Resonance for Mechanical Oscillator
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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 -
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Exercise:
The differential s for a driven series LC oscillator with damping and a driven mass on a spring with linear drag are formally equivalent: ddotqt + omega_e^ qt+ delta_e dotqt Q_cosomega t ddotyt + omega_m^ yt+ delta_m dotyt A_cosomega t with the angular frequencies for the undamped system omega_e fracsqrtLC labelome quad textrmand quad omega_m sqrtfrackm and the damping constant delta_e fracRL labeldee quad textrmand quad delta_m fracbetam Using the analogy between mechanical and electrical quantities derive the expressions for the resonance frequency of the mechanical oscillator. Calculate the numerical value for a mass mO an elastic constant kO and a drag coefficient beO.
Solution:
The displacement yt of the mass corresponds to the charge qt on the capacitor which is proportional to the capacitor voltage v_Ct. We know that the amplitude of the capacitor voltage has a maximum for omega_R sqrtomega_^-delta^ For the mechanical system we find omega_R omRF sqrtfrackm-fracbe^timesm^ omR approx resultomRP The angular frequency of the undamped system is omega_ omudF sqrtfrackm omud approx resultomudP
The differential s for a driven series LC oscillator with damping and a driven mass on a spring with linear drag are formally equivalent: ddotqt + omega_e^ qt+ delta_e dotqt Q_cosomega t ddotyt + omega_m^ yt+ delta_m dotyt A_cosomega t with the angular frequencies for the undamped system omega_e fracsqrtLC labelome quad textrmand quad omega_m sqrtfrackm and the damping constant delta_e fracRL labeldee quad textrmand quad delta_m fracbetam Using the analogy between mechanical and electrical quantities derive the expressions for the resonance frequency of the mechanical oscillator. Calculate the numerical value for a mass mO an elastic constant kO and a drag coefficient beO.
Solution:
The displacement yt of the mass corresponds to the charge qt on the capacitor which is proportional to the capacitor voltage v_Ct. We know that the amplitude of the capacitor voltage has a maximum for omega_R sqrtomega_^-delta^ For the mechanical system we find omega_R omRF sqrtfrackm-fracbe^timesm^ omR approx resultomRP The angular frequency of the undamped system is omega_ omudF sqrtfrackm omud approx resultomudP
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Exercise:
The differential s for a driven series LC oscillator with damping and a driven mass on a spring with linear drag are formally equivalent: ddotqt + omega_e^ qt+ delta_e dotqt Q_cosomega t ddotyt + omega_m^ yt+ delta_m dotyt A_cosomega t with the angular frequencies for the undamped system omega_e fracsqrtLC labelome quad textrmand quad omega_m sqrtfrackm and the damping constant delta_e fracRL labeldee quad textrmand quad delta_m fracbetam Using the analogy between mechanical and electrical quantities derive the expressions for the resonance frequency of the mechanical oscillator. Calculate the numerical value for a mass mO an elastic constant kO and a drag coefficient beO.
Solution:
The displacement yt of the mass corresponds to the charge qt on the capacitor which is proportional to the capacitor voltage v_Ct. We know that the amplitude of the capacitor voltage has a maximum for omega_R sqrtomega_^-delta^ For the mechanical system we find omega_R omRF sqrtfrackm-fracbe^timesm^ omR approx resultomRP The angular frequency of the undamped system is omega_ omudF sqrtfrackm omud approx resultomudP
The differential s for a driven series LC oscillator with damping and a driven mass on a spring with linear drag are formally equivalent: ddotqt + omega_e^ qt+ delta_e dotqt Q_cosomega t ddotyt + omega_m^ yt+ delta_m dotyt A_cosomega t with the angular frequencies for the undamped system omega_e fracsqrtLC labelome quad textrmand quad omega_m sqrtfrackm and the damping constant delta_e fracRL labeldee quad textrmand quad delta_m fracbetam Using the analogy between mechanical and electrical quantities derive the expressions for the resonance frequency of the mechanical oscillator. Calculate the numerical value for a mass mO an elastic constant kO and a drag coefficient beO.
Solution:
The displacement yt of the mass corresponds to the charge qt on the capacitor which is proportional to the capacitor voltage v_Ct. We know that the amplitude of the capacitor voltage has a maximum for omega_R sqrtomega_^-delta^ For the mechanical system we find omega_R omRF sqrtfrackm-fracbe^timesm^ omR approx resultomRP The angular frequency of the undamped system is omega_ omudF sqrtfrackm omud approx resultomudP
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