Question

Q5. Report the slope and uncertainty in the slope of the linear fit on your graph, include the correct units. Q6. Use your reported slope from the graph to calculate the spring constant k, and include the correct units. Show all work. Q7. Using propagation of uncertainties, it can be shown that the uncertainty in the spring constant k may be calculated using Using your graph for the uncertainty in the slope, along with your answer to Q6, calculate the uncertainty in the spring constant. Show all work.

m=2.14 +- 0.46, b=-0.0074+-0.0097, r=0.937. The graph wouldn't upload.

T^2 (s^2)	Weight k(g)
0.0016	0.1
0.0025	0.15
0.0036	0.2
0.0036	0.25
0.0064	0.3

Answer 1

The period of a spring-mass system is given by:

where, m is the mass and k is the spring constant

So, if we plot T² vs m, we will get a straight line and as the slope. If we fit a straight line y = mx + b, y being the T² values and x being the weight (mass) values, then m is the slope and b is the intercept

......... (i)

According to propagation of errors,

....... (ii)

where, is the error in the slope and is the error in k

As the way values of T² in the given table are progressing, I guess there is a typing mistake and the fourth value of T² is 0.0049 instead of 0.0036. So, the corrected table is:

T2 (s2)	Weight (g)
0.0016	0.1
0.0025	0.15
0.0036	0.2
0.0049	0.25
0.0064	0.3

You can fit a straight line to this data and find the slope and intercept using LINEST function in a spreadsheet software. You have to use the function wizard and choose LINEST function and check the array box. Note that T² are the Y values and weight are the X values.

Cell C2 gives the slope, C3 gives the error in slope, D2 gives the y intercept (b) and D3 gives the error in y intercept, C4 gives the value of R²

We see that:

..... slope

..... uncertainty in the slope

from eq (i),

...... spring constant k

from eq (ii),

..... uncertainty in k