M4 Damped and Forced Harmonic Oscillations Watch

JRichardson12
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This is question 9 from Exercise 4A of the old Edexcel M4 Textbook:

A light spring AB, having natural length a and modulus of elasticity 3amn^2, lies straight and at its natural length at rest on a horizontal table. A particle of mass m is attached to the end A.
The end B is then moved in a straight line in the direction AB with constant speed V. The resulting motion of the particle is resisted by a force of magnitude 4mnv where v is the speed of the particle. If x is the extension of the spring at time t , show that

\frac{d^2x}{dt^2}+4n\frac{dx}{dt  }+3n^2x=4nV

Obtain x in terms of t.


I have done the first part of the question have got an answer for the second part of the question of: x=\frac{2V}{3n}e^{-3nt}-\frac{2V}{n}e^{-nt}  +\frac{4V}{3n}, by using the initial conditions of x=0, \frac{dx}{dt}=0 when t=0.

However, the answer given in the book is x=\frac{V}{6n}(8-9e^{-nt}+e^{-3nt}) which seems to have used an initial condition of \frac{dx}{dt}=V when t=0.

So, which initial condition is correct for this situation?
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physicsmaths
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(Original post by JRichardson12)
This is question 9 from Exercise 4A of the old Edexcel M4 Textbook:

A light spring AB, having natural length a and modulus of elasticity 3amn^2, lies straight and at its natural length at rest on a horizontal table. A particle of mass m is attached to the end A.
The end B is then moved in a straight line in the direction AB with constant speed V. The resulting motion of the particle is resisted by a force of magnitude 4mnv where v is the speed of the particle. If x is the extension of the spring at time t , show that

\frac{d^2x}{dt^2}+4n\frac{dx}{dt  }+3n^2x=4nV

Obtain x in terms of t.


I have done the first part of the question have got an answer for the second part of the question of: x=\frac{2V}{3n}e^{-3nt}-\frac{2V}{n}e^{-nt}  +\frac{4V}{3n}, by using the initial conditions of x=0, \frac{dx}{dt}=0 when t=0.

However, the answer given in the book is x=\frac{V}{6n}(8-9e^{-nt}+e^{-3nt}) which seems to have used an initial condition of \frac{dx}{dt}=V when t=0.

So, which initial condition is correct for this situation?
Is it not just the dx/dt=V one ? Due to the information of a constant speed.


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JRichardson12
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(Original post by physicsmaths)
Is it not just the dx/dt=V one ? Due to the information of a constant speed.


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There's an example in the new book that's very similar where they take \frac{dx}{dt}=0:

But, then in the next question of the old book, they have another question where they have a initial condition equal to the constant speed.
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physicsmaths
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(Original post by JRichardson12)
There's an example in the new book that's very similar where they take \frac{dx}{dt}=0:

But, then in the next question of the old book, they have another question where they have a initial condition equal to the constant speed.
I have the book and i am currently working through M4 aswell,


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physicsmaths
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(Original post by JRichardson12)
There's an example in the new book that's very similar where they take \frac{dx}{dt}=0:

But, then in the next question of the old book, they have another question where they have a initial condition equal to the constant speed.
Which questions are you taking about ?


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JRichardson12
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(Original post by physicsmaths)
Which questions are you taking about ?


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Ah, sorry. I meant to be clearer. I mean question 10 of Exercise 4A of the old book on page 99, with the original question in my first post being question 9 of Exercise 4A.

Also, I mean the example on page 86 of the new book in case that was also unclear.
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