We did an experiment with a vertical spring-mass system. Here is an example of data collected:(adsbygoogle = window.adsbygoogle || []).push({});

Mass------------T (period)

200 g-----------.3 s

400g------------.5 s

My question is why does the period increase as the mass increases?

I know that the amp. and T are independent of each other, but how do you show this mathematically? I now it’s easy to show visually, but are there any ties in deriving this, along with the relationship that T = 1/sqrt(x)?

Then, my teacher mentioned some of the following derivisions. I understand a majority of it, but I don’t know how these two parts directly correspond with each other. Does this look familiar to anyone? I don’t quite follow the M_efficiency line, but I do get the calculus.

1st Part

F = -kx, where T = 2*pi*sqrt(M/k) and Mg = kL (L=x)

M*[(d^2*x)/(dt^2)] = -(k/M)x (second derivative)

(d^2*x)/(dt^2) = -w^2*x, where w (or omega) = 2*pi*f

x(t) = Asin(wt + phi)

This is to show that T is proportional to k, I guess.

2nd Part

M_efficiency = M + (M_spring/3)

k_spring = (1/2) integrate(dx/dt)^2 dm

Proportion

v/L = (dx/dt)/x

mass density p = dm/dx, where p = M_spring/L, where L is length

k_spring = (1/2)*definite integral(xv/L)^2 pdx L to 0

k_spring = ½*[(pL^3v^2)/(3L^2)] = ½*(M_spring/3)v^2

and somehow KE = ½[M + (M_spring/3)]*v^2. How do you get to this point?

Thanks for ANY help.

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# Hooke's Law and Simple Harmonic Motion

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