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The 3D Motion Work Calculator is provided in support of our Physics Tutorial: Work and Energy. Types of Energy though it can be used as a stand alone calculator to help check homework, and assignment or Physics work computations for accuracy. We provide details of the Work Formula used and some details to refresh your memory and support accurate calculations using this tool

The 3D Motion Work Calculator can take inputs in a mixed units and produce the work calculation in a single standardised unit that you can select. You can also change the unit of work for output and convert the work units as required. This is a very diverse work calculator that covers most of your equation needs.

**Note 1** All elements of the equation will be converted to the "Units for Calculation" automatically so that an even unit of measurement is applied throughout the formula during calculation of Work and to illustrate the steps in a logical manner.

**Note 2** You can change the units of work shown in the results, the final answer will always be shown in Joules and then converted to the desired unit of measurement for work.

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In Physics, Work (W) represents the scalar product between the force F exerted on an object and the Displacement Δx of the object due to the action of this force (or the linear distance the object is moved in the direction of the force).

In cases when the Force is not in the direction of the motion (displacement), we must consider only the component of the force that lies in the direction of motion when calculating the work. Look at the figure below.

In this case, only Fx = F × cos α contributes to the motion. Therefore, the work fone by the force F will be

W = F_{x} × ∆x = F × cos α × ∆x

When the object is moving in two or three dimensions, we use the equation of the distance r between two points

r = √**(∆x)**^{2} + (∆y)^{2} + (∆z)^{2}

Or

r = √**(x**_{f}-x_{i} )^{2} + (y_{f}-y_{i} )^{2} + (z_{f}-z_{i} )^{2}

We can also write the components of the force F according the three main directions. The figure is the same as well. The only difference is that we can write F instead of r. Thus, if we denote the angles of the force with these directions as α, β and γ respectively, we can write for the force components

F_{x} = |F| × cos α

F_{y} = |F| × cos β

F_{z} = |F| × cos γ

F

F

where |F| is the magnitude of the force F. It is calculated by

|F| = √**F**_{x}^{2} + F_{y}^{2} + F_{z}^{2}

= √**(|F| × cos α )**^{2} + (|F| × cos β )^{2} + (|F| × cos γ )^{2}

= √

Therefore, if we want to calculate the work done by the force in each direction, we can write

W_{x}=F_{x} × ∆r_{x}

= F_{x} × (x_{f}-x_{i} )

= |F| × cos α × (x_{f}-x_{i} )

= F

= |F| × cos α × (x

W_{y} = F_{y} × ∆r_{y}

= F_{y} × (y_{f}-y_{i} )

= |F| × cos β × (y_{f}-y_{i} )

= F

= |F| × cos β × (y

W_{z} = F_{z} × ∆r_{z}

= F_{z} × (z_{f}-z_{i} )

= |F| × cos γ × (z_{f}-z_{i} )

= F

= |F| × cos γ × (z

All the above-mentioned equations contribute in generating the equation of work W done by a force F in 3 dimensions (in space) in the most complicated scenario:

W = |F| × r

= |F| × √**(x**_{f}-x_{i} )^{2} + (y_{f}-y_{i} )^{2} + (z_{f}-z_{i} )^{2}

= |F| × √

Don't be afraid by this long formula as in general, most of these quantities are zero. For example when the object is moving only in one direction (for example only according the x-direction), all values of the coordinates in the other two directions are zero. Therefore, the expression for the displacement r becomes

r = r_{x} = √**(x**_{f}-x_{i} )^{2}

= x_{f}-x_{i}

= ∆x

= x

= ∆x

and that of the work W is

W = W_{x} = F_{x} × ∆x

As you see, the known formula Δx = xf - xi represents only a special case of the general expression for the displacement. It is used when the motion is taking place in a single direction.

This calculator is very simple to use. There is a box for the magnitude of force |F| and three separate boxes for the force components Fx, Fy and Fz according the directions x, y and z. These components are calculated through the respective cosines cos α, cos β and cos γ.

Also, there are specific boxes for the coordinates xi, xf, yi, yf, zi and zf, in which you can insert the given values. Then, the equation

W = |F| × √**(x**_{f}-x_{i} )^{2} + (y_{f}-y_{i} )^{2} + (z_{f}-z_{i} )^{2}

is used to calculate the work.

The unnecessary values appear zero by default in the calculator.

Also, you can insert the values of work and find the values of a certain coordinate or force.

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