WORK

Before beginning this section, one must be aware that there are various forms of work. Below, there is a chart that summarizes them. This section focuses on the mechanical form of work.

Forms of Work	General Expression for Work	Variables
Mechanical - Force	W_1 − 2 = ∫^s₂_s₁ F⋅ds	F: Force vector
		s: Displacement vector

Mechanical - Moment	W_1 − 2 = ∫^θ₂_θ₁ M⋅dθ	M: Moment vector
		θ: Rotation angle vector

Electrical	W_1 − 2 = Q∫^s₂_s₁ E⋅ds	Q: Charge of particle
		E: Electric field vector
		s: Displacement vector

Thermodynamic	W_1 − 2 = ∫^V₂_V₁pdV	p: Pressure
		V: Volume

Work, W, is a scalar quantity that measures the product of the force or moment exerted on an object and the resulting displacement of that object, is a measure of what an applied force accomplishes. The greater the force or moment applied, and the farther the object travels, the more work is done. Work is measured in units of joules (J) in the SI system, where 1 J = 1 Nm = 1 kg m² ⁄ s² or in ft-lbs in the English system. Thus, work is simply done when a force or moment causes a displacement on a system.

The work done by forces or moments are defined by the following formulas:

Work done by the total sum of forces, acting at the mass center, with a displacement s, is W = ∫∑F_c⋅ds_c
- For a force vector with constant magnitude, work simplifies to W = |F|cosθ|s₂ − s₁|, where |F| is the magnitude of the force vector, θ is the angle between the force vector and the displacement vector, s₁ and s₂ are the vector displacements along a path.
- For the force of weight, the work done would be W = − mgΔy, where ( − mg) is the force of the weight and Δy is the change in displacement, assuming the displacement and weight both act parallel to the y-axis.
- For a spring force, the work is done when the spring displacement changes (either compresses or stretches) to a position further than the unstretched position or |s₂| > |s₁|. Thus, work would be W = − 1 ⁄ 2k(s²₂ − s²₁), where the spring force 1 ⁄ 2ks and the displacement s are acting along the same axis.
Work done by the total sum of moments, acting at the mass center, W = ∫^θ₂_θ₁∑ M_c⋅dθ
- For a moment vector with constant magnitude in the direction of the displacement, this simplifies to W = M(θ₂ − θ₁).

SUMMARY POINTS

When both the force and displacements are in the same direction, work is simply the product of the magnitudes of force and displacement.
When the force and displacements are at an angle to each other, calculation of work requires the force component that acts in the direction of the displacement. Work in this case is simply the product of displacement and the component of the force that acts in the direction of the displacement.
When force and displacement are perpendicular, there is no work done.
Positive work is done when the force that contributing work acts in the same direction of the displacement. Negative work is done when the force acts in the opposite direction of the displacement.