ACTIVITY ON OSCILLATIONS' SYNTHESIS (EXPERIMENTAL SOLUTION) (2nd PROBLEM: SAME WIDTHS, DIFFERENT FREQUENCIES)     

The position of the body that performs the composite oscillation can be calculated as:
x(t) = x1(t) + x2(t)   

x[t_] := x1[t] + x2[t]

1) Firsty we are calculating the periods of the two oscillations T1 = (2π)/w1, T2 = (2π)/w2

T1 = N[2 * Pi/w1]

T2 = N[2 * Pi/w2]

6.

8.

2) We are calculating the values of x(t), t = 0 (0.2) T1+T2 and we are plotting the corresponding points (t, x(t)), t = 0 (0.2) T1+T2

TableForm[Table[{t, N[x[t]]}, {t, 0, T1 + T2, 0.2}], TableHeadings→ {None, {"t", "x(t)"}}, TableAlignments→Center, TableDirections→Row]

t 0 0.2 0.4 0.6 0.8 1. 1.2 1.4 1.6 1.8 2. 2.2 2.4 2.6 2.8 3. 3.2 3.4 3.6 3.8 4. 4.2 4.4 4.6 4.8 5. 5.2 5.4 5.6 5.8 6. 6.2 6.4 6.6 6.8 7. 7.2 7.4 7.6 7.8 8. 8.2 8.4 8.6 8.8 9. 9.2 9.4 9.6 9.8 10. 10.2 10.4 10.6 10.8 11. 11.2 11.4 11.6 11.8 12. 12.2 12.4 12.6 12.8 13. 13.2 13.4 13.6 13.8 14.
x(t) 0. 7.28692 14.3151 20.8355 26.6186 31.4626 35.2015 37.7106 38.9116 38.7749 37.3205 34.6167 30.7768 25.9549 20.3386 14.1421 7.59747 0.945077 -5.57537 -11.7342 -17.3205 -22.1498 -26.0708 -28.9702 -30.7768 -31.4626 -31.0432 -29.5758 -27.1559 -23.912 -20. -15.5955 -10.8864 -6.06443 -1.31744 3.17837 7.26543 10.8106 13.7101 15.8924 17.3205 17.9916 17.936 17.2145 15.9139 14.1421 12.0221 9.6854 7.26543 4.89087 2.67949 0.732636 -0.869308 -2.07031 -2.84079 -3.17837 -3.10719 -2.6759 -1.95439 -1.02954 -2.44929*10^^-15 1.02954 1.95439 2.6759 3.10719 3.17837 2.84079 2.07031 0.869308 -0.732636 -2.67949

ListPlot[Table[{t, N[x[t]]}, {t, 0, T1 + T2, 0.2}], AspectRatio→1, PlotStyle→ {AbsolutePointSize[5], RGBColor[1, 0, 0]}, AxesLabel→ {"t", "x(t)"}]

[Graphics:../HTMLFiles/Physics, Oscillations_128.gif]

-Graphics -

3) We are plotting the points (t, x(t)), t = 0 (0.1) 6(T1+T2)

ListPlot[Table[{t, N[x[t]]}, {t, 0, 6 * (T1 + T2), 0.1}], AspectRatio→1, PlotStyle→ {AbsolutePointSize[5], RGBColor[1, 0, 0]}, AxesLabel→ {"t", "x(t)"}]

[Graphics:../HTMLFiles/Physics, Oscillations_131.gif]

-Graphics -

4) We are plotting the points (t, x(t)), (t, x1(t)) and (t, x2(t))    for t ∈ [0, T1+T2]

[Graphics:../HTMLFiles/Physics, Oscillations_134.gif]

-Graphics -

Questions:
1) a)  Does the composite oscillation have constant frequency?


b) What is / are the value / s of the frequency?


2) a)  Does the composite oscillation have constant width?


b) What is / are the value / s of the width?


3)  Does the composite oscillation have constant period?


b) What is / are the value / s of the period?


EXERCISE:
Using the commands and programs presented above, study the same problem for another set of values of widths A1 = A2 and frequencies w1, w2.
Are the results on the former three questions similar?



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