A cube of side a is placed on rough surface (shown in the...

A cube of side $a$ is placed on rough surface (shown in the figure). Then:
(a) if $f<\frac{M g}{2}$, then the body will move in pure rotation and will not topple
(b) if $F>\frac{M g}{2}$, the body will topple
(c) if $\mu>0.5$, the body will move by toppling
(d) all the above

Key Concepts

Moment Equilibrium

Moment equilibrium involves balancing the torques (moments) acting on a body such that the net torque about any chosen pivot is zero. This principle is used to determine the conditions under which a body begins to rotate (topple) or remains in static balance, taking into account both the applied forces and the reactive moments from support forces like friction.

Toppling

Toppling is the process by which a body loses its equilibrium and rotates about an edge, resulting in it falling over. It occurs when the line of action of the gravitational force (through the center of mass) moves outside the area of support, causing an unbalanced moment that induces rotation about the pivot point.

Center of Mass

The center of mass of a body is the point at which its mass can be considered to be concentrated. In stability analysis, especially for objects resting on a surface, the location of the center of mass relative to the base of support is crucial in determining whether the body will overturn or remain stable.

Static Friction

Static friction is the resisting force present at the interface of two surfaces that prevents relative motion until a certain threshold force is exceeded. Its magnitude depends on the normal force and the coefficient of static friction, and it plays a key role in determining whether a body will slide or remain stationary when forces are applied.

Pure Rotation

Pure rotation refers to the motion of a rigid body rotating about a fixed axis without any translational movement. This concept is important in dynamics when analyzing cases where applied forces generate a net moment about a pivot point, leading to rotational motion in the absence of sliding.

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