Schwerpunkt Kreissegment
About points...
We associate a certain number of points with each exercise.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
About difficulty...
We associate a certain difficulty with each exercise.
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
Question
Solution
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\(\LaTeX\)
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Exercise:
Berechne den Schwerpunkt der in der Abbildung angegebenen Menge Omega. center tikzpicturescale. drawcolorgreen!!black -latex -.--. noderight x; drawcolorgreen!!black -latex -.--. nodeabove y; drawcolorblue!!white arc::; drawcolorblue!!white --; filldrawfillblue!!white opacity. arc:: -- cycle; drawcolorgreen!!black -. -- . nodeleft ; drawcolorgreen!!black -. -- . nodebelow ; tikzpicture center
Solution:
Da keine Angabe an der Massichte steht setzt man rho equiv . Die Masse von Omega lautet somit M _Omegaddmuxy. Man schreibt Omega als Normalbereich. Dazu betrachtet man die Abbildung wählt eine der beiden Variablen aus z.B. y und glqq vergisstgrqq die andere Variable in diesem Fall also x. Man liest leq yleq ab äusserste Grenzen. Dann fixiert man y und schaut was mit x passiert. x liegt zwischen der Geraden -y und der Kreislinie sqrt-y^. Somit Omega xymid leq yleq -yleq xleq sqrt-y^. Mit der Formel für Normalbereiche Erweiterung des Satzes von Fubini findet man dann: M _Omegaddmuxy _^ ddy _-y^sqrt-y^ ddx _^ ddy leftsqrt-y^-+yright left fracysqrt-y^+fracarcsiny-y+fracy^right_^ fracpi- Die x-Komponente des Schwerpunktes Sx_Sy_S lautet damit: x_S fracM_Omegaxddmuxy fracM_^ ddy _-y^sqrt-y^ ddxx fracM_^ ddy leftfracx^right_-y^sqrt-y^ fracM_^ ddy frac-y^--y^ fracM_^ ddy y-y^ fracMleftfrac-fracright fracM fracpi-. Für die y-Komponente gilt: y_S fracM_Omegayddmuxy fracM_^ ddyy _-y^sqrt-y^ ddx fracM_^ ddy leftysqrt-y^-y+y^right fracMleft-frac-y^frac+fracy^-fracy^right_^ fracMleft frac+frac-fracright fracM fracpi-. Damit folgt insgesamt: S x_Sy_S leftfracpi- fracpi-right.
Berechne den Schwerpunkt der in der Abbildung angegebenen Menge Omega. center tikzpicturescale. drawcolorgreen!!black -latex -.--. noderight x; drawcolorgreen!!black -latex -.--. nodeabove y; drawcolorblue!!white arc::; drawcolorblue!!white --; filldrawfillblue!!white opacity. arc:: -- cycle; drawcolorgreen!!black -. -- . nodeleft ; drawcolorgreen!!black -. -- . nodebelow ; tikzpicture center
Solution:
Da keine Angabe an der Massichte steht setzt man rho equiv . Die Masse von Omega lautet somit M _Omegaddmuxy. Man schreibt Omega als Normalbereich. Dazu betrachtet man die Abbildung wählt eine der beiden Variablen aus z.B. y und glqq vergisstgrqq die andere Variable in diesem Fall also x. Man liest leq yleq ab äusserste Grenzen. Dann fixiert man y und schaut was mit x passiert. x liegt zwischen der Geraden -y und der Kreislinie sqrt-y^. Somit Omega xymid leq yleq -yleq xleq sqrt-y^. Mit der Formel für Normalbereiche Erweiterung des Satzes von Fubini findet man dann: M _Omegaddmuxy _^ ddy _-y^sqrt-y^ ddx _^ ddy leftsqrt-y^-+yright left fracysqrt-y^+fracarcsiny-y+fracy^right_^ fracpi- Die x-Komponente des Schwerpunktes Sx_Sy_S lautet damit: x_S fracM_Omegaxddmuxy fracM_^ ddy _-y^sqrt-y^ ddxx fracM_^ ddy leftfracx^right_-y^sqrt-y^ fracM_^ ddy frac-y^--y^ fracM_^ ddy y-y^ fracMleftfrac-fracright fracM fracpi-. Für die y-Komponente gilt: y_S fracM_Omegayddmuxy fracM_^ ddyy _-y^sqrt-y^ ddx fracM_^ ddy leftysqrt-y^-y+y^right fracMleft-frac-y^frac+fracy^-fracy^right_^ fracMleft frac+frac-fracright fracM fracpi-. Damit folgt insgesamt: S x_Sy_S leftfracpi- fracpi-right.
