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work in classic mechanics (Definition)

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 $



"work in classic mechanics" is owned by bloftin. [ full author list (2) | owner history (1) ]

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Cross-references: scalar product, parameters, scalar, energy, velocity, formula, kinematic, acceleration, motion, object, kinetic energy, work

This is version 37 of work in classic mechanics, born on 2006-07-18, modified 2009-01-16.
Object id is 199, canonical name is WorkInClassicMechanics.
Accessed 1464 times total.

Classification:
Physics Classification: 40. (ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID MECHANICS)
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