Coupled Two-Mass System
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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.
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In physics exercises, we try to follow this pattern:
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Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
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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 -
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Exercise:
Determine the eigenvalues for a coupled two-mass system with m_m m_ m k_K and k_K. Calculate the normal angular frequencies for mmO KKO and kkO
Solution:
The general coefficient matrix for the coupled two-mass system is A pmatrix -frack_+km_ & frackm_ frackm_ & -frack_+km_ pmatrix pmatrix -fracK+km & frackm frackm & -fracK+km pmatrix The trace and determinant of this matrix are tau -fracK+km+fracK+km -fracK+k+K+km -fracK+km Delta fracK+kK+km^-frack^m^ fracK^+Kk+k^-k^m^ fracK^+Kkm^ It follows for the eigenvalues lambda fracleft-fracK+kmpmsqrtfracK+k^m^-fracK^+Kkm^right frac-K+kpmsqrtK^+Kk+k^-K^-Kkm frac-K+kpm km Longrightarrow lambda_ result-fracKm lambda_ result-fracK+km The normal frequencies are omega_ omsF sqrtfractimesKm oms approx resultomsP omega_ omfF sqrtfractimesK+timeskm omf approx resultomfP The eigenvectors are verify! hat x_ pmatrix pmatrix hat x_ pmatrix- pmatrix In the symmetrical mode hat x_ the second mass is twice as heavy but it also experiences twice the force for the same displacement. In the antisymmetrical mode hat x_ the amplitudes for the two masses are no longer the same.
Determine the eigenvalues for a coupled two-mass system with m_m m_ m k_K and k_K. Calculate the normal angular frequencies for mmO KKO and kkO
Solution:
The general coefficient matrix for the coupled two-mass system is A pmatrix -frack_+km_ & frackm_ frackm_ & -frack_+km_ pmatrix pmatrix -fracK+km & frackm frackm & -fracK+km pmatrix The trace and determinant of this matrix are tau -fracK+km+fracK+km -fracK+k+K+km -fracK+km Delta fracK+kK+km^-frack^m^ fracK^+Kk+k^-k^m^ fracK^+Kkm^ It follows for the eigenvalues lambda fracleft-fracK+kmpmsqrtfracK+k^m^-fracK^+Kkm^right frac-K+kpmsqrtK^+Kk+k^-K^-Kkm frac-K+kpm km Longrightarrow lambda_ result-fracKm lambda_ result-fracK+km The normal frequencies are omega_ omsF sqrtfractimesKm oms approx resultomsP omega_ omfF sqrtfractimesK+timeskm omf approx resultomfP The eigenvectors are verify! hat x_ pmatrix pmatrix hat x_ pmatrix- pmatrix In the symmetrical mode hat x_ the second mass is twice as heavy but it also experiences twice the force for the same displacement. In the antisymmetrical mode hat x_ the amplitudes for the two masses are no longer the same.
Meta Information
Exercise:
Determine the eigenvalues for a coupled two-mass system with m_m m_ m k_K and k_K. Calculate the normal angular frequencies for mmO KKO and kkO
Solution:
The general coefficient matrix for the coupled two-mass system is A pmatrix -frack_+km_ & frackm_ frackm_ & -frack_+km_ pmatrix pmatrix -fracK+km & frackm frackm & -fracK+km pmatrix The trace and determinant of this matrix are tau -fracK+km+fracK+km -fracK+k+K+km -fracK+km Delta fracK+kK+km^-frack^m^ fracK^+Kk+k^-k^m^ fracK^+Kkm^ It follows for the eigenvalues lambda fracleft-fracK+kmpmsqrtfracK+k^m^-fracK^+Kkm^right frac-K+kpmsqrtK^+Kk+k^-K^-Kkm frac-K+kpm km Longrightarrow lambda_ result-fracKm lambda_ result-fracK+km The normal frequencies are omega_ omsF sqrtfractimesKm oms approx resultomsP omega_ omfF sqrtfractimesK+timeskm omf approx resultomfP The eigenvectors are verify! hat x_ pmatrix pmatrix hat x_ pmatrix- pmatrix In the symmetrical mode hat x_ the second mass is twice as heavy but it also experiences twice the force for the same displacement. In the antisymmetrical mode hat x_ the amplitudes for the two masses are no longer the same.
Determine the eigenvalues for a coupled two-mass system with m_m m_ m k_K and k_K. Calculate the normal angular frequencies for mmO KKO and kkO
Solution:
The general coefficient matrix for the coupled two-mass system is A pmatrix -frack_+km_ & frackm_ frackm_ & -frack_+km_ pmatrix pmatrix -fracK+km & frackm frackm & -fracK+km pmatrix The trace and determinant of this matrix are tau -fracK+km+fracK+km -fracK+k+K+km -fracK+km Delta fracK+kK+km^-frack^m^ fracK^+Kk+k^-k^m^ fracK^+Kkm^ It follows for the eigenvalues lambda fracleft-fracK+kmpmsqrtfracK+k^m^-fracK^+Kkm^right frac-K+kpmsqrtK^+Kk+k^-K^-Kkm frac-K+kpm km Longrightarrow lambda_ result-fracKm lambda_ result-fracK+km The normal frequencies are omega_ omsF sqrtfractimesKm oms approx resultomsP omega_ omfF sqrtfractimesK+timeskm omf approx resultomfP The eigenvectors are verify! hat x_ pmatrix pmatrix hat x_ pmatrix- pmatrix In the symmetrical mode hat x_ the second mass is twice as heavy but it also experiences twice the force for the same displacement. In the antisymmetrical mode hat x_ the amplitudes for the two masses are no longer the same.
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