inkyvoyd
  • inkyvoyd
A m kg block slides along a horizontal surface with friction. The block has a speed v when it strikes a massless spring head-on (as in the figure).
Physics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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inkyvoyd
  • inkyvoyd
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inkyvoyd
  • inkyvoyd
What minimum value of the coefficient of static friction, ┬Ás, will assure that the spring remains compressed at the maximum compressed position? Assume the spring has a spring constant k.
inkyvoyd
  • inkyvoyd
I did the system 1/2 mv^2=1/2kx^2+u m g x for energy conservation

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inkyvoyd
  • inkyvoyd
u m g=k x for spring remaining compressed at the maximum compressed position
Empty
  • Empty
Cool this looks fun, but the thing is we can't use energy conservation here because friction isn't a conservative force, so it's path dependent and not a state function.
inkyvoyd
  • inkyvoyd
fudge
Empty
  • Empty
Well let's try to piece this together, it looks like we can lay out a few variables and stuff and see what happens. Give me a sec to fiddle with some symbols on my paper here.
inkyvoyd
  • inkyvoyd
the first part of the problem specified the static friction, and asked what the compression would be, and I got that right using \(\large 1/2 mv^2=\int^x_0kx+umg \text{ }ds\) which is how I got \(1/2mv^2=1/2kx^2+umgx\) then I solved for x and put that in, and it said it was right... did I get it right cause I was lucky even though the method is invalid, or did that integral exhibit path dependency?
Empty
  • Empty
Yeah this is fine cause you are only going to the maximum, now if it were to go past that and say go back to spring equilibrium, then you'd feel the friction force again on the way back. So this integral is valid yeah.
Empty
  • Empty
So wait, is u your coefficient of static friction?
inkyvoyd
  • inkyvoyd
yeah
Empty
  • Empty
I think we might be able to take the derivative of u with respect to x and set it equal to 0 to find the minimum of u?
inkyvoyd
  • inkyvoyd
(sorry for delayed responses, I'm working on other parts of this assignment that aren't so conceptual and more tedious)
Empty
  • Empty
Yeah sure I'm just kinda having fun here anyways I haven't really touched this sorta stuff in a while. I'm not sure about this derivative though, I guess because v and x will be a constant at the maximum.
inkyvoyd
  • inkyvoyd
well I got du/dx=-(kx+umg)/(mgx)
inkyvoyd
  • inkyvoyd
which, as expected, gives me kx=umg hahaha
inkyvoyd
  • inkyvoyd
then I did the system; 1/2 mv^2=1/2kx^2+u m g x kx=umg and got u=(v sqrt(k))/(g sqrt(3m))
inkyvoyd
  • inkyvoyd
which... is wrong
inkyvoyd
  • inkyvoyd
oh wait that's stupid, that assumes u_s=u_k which is totally not true I'm stoopid
Astrophysics
  • Astrophysics
That should work
Astrophysics
  • Astrophysics
\[kx= u mg\]
inkyvoyd
  • inkyvoyd
yeah I got confused cause I assumed u_s=u_k for some reason and then subsituted that equation into the energy equation.
Astrophysics
  • Astrophysics
Ah ok I see
inkyvoyd
  • inkyvoyd
figured it out now tho
inkyvoyd
  • inkyvoyd
thanks
Astrophysics
  • Astrophysics
Cool question though, thanks for sharing :)

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