[Edexcel M4] Proof that SHM is damped?
Hey everyone, came across this question in the exam style paper at the back of the M4 textbook (any of you who have used this will know that they aren't exactly stellar on their explanations of their answers, and due to the outdated LiveText software I can't actually view Solution Bank):
"A particle moves along a straight line. At time the displacement of the particle from a fixed point O is , where:
Show that the particle performs damped harmonic motion."
^N.B: I believe there to be a mistake here; I think that they mean instead of .
Now, I know the conditions for the various types of damped harmonic motion based on the nature of solutions to the auxiliary equation of the above differential equation, however I can't see how I'd show that the motion is damped. The question is five marks so I don't believe saying "There is a force proportional to the speed blah blah blah.." would cut it, and nor would saying "here's your auxiliary equations, here's the roots, that means blah blah blah..." any thoughts?
"A particle moves along a straight line. At time the displacement of the particle from a fixed point O is , where:
Show that the particle performs damped harmonic motion."
^N.B: I believe there to be a mistake here; I think that they mean instead of .
Now, I know the conditions for the various types of damped harmonic motion based on the nature of solutions to the auxiliary equation of the above differential equation, however I can't see how I'd show that the motion is damped. The question is five marks so I don't believe saying "There is a force proportional to the speed blah blah blah.." would cut it, and nor would saying "here's your auxiliary equations, here's the roots, that means blah blah blah..." any thoughts?
just as you say, the given equation (but with 3 multiplying x) is the standard equation for unforced, underdamped SHM.
i think the key must be (question should really, i think, say "verify that...") to show that, if we have , then since , this verifies the system is underdamped, together with the solution that, when t is small, has large(ish) oscillations, but dies the larger t gets
- other than that, i really can`t see what else you can say to answer it!
i think the key must be (question should really, i think, say "verify that...") to show that, if we have , then since , this verifies the system is underdamped, together with the solution that, when t is small, has large(ish) oscillations, but dies the larger t gets
- other than that, i really can`t see what else you can say to answer it!
Original post by Ktulu666
Hey everyone, came across this question in the exam style paper at the back of the M4 textbook (any of you who have used this will know that they aren't exactly stellar on their explanations of their answers, and due to the outdated LiveText software I can't actually view Solution Bank):
"A particle moves along a straight line. At time the displacement of the particle from a fixed point O is , where:
Show that the particle performs damped harmonic motion."
^N.B: I believe there to be a mistake here; I think that they mean instead of .
Now, I know the conditions for the various types of damped harmonic motion based on the nature of solutions to the auxiliary equation of the above differential equation, however I can't see how I'd show that the motion is damped. The question is five marks so I don't believe saying "There is a force proportional to the speed blah blah blah.." would cut it, and nor would saying "here's your auxiliary equations, here's the roots, that means blah blah blah..." any thoughts?
"A particle moves along a straight line. At time the displacement of the particle from a fixed point O is , where:
Show that the particle performs damped harmonic motion."
^N.B: I believe there to be a mistake here; I think that they mean instead of .
Now, I know the conditions for the various types of damped harmonic motion based on the nature of solutions to the auxiliary equation of the above differential equation, however I can't see how I'd show that the motion is damped. The question is five marks so I don't believe saying "There is a force proportional to the speed blah blah blah.." would cut it, and nor would saying "here's your auxiliary equations, here's the roots, that means blah blah blah..." any thoughts?
The AE is x^2+2x+3, which has roots . So the general solution is and the particle exhibits damped harmonic motion.
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