According to the Principle of Resolution, the algebraic sum of the resolved parts of multiple forces in a given direction is equal to:

Explanation:

Principle of Resolution in Mechanics

The Principle of Resolution is a fundamental concept in mechanics that deals with resolving forces into their components. According to this principle, the algebraic sum of the resolved parts of multiple forces in a given direction is equal to the resolved part of their resultant force in the same direction. This principle is crucial for understanding and solving problems related to equilibrium and force analysis in mechanical systems.

When multiple forces act on a body, each force can be resolved into components along specified directions, usually along the coordinate axes (e.g., x and y axes in a two-dimensional plane). The resolution of forces simplifies the analysis by breaking down complex force systems into simpler, manageable parts.

Detailed Solution:

Consider a system where several forces are acting on a body. To analyze the effect of these forces, we can resolve each force into its components along a particular direction. Let's denote the forces acting on the body as F₁, F₂, F₃, ..., F_n. Each of these forces can be resolved into components along the x and y directions (or any other chosen directions).

According to the Principle of Resolution, the algebraic sum of the resolved parts of these forces in the x direction is equal to the resolved part of their resultant force in the x direction. Mathematically, this can be expressed as:

R_x = ΣF_ix

where R_x is the x-component of the resultant force, and ΣF_ix represents the sum of the x-components of all individual forces.

Similarly, the algebraic sum of the resolved parts of these forces in the y direction is equal to the resolved part of their resultant force in the y direction:

R_y = ΣF_iy

where R_y is the y-component of the resultant force, and ΣF_iy represents the sum of the y-components of all individual forces.

The resultant force R can be found by combining its components along the x and y directions using the Pythagorean theorem:

R = √(R_x² + R_y²)

The direction of the resultant force can be determined using trigonometry:

θ = tan^-1(R_y/R_x)

Hence, the algebraic sum of the resolved parts of multiple forces in a given direction equals the resolved part of their resultant force in the same direction