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)

$\displaystyle a = \frac{v_f{}^2 - v_i{}^2}{2\Delta x}$

When inserting the previous equation into Newton's second law:

$\displaystyle \Sigma F_x = \frac{{m(v_f{}^2 - v_i{}^2)}}{2\Delta x}$

And after a few algebric actions we get:

$\displaystyle \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$ .

$\displaystyle W = \vert\nabla F\vert\cdot \vert\nabla \Delta S\vert\cos \theta $



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