User:Egm6321.f12.team4.lu/Homework1 R1.3

=Homework1 Problem R1.3 – Dimensional Analysis=

Given
Equation of motion (EOM) of wheel/magnet is expressed as

$$C_{3}\left ( Y^{1},t \right )\ddot{Y}+C_{2}\left ( Y^{1},t \right )(\dot{Y}^{1})^{2}+C_{1}\left ( Y^{1},t \right )\dot{Y}^{1}+C_{0}\left ( Y^{1},t \right )=0$$                             (1.3.1)

Where

$$C_{0}\left ( Y^{1},t \right )=-F^{1}[1-\overline{R}U_{SS}^{2}\left ( Y^{1},t \right )]-F^{2}U_{S}^{2}-\frac{T}{R}+M[(1-\overline{R}U_{SS}^{2})(U_{tt}^{1}-\overline{R}U_{Stt}^{2})+U_{S}^{2}U_{tt}^{2}]$$          (1.3.2)

Find
Analyze the dimension of all terms in the equation, and provide the physical meaning.

Solution
 Dimensional analysis: 


 * M: mass, [kg]


 * L: length, [m]


 * T: time, [s]

 First term 


 * $$F^{1}[1-\overline{R}U_{SS}^{2}\left ( Y^{1},t \right )]$$

Where


 * $$[F]=MLT^{-2}$$


 * $$[1]=1$$


 * $$[\overline{R}]=L$$


 * $$[U_{SS}^{2}\left ( Y^{1},t \right )]=L^{-1}$$


 * $$[\overline{R} U_{SS}^{2}\left ( Y^{1},t \right )]=1$$

We have

$$[F^{1}[1-\overline{R}U_{SS}^{2}\left ( Y^{1},t \right )]]= MLT^{-2}$$                       (1.3.3)

 Second term 


 * $$F^{2}U_{S}^{2}$$

Where


 * $$[F^{2}]=MLT^{-2}$$


 * $$[U_{S}^{2}]=1$$

We have

$$[F^{2}U_{S}^{2}]= MLT^{-2}$$                                        (1.3.4)

 Third term 


 * $$\frac{T}{R}$$

Where


 * $$[T]=MLT^{-1}$$


 * $$[R]= L$$

We have

$$[\frac{T}{R}]= MLT^{-2}$$                                           (1.3.5)

 Fourth term 


 * $$M[(1-\overline{R}U_{SS}^{2})(U_{tt}^{1}-\overline{R}U_{Stt}^{2})+U_{S}^{2}U_{tt}^{2}]$$

Where


 * $$[M]=M$$


 * $$[1]=1$$


 * $$[{R}U_{SS}^{2}]=1$$


 * $$[\overline{R}U_{Stt}^{2}]=T^{-2}$$


 * $$U_{S}^{2}U_{tt}^{2}=L T^{-2}$$

We have

$$M[(1-\overline{R}U_{SS}^{2})(U_{tt}^{1}-\overline{R}U_{Stt}^{2})+U_{S}^{2}U_{tt}^{2}]= MLT^{-2}$$       (1.3.6)

 Physical meaning 

We know that, the dimension of the force is $$MLT^{-2}$$, so the equations from (1.3.3),(1.3.4),(1.3.5),(1.3.6) are showing force from different part of the physical aspect.

In conclusion, the external forces F1, F2 and moment T and mass M contribute to each element from the physical aspect.