To find the period of the simple harmonic motion given by the equation [tex]\( d = 9 \cos \left(\frac{\pi}{2} t\right) \)[/tex], follow these steps:
1. Identify the standard form of the cosine function: The general form for a cosine function in simple harmonic motion is [tex]\( d = A \cos(Bt + C) \)[/tex], where [tex]\( A \)[/tex] is the amplitude, [tex]\( B \)[/tex] is the angular frequency, [tex]\( t \)[/tex] is time, and [tex]\( C \)[/tex] is the phase shift.
2. Determine the angular frequency [tex]\( B \)[/tex]: In the given equation [tex]\( d = 9 \cos \left(\frac{\pi}{2} t\right) \)[/tex], the term [tex]\(\frac{\pi}{2}\)[/tex] is the angular frequency [tex]\( B \)[/tex].
3. Use the formula for finding the period: The period [tex]\( T \)[/tex] of a cosine function [tex]\( \cos(Bt) \)[/tex] is given by the formula:
[tex]\[
T = \frac{2\pi}{B}
\][/tex]
4. Substitute the angular frequency [tex]\( B \)[/tex] into the period formula: Since [tex]\( B = \frac{\pi}{2} \)[/tex], substitute this into the formula:
[tex]\[
T = \frac{2\pi}{\frac{\pi}{2}}
\][/tex]
5. Simplify the expression: To simplify the period [tex]\( T \)[/tex]:
[tex]\[
T = \frac{2\pi}{\frac{\pi}{2}} = 2\pi \cdot \frac{2}{\pi} = 4
\][/tex]
Therefore, the period [tex]\( T \)[/tex] of the simple harmonic motion given by the equation [tex]\( d = 9 \cos \left(\frac{\pi}{2} t\right) \)[/tex] is [tex]\( 4 \)[/tex] units.
