Introduction:

In this problem, we have an equilateral prism resting on a rough horizontal surface. We need to determine the minimum force required to topple the prism without it sliding. The prism has a mass of m and the coefficient of friction between the prism and the surface is given as μ.

Analysis:

To calculate the minimum force required to topple the prism, we need to consider the forces acting on it. There are two main forces involved:

1. The gravitational force acting downwards, which can be represented as Fg = mg, where m is the mass of the prism and g is the acceleration due to gravity (approximately 9.8 m/s²).

2. The frictional force acting horizontally, opposing the applied force. This force can be calculated using the equation Ff = μN, where N is the normal force.

Normal Force:

Before we can calculate the frictional force, we need to determine the normal force. The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case, the normal force will be equal to the gravitational force acting downwards.

Frictional Force:

The frictional force is given by the equation Ff = μN, where Ff is the frictional force and μ is the coefficient of friction. In this case, the frictional force will act in the opposite direction to the applied force.

Conditions for Toppling:

To prevent the prism from sliding before toppling, the frictional force should be greater than or equal to the force component trying to slide the prism. This force component can be calculated as Fslide = F * sin(θ), where θ is the angle between the applied force and the horizontal surface.

Minimum Force for Toppling:

The minimum force required to topple the prism is the force at which the frictional force is equal to the force component trying to slide the prism. This can be expressed as Ff = Fslide.

Calculation:

Since Ff = μN and N = mg, we can substitute these values into the equation Ff = Fslide.

μmg = F * sin(θ)

Rearranging the equation, we can solve for the minimum force required to topple the prism:

F = μmg / sin(θ)

Conclusion:

In conclusion, the minimum force required to topple the prism without it sliding can be calculated using the equation F = μmg / sin(θ), where F is the force, μ is the coefficient of friction, m is the mass of the prism, g is the acceleration due to gravity, and θ is the angle between the applied force and the horizontal surface.