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A pendulum swings back and forth. The angular displacement of the pendulum
from its rest position after t seconds is given by the function 0 = 15 cos(3mt), where
0 is measures in degrees.


Sagot :

The motion of the pendulum is a repetitive motion and the time at which

the displacement is maximum, can be found by differentiation.

[tex]The \ times \ in \ seconds \ are; -1\dfrac{2}{3}, \ -1, \ 0, \ \dfrac{1}{3}, \ \dfrac{2}{3}, \ 1\dfrac{1}{3} \\[/tex]

Reasons:

The given function for the angular displacement from the rest location, θ(t),

is presented as follows;

θ(t) = 15·cos(3·π·t)

Where;

t = The time of displacement of the pendulum

Required:

To find the times, t, when θ(t) is greatest

Solution:

When θ(t) is greatest, θ'(t) = 0

Therefore;

[tex]\dfrac{d}{dt} \theta(t) = \dfrac{d}{dt} \left(15 \cdot cos(3 \cdot \pi \cdot t) \right) = -15 \cdot 3 \cdot sin (3 \cdot t \cdot \pi ) = 0[/tex]

When sin(3·t·π) = 0, we have;

[tex]t = -\dfrac{2 \cdot n_1 - 1}{3}[/tex] or [tex]t = \dfrac{2 \cdot n_1 }{3}[/tex]

Where;

n₁ = An integer

Therefore, the times, t, in seconds are;

[tex]t = \dfrac{1}{3}, \ -1, \ -1\dfrac{2}{3}, ...[/tex]

[tex]t = 0, \ \dfrac{2}{3}, \ 1\dfrac{1}{3}, ...[/tex]

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