The time (in seconds) that it take the pendulum to complete one full cycle is given by: Option C: 6 (seconds)
What is the period of sine function?
Trigonometric functions are periodic. They repeat their values after some specific constant input interval (constant depends on the trigonometric function chosen).
For sine function, its period is 2π
Thus, we get:
[tex]y = \sin(x) = \sin(x+2\pi) = \sin(x + 2\pi n) ; n \in \mathbb Z[/tex] integer, and y is some constant at some constant value of x)
For this case, the pendulum's motion is modeled by:
[tex]P(t) = -10\sin\left(\dfrac{\pi}{3} t\right)[/tex]
Let it takes 'd' seconds to do one full cycle, then we get:
[tex]P(t) = P(t+d) = P(t+2d) = P(t + nd); n \in \mathbb Z - \mathbb Z^{-}[/tex] (n is non-negative integer).
Using the formula, we get:
[tex]-10\sin\left(\dfrac{\pi}{3} t\right) = -10\sin\left(\dfrac{\pi}{3} (t + nd)\right)\\\sin\left(\dfrac{\pi}{3} t\right) =\sin\left(\dfrac{\pi}{3}t + \dfrac{\pi}{3}nd\right)[/tex]
But we know that:
[tex]\sin\left(\dfrac{\pi}{3} t\right) =\sin\left(\dfrac{\pi}{3}t +2\pi n\right)[/tex]
Thus, we get:
[tex]2\pi n = \dfrac{\pi}{3}nd\\[/tex] (compared for non-negative values of n)
This gives:
[tex]d = \dfrac{2\pi n \times 3}{\pi n} = 6[/tex] (in seconds)
Thus, the time (in seconds) that it take the pendulum to complete one full cycle is given by: Option C: 6 (seconds)
Learn more about periodic functions here:
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