In an experiment similar to the one you did in lab the follo

In an experiment similar to the one you did in lab, the following data was recorded for a spring obeying Hooke\'s Law. After a careful inspection of the data, formally report your best estimates for the spring constant k in N/m, and the equilibrium position r_e in meters. See the Excel sheet provided for a copy of this table. b) What was the linear correlation coefficient, R^2, for the data you used to find k and r_e? R^2 = (To 4 decimal places) c) As in lab, this 9.96 gram spring and an attached 81.76 gram brass cylinder is spun in a centripetal motion device. After the analysis, an experimental effective mass of system is found. m_eff, exp = 88.23 g. The spring is oddly shaped however, and we are not sure how much it contributes to the effective mass. We still assume theoretical effective mass is m_eff, acc = m_cyl + n = m_s where we want to find n. Perform a series of percent fractional errors scanning these possible n\'s: 0, 1/4, 1/3, 1/2, 2/3, 3/4, and 1. (You just need one sample calculation.) What seems to be the most likely value for n?

Solution

ans (a)

For the mass spring system:

Applied weight w = k(r-r_e) = kr-kr_e ---(1)

where k is the spring constant and r is the extension.

This equation is of the form y=mx+b

Here, the sprign constant k= ( W2-W1)/ (r2-r1)

Choosing w2=24N r2=0.1700m and w1=20 N , r1=0.1439

we get: k =156.2567 N/m

Now using equ(i)

re= r- w/k

Using the data: r2=0.17 m , w2=24 kg and the calculated value of k, we get

re=0.17 - (24/156.2567)=0.0134 m

Ans (b)

From the graph: the coefficient of regression R2=0.6046

Ans (c)

Effective mass (actual) of the system is defined as : m_eff,acc= m_Cylin + n* m_s, --(2)

Now percent fractional error= ( m_{eff ,acc}-m_eff,exp)/ m_eff,acc X100 % --(3)

Obeserved m_eff,exp= 88.23g --(4)

For the different values of n, m_{eff,acc ,} are found from equ(2) and PFE are calculated

n meff,acc=81.27 +n*9.96 PFE

0 81.27 8.56

1/4 83.76 5.34

1/3 84.59 4.30

1/2 86.25 2.30

2/3 87.91 0.36

3/4 88.74 0.57

1 91.23 3.29

The likely value of n is 2/3