Completely lost on oscillations problem

  • #1
265
2

Homework Statement


Two springs are joined and connected to a mass m such that they are all in a straight line. The two springs are connected first and then the mass last so that all three are in a row. If the springs have a stiffness of k1 and then k2, find the frequency of oscillation of m.


Homework Equations


[itex] T = 2 \pi \sqrt{\frac{m}{k}} [/itex]

The Attempt at a Solution



so i tried making an F=ma for the mass and spring 1 (which is said was the spring closer to the mass)...

F=ma system mass

[itex] F_{el} = ma [/itex]
[itex] k_1 x_1 = ma [/itex]
max acceration happens at aplitude:
[itex] k_1 x_1 = mA \omega ^2 [/itex]

F=ma system spring 1.

[itex] F_{el mass} - F _{el 2} = M_{s1} [/itex]
i am assuming the spring is massless ( i think we can do that)
so [itex] F_{elmass} = F_{el 2} [itex]
[itex] k_1 x_1 = k_2 x_2 [/itex]

i suppose [itex] x_1 + x_2 = A [/itex] when both x's are at maximum. ...
so [itex] \frac {k_1}{k_1} = k_2 (A-x_1) [/itex]

[itex] x_1 = \frac {A}{ \frac {k_1}{k_2} + 1 } [/itex]

go back the the last equation we got in f=ma system mass and subtitute in the x_1 we just found....

the A's cancel out and after we simply we get:

[itex] \omega = \sqrt{ \frac{k_1 k_2}{ (k_1 + k_1) m }} [/itex] ////
and to get F... just divide it by 2pi... right?
is this even correct?
 

Answers and Replies

  • #2
14
2
try it with energy method you would be albe to solve it. if not reply i will post the solution.
 
  • #3
SammyS
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Science Advisor
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try it with energy method you would be albe to solve it. if not reply i will post the solution.
Be careful about this in the homework section of PF !
 
  • #4
265
2
Be careful about this in the homework section of PF !
do you see a problem with my solution?
 
  • #5
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,317
1,007

Homework Statement


Two springs are joined and connected to a mass m such that they are all in a straight line. The two springs are connected first and then the mass last so that all three are in a row. If the springs have a stiffness of k1 and then k2, find the frequency of oscillation of m.

Homework Equations


[itex] T = 2 \pi \sqrt{\frac{m}{k}} [/itex]

The Attempt at a Solution


so i tried making an F=ma for the mass and spring 1 (which is said was the spring closer to the mass)...
F=ma system mass
[itex] F_{el} = ma [/itex]
[itex] k_1 x_1 = ma [/itex]
max acceration happens at aplitude:
[itex] k_1 x_1 = mA \omega ^2 [/itex]

F=ma system spring 1.
[itex] F_{el mass} - F _{el 2} = M_{s1} [/itex]
i am assuming the spring is massless ( i think we can do that)
so [itex] F_{elmass} = F_{el 2} [itex]
[itex] k_1 x_1 = k_2 x_2 [/itex]

i suppose [itex] x_1 + x_2 = A [/itex] when both x's are at maximum. ...
so [itex] \frac {k_1}{k_1} = k_2 (A-x_1) [/itex]

[itex] x_1 = \frac {A}{ \frac {k_1}{k_2} + 1 } [/itex]

go back the the last equation we got in f=ma system mass and subtitute in the x_1 we just found....

the A's cancel out and after we simply we get:

[itex] \omega = \sqrt{ \frac{k_1 k_2}{ (k_1 + k_1) m }} [/itex] ////
and to get F... just divide it by 2pi... right?
is this even correct?
toesockshoe,

I haven't examined your entire solution, but I'm pretty sure that your final answer IS correct !

Two springs connected in that manner have an effective spring constant of ##\displaystyle\ k_\text{eff}=\frac{k_1\,k_2}{k_1+k_2}\ .##
 
Last edited:

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