Question

A pendulum has max displacement of A=2.5m and frequency=6Hz. Displacement at time x is x=Asin(wt).where A is max displacement and w is angular velocity 2pief.
Find time when displacement is 90cm.

Using a Power series expansion of above equation verify a time of 0.0416seconds for displacement to reach 250cm

Answer 1

Answer on Question #49389, Engineering, Other

Task:

A pendulum has max displacement of $A = 2.5\,\mathrm{m}$ and frequency $= 6\,\mathrm{Hz}$. Displacement at time $x$ is $x = \mathrm{Asin}(wt)$ where A is max displacement and w is angular velocity 2pief. Find time when displacement is 90cm.

Using a Power series expansion of above equation verify a time of 0.0416 seconds for displacement to reach 250cm

Answer:

$$A = 2.5\,\mathrm{m}; \quad f = 6\,\mathrm{Hz}; \quad x_1 = 90\,\mathrm{cm};$$ $$t_2 = 0.0416; \quad x_2 = 250\,\mathrm{cm}$$ $$x = \mathrm{Asin}(wt) = \mathrm{Asin}(2\pi ft)$$

thus time when displacement is $90\,\mathrm{cm}$: $t_1 = \frac{1}{2\pi f} \arcsin \frac{x_1}{A} = \frac{1}{2\pi \cdot 6} \arcsin \frac{0.9}{2.5} \approx 0.00977\,\mathrm{s}$

Using a Power series expansion of above equation: $\sin wt = wt - \frac{(wt)^3}{6} + O[t]^5$

$$x_0 = A \sin wt_2 = A \left[ wt_2 - \frac{(wt_2)^3}{6} \right] = A \left[ 2\pi ft_2 - \frac{(2\pi ft_2)^3}{6} \right] = 2.5 \left[ 6.28 \cdot 6 \cdot 0.0416 - \frac{(6.28 \cdot 6 \cdot 0.0416)^3}{6} \right] = 1.643\,\mathrm{m}$$

So $x_0 \neq x_2$ and $t_2$ is incorrect.

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