by the gravitational force (acting through the center of mass) and the applied force (acting at a specified height) about the pivot edge determines when the object will begin to rotate.

Center of Mass

The center of mass is the point at which the entire mass of an object can be thought to be concentrated and where gravitational force effectively acts. For uniform objects like a cube, the location of the center of mass is geometrically symmetric, and its position relative to the pivot point is crucial in assessing when the gravitational torque can no longer counterbalance an applied torque.

Pivot Point

The pivot point, which is the edge of contact with the table in the tipping scenario, is the point about which rotation occurs when the cube tilts. Analyzing moments about this specific point helps in determining the conditions under which the applied force generates enough torque to initiate tipping.

Moment Arm

The moment arm is the shortest (perpendicular) distance from the line of action of a force to the pivot point. It is a key parameter in rotational dynamics because the effectiveness of a force in producing torque depends directly on the length of its moment arm. When calculating tipping conditions, comparing the moment arms of the gravitational force and the applied force is essential.

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