Damping Ratio
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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 -
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Exercise:
How can the damping of the following oscillations with linear damping i.e. exponentially decreasing envelope be characterised? abcliste abc For a mass mbO on a spring kkbO with a damping coefficient debO. abc For an oscillator with period TaO whose amplitude decreases from AaO to BaO in taO. abc For an oscillator whose amplitude decreases to one half of the initial value during one period. abc For an LC oscillator with inductance LdO capacitance CdO and resistance RdO. abcliste
Solution:
abcliste abc The damping ratio is zeta fracdeltaomega_ zebF deb times sqrtfracmbkb zebP Since zeta the oscillator is overdamped. abc The damping coefficient is given by At A_ e^-delta t Longrightarrow -delta t lnfracAtA_ Longrightarrow delta deaF The damping ratio is zeta fracdeltaomega_ approx zeaF fraclnAa/Ba times Tapitimes ta zeaP Because zeta ll the oscillation is only weakly damped. abc The half-life for the exponential decay corresponds to T_/ fracln delta Since the half-life is equal to the period it follows for the damping coefficient delta fracln T_/ fracln T The damping ratio is zeta fracdeltaomega_ approx fracln Tpi T zecF zecP This is significantly smaller than so the oscillation is weakly damped. abc The damping ratio is zeta fracdeltaomega_ fracfracRLfracsqrtLC zedF fracRdtimes sqrtfracCdLd zedP This is clearly overdamped. abcliste
How can the damping of the following oscillations with linear damping i.e. exponentially decreasing envelope be characterised? abcliste abc For a mass mbO on a spring kkbO with a damping coefficient debO. abc For an oscillator with period TaO whose amplitude decreases from AaO to BaO in taO. abc For an oscillator whose amplitude decreases to one half of the initial value during one period. abc For an LC oscillator with inductance LdO capacitance CdO and resistance RdO. abcliste
Solution:
abcliste abc The damping ratio is zeta fracdeltaomega_ zebF deb times sqrtfracmbkb zebP Since zeta the oscillator is overdamped. abc The damping coefficient is given by At A_ e^-delta t Longrightarrow -delta t lnfracAtA_ Longrightarrow delta deaF The damping ratio is zeta fracdeltaomega_ approx zeaF fraclnAa/Ba times Tapitimes ta zeaP Because zeta ll the oscillation is only weakly damped. abc The half-life for the exponential decay corresponds to T_/ fracln delta Since the half-life is equal to the period it follows for the damping coefficient delta fracln T_/ fracln T The damping ratio is zeta fracdeltaomega_ approx fracln Tpi T zecF zecP This is significantly smaller than so the oscillation is weakly damped. abc The damping ratio is zeta fracdeltaomega_ fracfracRLfracsqrtLC zedF fracRdtimes sqrtfracCdLd zedP This is clearly overdamped. abcliste
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Exercise:
How can the damping of the following oscillations with linear damping i.e. exponentially decreasing envelope be characterised? abcliste abc For a mass mbO on a spring kkbO with a damping coefficient debO. abc For an oscillator with period TaO whose amplitude decreases from AaO to BaO in taO. abc For an oscillator whose amplitude decreases to one half of the initial value during one period. abc For an LC oscillator with inductance LdO capacitance CdO and resistance RdO. abcliste
Solution:
abcliste abc The damping ratio is zeta fracdeltaomega_ zebF deb times sqrtfracmbkb zebP Since zeta the oscillator is overdamped. abc The damping coefficient is given by At A_ e^-delta t Longrightarrow -delta t lnfracAtA_ Longrightarrow delta deaF The damping ratio is zeta fracdeltaomega_ approx zeaF fraclnAa/Ba times Tapitimes ta zeaP Because zeta ll the oscillation is only weakly damped. abc The half-life for the exponential decay corresponds to T_/ fracln delta Since the half-life is equal to the period it follows for the damping coefficient delta fracln T_/ fracln T The damping ratio is zeta fracdeltaomega_ approx fracln Tpi T zecF zecP This is significantly smaller than so the oscillation is weakly damped. abc The damping ratio is zeta fracdeltaomega_ fracfracRLfracsqrtLC zedF fracRdtimes sqrtfracCdLd zedP This is clearly overdamped. abcliste
How can the damping of the following oscillations with linear damping i.e. exponentially decreasing envelope be characterised? abcliste abc For a mass mbO on a spring kkbO with a damping coefficient debO. abc For an oscillator with period TaO whose amplitude decreases from AaO to BaO in taO. abc For an oscillator whose amplitude decreases to one half of the initial value during one period. abc For an LC oscillator with inductance LdO capacitance CdO and resistance RdO. abcliste
Solution:
abcliste abc The damping ratio is zeta fracdeltaomega_ zebF deb times sqrtfracmbkb zebP Since zeta the oscillator is overdamped. abc The damping coefficient is given by At A_ e^-delta t Longrightarrow -delta t lnfracAtA_ Longrightarrow delta deaF The damping ratio is zeta fracdeltaomega_ approx zeaF fraclnAa/Ba times Tapitimes ta zeaP Because zeta ll the oscillation is only weakly damped. abc The half-life for the exponential decay corresponds to T_/ fracln delta Since the half-life is equal to the period it follows for the damping coefficient delta fracln T_/ fracln T The damping ratio is zeta fracdeltaomega_ approx fracln Tpi T zecF zecP This is significantly smaller than so the oscillation is weakly damped. abc The damping ratio is zeta fracdeltaomega_ fracfracRLfracsqrtLC zedF fracRdtimes sqrtfracCdLd zedP This is clearly overdamped. abcliste
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