PlanetPhysics/Work in Classic Mechanics

Work is defined as the change of kinetic energy of an object caused by a force along a distance.

Work is commonly denoted by the latter $$W$$

The SI unit for work is joule [J] which is the same as $$\frac{kg\cdot m^2}{s^2}$$ in SI base units.

When focusing on an object moving along a straight line under the effect of constant forces $$\Sigma F$$. Let's define a $$x$$ axis along the line of motion.

According to Newton: $$\Sigma F_x$$ = $$ma_x$$

The acceleration $$a$$ is constant (the sum of forces is constant, and so the following kinematic formula is relevant: v_f{}^2 = v_i{}^2 + 2a \Delta x $$ ($$v_f$$-The final velocity, $$v_i-The initial velocity) $$ a = \frac{v_f{}^2 - v_i{}^2}{2\Delta x}$$

When inserting the previous equation into Newton's second law: $$ \Sigma F_x = \frac{2\Delta x}$$And after a few algebric actions we get: $$ \Sigma F_x \Delta x = 0.5mv_f{}^2 - 0.5mv_i{}^2 = \Delta E_k $$From that we can conclude that $$ W = F\Delta x $$ ; $$F_x\Delta x$$ is the work done by the force $$F_x$$ along the route $$\Delta x$$

Work like energy is a scalar but is defined as the product of two vectorial parameters: $$\nabla F\cdot \nabla \Delta S$$ and so in a two dimentional space work is defined as the scalar product of force $$\nabla F$$ and change in place $$\nabla \Delta S$$. $$ W = |\nabla F|\cdot |\nabla \Delta S|\cos \theta $$