Modeling Exercise 6.3.3 Count the number of oscillations that the spring goes through over the course of the data set, and use it to estimate the period P^(**) of this damped spring-mass system. What is the corresponding value for omega ?
Modeling Exercise 6.3.4 The value of alpha in (4.118) dictates the rate of decay of the oscillations. Find a good choice for alpha . Hint: take a guess at alpha , then plot y(t) in (4.118) with d_(2)=0 and the values you obtained for d_(1) and omega ; then adjust alpha .
Modeling Exercise 6.3.5 Given that
alpha =(c)/(2m), and ,omega =(sqrt(4mk-c^(2)))/(2m),
and that you know m=0.2kg, come up with estimates for c and k.
The spring constant was determined experimentally to be k=17.306, by plotting force versus displacement data for a variety of different masses and using Hooke's law to fit a linear relationship F=k*x, where x is the displacement of the mass and F is the force in newtons necessary to obtain that displacement. How does your estimate of k compare?
Modeling Exercise 6.3.6 Suppose that m itself had not been measured. Would it be possible to estimate all three parameters, m,c, and k, from the data? If so, how? If not, why not?
In the project "Frequency Analysis of Signals" in Section 8.5.4 we develop far more efficient techniques for estimating the period and frequency of oscillatory signals like that of this spring-mass system.
keacosine Modeling Exercise 6.3.3 Count the number of oscillations that the spring goes through over the is the corresponding value for w? course of the data set, and use it to estimate the period P* of this damped spring-mass system. What Modeling Exercise 6.3.4 The value of a in (4.118dictates the rate of decay of the oscillations Find a good choice for . Hint: take a guess at , then plot y(t) in (4.118) with d = 0 and the values you obtained for d and w;then adjust Modeling Exercise 6.3.5 Given that C 2m and = V4mk-c2 2m and that you know m=0.2 kg,come up with estimates for c and k The spring constant was determined experimentally to be k = 17.306, by plotting force versus displacement data for a variety of different masses and using Hooke's law to fit a linear relationship F = k - x, where x is the displacement of the mass and F is the force in newtons necessary to obtain that displacement. How does your estimate of k compare? Modeling Exercise 6.3.6 Suppose that m itself had not been measured. Would it be possible to estimate all three parameters, m, c, and k, from the data? If so, how? If not, why not? In the project Frequency Analysis of Signals" in Section 8.5.4 we develop far more efficient techniques for estimating the period and frequency of oscillatory signals like that of this spring-mass system.