**Question Analysis:**

In this question, we are given a solid cube placed on a rough horizontal surface. The coefficient of friction between the cube and the surface is given as μ, where μ < 1/2.="" we="" need="" to="" find="" the="" maximum="" acceleration="" with="" which="" the="" cube="" can="" move="" without="" toppling="" />

**Solution:**

To solve this problem, we need to consider the forces acting on the cube and analyze the conditions for toppling.

**Forces Acting on the Cube:**

1. Weight (mg): The weight of the cube acts vertically downwards.
2. Normal Force (N): The normal force exerted by the surface acts vertically upwards and balances the weight of the cube.
3. Frictional Force (f): The frictional force acts parallel to the surface opposing the motion of the cube.

**Conditions for Toppling:**

For the cube to topple, the torque produced by the weight of the cube about its edge must exceed the torque produced by the frictional force about the same edge.

**Torque due to Weight (τw):**

The torque due to the weight of the cube about its edge is given by τw = mg * h, where h is the height of the cube.

**Torque due to Friction (τf):**

The torque due to the frictional force about the same edge is given by τf = f * d, where d is the distance between the point of application of the frictional force and the edge.

**Condition for No Toppling:**

For the cube to move without toppling, the torque due to the weight must be less than or equal to the torque due to the friction.

τw ≤ τf

mg * h ≤ f * d

**Maximum Frictional Force (fmax):**

The maximum frictional force is given by fmax = μ * N, where N is the normal force.

**Finding Maximum Acceleration (amax):**

The maximum acceleration can be found using Newton's second law:

f = ma

Substituting the value of fmax, we get:

fmax = μ * N = μ * mg

amax = fmax / m = μ * g

Therefore, the maximum acceleration with which the cube can move without toppling is given by amax = μ * g.

**Final Answer:**

The maximum acceleration with which the cube can move without toppling is μ * g.