How Is Damping Calculated in a Spring-Mass System?

[miann]

Homework Statement

A egg with a mass 0.0500kg is suspended from a spring with a k = 25.0N/m... initially it starts off with a displacement of x = 0.300m but a force F = -bv acts against the egg and after t= 5.00s the displacement is x = 0.100m

Homework Equations

F = -bv
F = -kx
-kx = ma

w = $$\sqrt{}K/m$$

ma = -kx - bv

The Attempt at a Solution

Well our prof didnt show up for class... but the homework is still due and i really have no idea were to start... i have never worked with damping... but i found out that

b = kg/s

Because F/v = kg/s

So if u calculate F at x = 0.300m you get 7.5N and then 2.5N for 0.1m...

thats a difference of 5... and that would be 1N/s loss when divided by time... which would make sense for the overall damping effect... (well I'm guessing... again never learned anything about damping) but if that is right... what 'v' would i use... because it is always changing as the spring looses energy...

So ya, I don't knoe if I'm even going in the right direction... but i am completely stumped on this one.

 

[Proggle]

ma = -kx - bv

This is a differential equation. Treat it as such.

 

[miann]

all I can think of is

x(t) = Acos(wt + $$\phi$$)

v(t) = -wAsin(wt + $$\phi$$)

a(t) = -w$$^{2}$$Acos(wt + $$\phi$$)

 

[miann]

Well... i can set it up as

-kx - b$$dx/dt$$ = m$$d^{2}x/dt^{2}$$

 

[miann]

but i can't think of the second equation so i can solve...

 

[miann]

sorry i haven't done differential equations for about a year.. and even then they were easy ones... I'm trying to figure it out but i have no idea were to start.

 

[Mindscrape]

Typically you want guess exponentials. x = Ae^(rt)

 

[Proggle]

Try
http://www.krellinst.org/UCES/archive/modules/diffeq/node4.html

 

[miann]

kx = -bv

kx - -b$$\frac{dx}{dt}$$

$$\int^{0.1}_{0.3}$$$$\frac{dx}{x}$$ = -$$\int^{t}_{0}$$$$\frac{k}{b}$$dt


ln(0.1) - ln(0.3) = $$\frac{k}{b}$$(5.0s)

1.1 = $$\frac{(25.0N/m)}{b}$$(5.0s)

b = $$\frac{(25.0N/m)(5.0s)}{1.1}$$

b = 113.64 N$$\bullet$$s/m


is that right?