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A 1 lb weight is suspended from a spring. Let y give the deflection (in inches) of the weight from its static deflection position, where “up” is the positive direction for y. If the static deflection is 24 in, find a differential equation for y. Solve, and determine the period and frequency of the SHM of the weight if it is set in motion.

To my understanding, the initial condition is y(0) = 24. I have a differential equation of the form $my'' + ky = 0$, and I know that m = 1, but how do I find k?

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  • $\begingroup$ The static deflection of 24 inches is not your initial condition. Use this deflection to calculate the spring constant by drawing a force diagram for the mass: gravity pulling the weight down has to balance the spring force pulling it up. $\endgroup$ Oct 25, 2012 at 7:28
  • $\begingroup$ We have kx = mg, so k = mg/x, correct? Do i just use m = 1 and g = 9.8? Do we express k in terms of x? $\endgroup$ Oct 25, 2012 at 7:36
  • $\begingroup$ That's the idea, but you need to think about your units. g = 9.8 uses meters and kilograms. The extension you have is in inches and the mass you have is in pounds. $\endgroup$ Oct 25, 2012 at 7:41
  • $\begingroup$ So can I do $k = \frac{mg}{x}$ => $k = \frac{0.4536 kg * 386 in}{x *kg}$ so that $k = 175$ in? $\endgroup$ Oct 25, 2012 at 7:45
  • $\begingroup$ So the differential equation will become: $y(x) = \alpha_1 \cos (\sqrt{175}x) + \alpha_2 \sin(\sqrt{175}x)$ But how do I solve for a1 and a2? $\endgroup$ Oct 25, 2012 at 7:45

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Hint: The characteristic equation for your differential equation is: $$ m\lambda^2 + k = 0$$ which is equivalent to $ \lambda^2 + \frac{k}{m} = 0$. Therefore, $\lambda = \pm \sqrt{\frac{k}{m}}$. And hence the solutions for the differential equation are: $$ y(x) = \alpha_1\cos \left (\sqrt{\frac{k}{m}}x \right) + \alpha_2\sin \left (\sqrt{\frac{k}{m}}x \right) $$ where $\alpha_i \in \mathbb{R}$.

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  • $\begingroup$ Ok, now do I solve for k and m? Particularly, do I have that k = 1/24 and W = mg, thus m = W/g or 1/386.4, since we want units of gravity in inches? $\endgroup$ Oct 25, 2012 at 7:32
  • $\begingroup$ You know $m =1$, so $y$ becomes a simpler equation. Now use the data you have to find $k$! $\endgroup$
    – ILoveMath
    Oct 25, 2012 at 7:34
  • $\begingroup$ Is the "data I have" just the initial condition of y(0) = 24? If so, How can I solve for two unknowns if I don't have an initial condition for the first derivative? $\endgroup$ Oct 25, 2012 at 7:35

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