ACTIVITY ON OSCILLATIONS' SYNTHESIS (EXPERIMENTAL SOLUTION) (1st PROBLEM: DIFFERENT WIDTHS, SAME 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 x1(t), t = 0 (0.5) T1+T2 and we are plotting the corresponding points (t, x1(t)), t = 0 (0.5) T1+T2

TableForm[Table[{t, N[x1[t]]}, {t, 0, T1 + T2, 0.5}], TableHeadings→ {None, {"t", "x1(t)"}}, TableAlignments→Center, TableDirections→Row]

t 0 0.5 1. 1.5 2. 2.5 3. 3.5 4. 4.5 5. 5.5 6. 6.5 7. 7.5 8. 8.5 9. 9.5 10. 10.5 11. 11.5 12. 12.5 13. 13.5 14.
x1(t) 0. 10. 17.3205 20. 17.3205 10. 2.44929*10^^-15 -10. -17.3205 -20. -17.3205 -10. -4.89859*10^^-15 10. 17.3205 20. 17.3205 10. 7.34788*10^^-15 -10. -17.3205 -20. -17.3205 -10. -9.79717*10^^-15 10. 17.3205 20. 17.3205

ListPlot[Table[{t, N[x1[t]]}, {t, 0, T1 + T2, 0.5}], AspectRatio→1, PlotStyle→ {AbsolutePointSize[5], RGBColor[1, 0, 0]}, AxesLabel→ {"t", "x1(t)"}]

[Graphics:../HTMLFiles/Physics, Oscillations_103.gif]

-Graphics -

3) We are calculating the values of x2(t), t = 0 (0.5) T1+T2 and we are plotting the corresponding points (t, x2(t)), t = 0 (0.5) T1+T2

TableForm[Table[{t, N[x2[t]]}, {t, 0, T1 + T2, 0.5}], TableHeadings→ {None, {"t", "x2(t)"}}, TableAlignments→Center, TableDirections→Row]

t 0 0.5 1. 1.5 2. 2.5 3. 3.5 4. 4.5 5. 5.5 6. 6.5 7. 7.5 8. 8.5 9. 9.5 10. 10.5 11. 11.5 12. 12.5 13. 13.5 14.
x2(t) 0. 7.65367 14.1421 18.4776 20. 18.4776 14.1421 7.65367 2.44929*10^^-15 -7.65367 -14.1421 -18.4776 -20. -18.4776 -14.1421 -7.65367 -4.89859*10^^-15 7.65367 14.1421 18.4776 20. 18.4776 14.1421 7.65367 7.34788*10^^-15 -7.65367 -14.1421 -18.4776 -20.

ListPlot[Table[{t, N[x2[t]]}, {t, 0, T1 + T2, 0.5}], AspectRatio→1, PlotStyle→ {AbsolutePointSize[5], RGBColor[1, 0, 0]}, AxesLabel→ {"t", "x2(t)"}]

[Graphics:../HTMLFiles/Physics, Oscillations_107.gif]

-Graphics -

4) We are calculating the values of x(t), t = 0 (0.5) T1+T2 and we are plotting the corresponding points (t, x(t)), t = 0 (0.5) T1+T2

TableForm[Table[{t, N[x[t]]}, {t, 0, T1 + T2, 0.5}], TableHeadings→ {None, {"t", "x(t)"}}, TableAlignments→Center, TableDirections→Row]

t 0 0.5 1. 1.5 2. 2.5 3. 3.5 4. 4.5 5. 5.5 6. 6.5 7. 7.5 8. 8.5 9. 9.5 10. 10.5 11. 11.5 12. 12.5 13. 13.5 14.
x(t) 0. 17.6537 31.4626 38.4776 37.3205 28.4776 14.1421 -2.34633 -17.3205 -27.6537 -31.4626 -28.4776 -20. -8.47759 3.17837 12.3463 17.3205 17.6537 14.1421 8.47759 2.67949 -1.52241 -3.17837 -2.34633 -2.44929*10^^-15 2.34633 3.17837 1.52241 -2.67949

ListPlot[Table[{t, N[x[t]]}, {t, 0, T1 + T2, 0.5}], AspectRatio→1, PlotStyle→ {AbsolutePointSize[5], RGBColor[1, 0, 0]}, AxesLabel→ {"t", "x(t)"}]

[Graphics:../HTMLFiles/Physics, Oscillations_111.gif]

-Graphics -

5) We are plotting the points (t, x(t)), t = 0 (0.1) 4(T1+T2)

ListPlot[Table[{t, N[x[t]]}, {t, 0, 4 * (T1 + T2), 0.1}], AspectRatio→1, PlotStyle→ {AbsolutePointSize[5], RGBColor[1, 0, 0]}, AxesLabel→ {"t", "x(t)"}]

[Graphics:../HTMLFiles/Physics, Oscillations_114.gif]

-Graphics -

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

[Graphics:../HTMLFiles/Physics, Oscillations_117.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 frequencies w1 = w2 and widths A1, A2.
Are the results on the former three questions similar?



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