3. A particle is subjected to an impulsive force that acts over the timescale-T < t

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3. A particle is subjected to an impulsive force that acts over the timescale-T < t <T
and is zero otherwise. It rises, and then declines, linearly with time according to
F(t) =
Fo
[ Po (1 + 1/1)
| Fo (1-2)
-T<t≤0
0≤t≤T
where Fo is a constant. Show that the Fourier transform of this force is,
1
F (W) = √ √ Lx
√ √ F(t)e-ist dt
√2π
and express A in terms of Fo
Sketch | F(t) against t, and |Â (w)|2 against w
sin² (wT/2)
= A-
w²T
Explain how the features of these diagrams are related to each other and describe
what happens as T → 0
Transcribed Image Text:3. A particle is subjected to an impulsive force that acts over the timescale-T < t <T and is zero otherwise. It rises, and then declines, linearly with time according to F(t) = Fo [ Po (1 + 1/1) | Fo (1-2) -T<t≤0 0≤t≤T where Fo is a constant. Show that the Fourier transform of this force is, 1 F (W) = √ √ Lx √ √ F(t)e-ist dt √2π and express A in terms of Fo Sketch | F(t) against t, and |Â (w)|2 against w sin² (wT/2) = A- w²T Explain how the features of these diagrams are related to each other and describe what happens as T → 0
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