Meta Information
Exercise:
Berechne den Schwerpunkt der in der Abbildung angegebenen Menge Omega. center tikzpicturescale. drawcolorgreen!!black -latex -.--. noderight x; drawcolorgreen!!black -latex -.--. nodeabove y; drawcolorblue!!white arc::; drawcolorblue!!white --; filldrawfillblue!!white opacity. arc:: -- cycle; drawcolorgreen!!black -. -- . nodeleft ; drawcolorgreen!!black -. -- . nodebelow ; tikzpicture center
Solution:
Da keine Angabe an der Massichte steht setzt man rho equiv . Die Masse von Omega lautet somit M _Omegaddmuxy. Man schreibt Omega als Normalbereich. Dazu betrachtet man die Abbildung wählt eine der beiden Variablen aus z.B. y und glqq vergisstgrqq die andere Variable in diesem Fall also x. Man liest leq yleq ab äusserste Grenzen. Dann fixiert man y und schaut was mit x passiert. x liegt zwischen der Geraden -y und der Kreislinie sqrt-y^. Somit Omega xymid leq yleq -yleq xleq sqrt-y^. Mit der Formel für Normalbereiche Erweiterung des Satzes von Fubini findet man dann: M _Omegaddmuxy _^ ddy _-y^sqrt-y^ ddx _^ ddy leftsqrt-y^-+yright left fracysqrt-y^+fracarcsiny-y+fracy^right_^ fracpi- Die x-Komponente des Schwerpunktes Sx_Sy_S lautet damit: x_S fracM_Omegaxddmuxy fracM_^ ddy _-y^sqrt-y^ ddxx fracM_^ ddy leftfracx^right_-y^sqrt-y^ fracM_^ ddy frac-y^--y^ fracM_^ ddy y-y^ fracMleftfrac-fracright fracM fracpi-. Für die y-Komponente gilt: y_S fracM_Omegayddmuxy fracM_^ ddyy _-y^sqrt-y^ ddx fracM_^ ddy leftysqrt-y^-y+y^right fracMleft-frac-y^frac+fracy^-fracy^right_^ fracMleft frac+frac-fracright fracM fracpi-. Damit folgt insgesamt: S x_Sy_S leftfracpi- fracpi-right.
Berechne den Schwerpunkt der in der Abbildung angegebenen Menge Omega. center tikzpicturescale. drawcolorgreen!!black -latex -.--. noderight x; drawcolorgreen!!black -latex -.--. nodeabove y; drawcolorblue!!white arc::; drawcolorblue!!white --; filldrawfillblue!!white opacity. arc:: -- cycle; drawcolorgreen!!black -. -- . nodeleft ; drawcolorgreen!!black -. -- . nodebelow ; tikzpicture center
Solution:
Da keine Angabe an der Massichte steht setzt man rho equiv . Die Masse von Omega lautet somit M _Omegaddmuxy. Man schreibt Omega als Normalbereich. Dazu betrachtet man die Abbildung wählt eine der beiden Variablen aus z.B. y und glqq vergisstgrqq die andere Variable in diesem Fall also x. Man liest leq yleq ab äusserste Grenzen. Dann fixiert man y und schaut was mit x passiert. x liegt zwischen der Geraden -y und der Kreislinie sqrt-y^. Somit Omega xymid leq yleq -yleq xleq sqrt-y^. Mit der Formel für Normalbereiche Erweiterung des Satzes von Fubini findet man dann: M _Omegaddmuxy _^ ddy _-y^sqrt-y^ ddx _^ ddy leftsqrt-y^-+yright left fracysqrt-y^+fracarcsiny-y+fracy^right_^ fracpi- Die x-Komponente des Schwerpunktes Sx_Sy_S lautet damit: x_S fracM_Omegaxddmuxy fracM_^ ddy _-y^sqrt-y^ ddxx fracM_^ ddy leftfracx^right_-y^sqrt-y^ fracM_^ ddy frac-y^--y^ fracM_^ ddy y-y^ fracMleftfrac-fracright fracM fracpi-. Für die y-Komponente gilt: y_S fracM_Omegayddmuxy fracM_^ ddyy _-y^sqrt-y^ ddx fracM_^ ddy leftysqrt-y^-y+y^right fracMleft-frac-y^frac+fracy^-fracy^right_^ fracMleft frac+frac-fracright fracM fracpi-. Damit folgt insgesamt: S x_Sy_S leftfracpi- fracpi-right.
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Schwerpunkt 3 by uz