the mass of the glider (m) and the effective spring constant (k).With a balance you measure the mass of the glider to be m = 0.253kg. [We ignore the mass of the springs]

d (m) F (N)

0.4293 0.212
0.4677 0.434
0.5062 0.657
0.5445 0.879

1. Use linear least squares to find the slope of this force vs.position data. From this slope, calculate the effective springconstant k:
N/M
2. Calculate the natural frequency fF you should expect for thismechanical oscillator:
Hz

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Atoms of mass M are in a linear chain with lattice spacing a,coupled with nearest neighbor forces characterized by springconstant C. According to the textbook, the phonon frequencies are =(4C/M)1/2 |sin Â½ Ka|
where K is the wavenumber of the phonon. We modify this system byadding another spring of spring constant Câ to each atom, so thatthis spring couples the atoms to some heavy mass that cannot move.Therefore we will have an additional restoring force on atom s,described by F's= -C'us, where us is the displacement.

a) Calculate the phonon frequencies ?????for this modified system.Make a graph of the result for Câ=C/10, Câ=C and Câ=10C

b) Look at the frequency at K=0. If you did the calculation right,the result should be ?=(Câ/M)1/2 . Explain this result.

An oscillator of mass m = 2 kg, whose natural frequency ofoscillation is 10 radians/s , feels a force when the oscillator isin motion,
F = -k x - m? d x/dt.
What is the value of the spring constant k?

1. 50 N/m
2. 150 N/m
3. 100 N/m
4. 250 N/m
5. 200 N/

At what value of ? will the damping rate of
the spring be ?/2, without any oscillations of
the spring?
1. 50 s^-1
2. 40 s^-1
3. 20 s^-1
4. 10 s^-1
5. 30 s^-1

I am fairly lost on this problem, so as much explanation aspossible would be greatly appreciated.

A group of students performed the "SpringâMass" experiment usinga spring of non-negligible mass Ms kg and spring constant k N/m.The students determined the period of oscillation for sevendifferent values of the hanging mass m. They obtained the followingresults:
m (g) T (s)
50 0.147
70 0.167
90 0.185
110 0.201
130 0.216
150 0.230
170 0.244

Note: In lab the spring's mass was negligible but here you need toinclude 1/3 of Ms with the hanging mass. Consider how this willchange the equation for the springâmass system.
(a) Use Excel to construct a spreadsheet and do the following. (Youwill not submit this spreadsheet. However, the results will beneeded later in this problem.)
(i) Enter the above data.
(ii) Create a graph of T2 vs. m.
(iii) Use the LINEST function to determine the slope, y interceptand the associated uncertainties.

(b) Record your values below.
slope = Â±
y-intercept = Â±

(c) What is the formula for the uncertainty in k in terms of theuncertainty in the slope? (Use the following variable as necessary:k and Ï_s for the uncertaintyin the slope.)
Ï_k =

(d) What is the formula for the uncertainty in Ms the mass of thespring, in terms of the uncertainty in the y intercept andÏ_y? (Use the following asnecessary: Ms, yi for the y intercept,Ï_s, and slope.)
Ï_Ms =

(e) From your value of the slope determine the spring constant k ofthe spring.
k = Â±

(f) From your value of the intercept determine the mass Ms of thespring.
Ms = Â±

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