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Physics Lesson 5.1.3 - Calculation of Work in 2 and 3 Dimensions
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Welcome to our Physics lesson on Calculation of Work in 2 and 3 Dimensions, this is the third lesson of our suite of physics lessons covering the topic of Work and Energy. Types of Energy, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.
Calculation of Work in 2 and 3 Dimensions
When the object is moving in two or three dimensions, we use the equation of the distance r between two points
Or
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
Fy = |F| × cos β
Fz = |F| × cos γ
where |F| is the magnitude of the force F. It is calculated by the equation
= √( |F| × cos α )2 + ( |F| × cos β )2 + ( |F| × cos γ )2
= √ |F|2 × (cos2α +cos2β+cos2γ
= √|F|2
= |F|
The last expression is true because
Therefore, if we want to calculate the work done by the force in each direction, we can write
= Fx × (xf - xi )
= |F| × cos α × (xf - xi )
= Fy × (yf - yi )
= |F| × cos β × (yf - yi )
= Fz × (zf - zi )
= |F| × cos γ × (zf - zi )
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:
= |F| × √(xf - xi )2 + (yf - yi )2 + (zf - zi )2
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
= xf - xi
= ∆x
and that of the work W is
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. You can use the 3D Motion Work Calculator to calculate and check your own results when performing this Physics equation.
You have reached the end of Physics lesson 5.1.3 Calculation of Work in 2 and 3 Dimensions. There are 5 lessons in this physics tutorial covering Work and Energy. Types of Energy, you can access all the lessons from this tutorial below.
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