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Springs: speed and acceleration

  1. Apr 24, 2008 #1
    1. The problem statement, all variables and given/known data
    A 0.450 kg object attached to a spring with a force constant of 8.00 N/m vibrates in simple harmonic motion with an amplitude of 11.0 cm.
    Calculate the maximum value (magnitude) of its speed and acceleration.[/B]

    3. The attempt at a solution

    I found both w and t.

    [tex] \omega = 4.216[/tex]
    [tex] t = 0.145[/tex]

    To get t I solved this equation for t.

    [tex] x(t) = Acos(\omega t + \phi)[/tex]
    [tex]t = \frac{1}{4.216}cos^-1(\frac{x}{11})[/tex]

    With that value of t, I solved for v. I got a negative number which is incorrect. However, if at the start I use sine instead of cosine for position i get a different value for t, and get the same answer for v, except positive, which is correct. But why would I use sine for position rather then cosine, and how would I know? Also why would it be positive or negative?
  2. jcsd
  3. Apr 24, 2008 #2


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    Why are you solving for 't'. What t? Just put x(t)=A*cos(omega*t). Then v(t)=x'(t) and a(t)=x''(t). Work those out. Now to the find the maxes you don't even have to worry about the trig function. The max of either sin or cos is 1.
  4. Apr 24, 2008 #3
    I am sorry... I left off part of the question...

    (b) Calculate the speed and acceleration when the object is 9.00 cm from the equilibrium position.

    That is why I was finding t
  5. Apr 24, 2008 #4


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    Ah. In that case the problem doesn't give you enough information to determine a sign for v or a. The object could be 9 cm from equilibrium and either moving away from or towards equilibrium. They seem to just want the magnitude of v and a. Not the sign. You can use either sin or cos. They only differ by a change the phase angle (phi).
  6. Apr 24, 2008 #5


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    As an alternative, you still are not required to find the time for part b. You can use conservation of energy to find the speed and a force diagram for the magnitude of the acceleration.
  7. Apr 24, 2008 #6
    Yes thats what thought, but I submitted Acceleration with a postive sign, and it was counted wrong, but with a negative sign correct.

    I'll try this method thank you.
  8. Apr 24, 2008 #7
    Also, the thing I didn't understand was... I have a solutions manual and this problem was in there. I checked the book and it started with x(t) = Asin(wt + phi), rather then cosine. Which gave the correct sign for direction. But how would you know whether to use cosine or sine? Generally you use cosine don't you?
  9. Apr 24, 2008 #8


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    That's silly. If they had wanted you to get that specific setup, they should have told you. sin(x+pi/2)=cos(x). There is no difference except for the phase. Even if you use sin, there are 4 times every cycle when the object is 9cm away from equilibrium - and they have all possible combinations of signs for a and v.
  10. Apr 24, 2008 #9
    Well, Thanks for your help.
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