To determine the period of the simple harmonic motion given by the equation [tex]\( d = 9 \cos \left(\frac{\pi}{2} t\right) \)[/tex], we can follow these steps:
1. Identify the general form of the cosine function in simple harmonic motion:
The equation for simple harmonic motion is generally written as:
[tex]\[ d = A \cos(Bt + C) \][/tex]
where:
- [tex]\(A\)[/tex] is the amplitude,
- [tex]\(B\)[/tex] affects the period [tex]\(T\)[/tex],
- [tex]\(C\)[/tex] is the phase shift,
- [tex]\(t\)[/tex] is time.
2. Extract the value of [tex]\(B\)[/tex]:
In the equation [tex]\( d = 9 \cos \left(\frac{\pi}{2} t\right) \)[/tex], we can see that the coefficient of [tex]\(t\)[/tex] inside the cosine function is [tex]\(\frac{\pi}{2}\)[/tex]. Thus, in this case, [tex]\(B = \frac{\pi}{2}\)[/tex].
3. Understand the relationship between [tex]\(B\)[/tex] and the period [tex]\(T\)[/tex]:
The period [tex]\(T\)[/tex] of a cosine function [tex]\( \cos(Bt) \)[/tex] is defined as the time it takes for the function to complete one full cycle. The relationship between [tex]\(B\)[/tex] and the period [tex]\(T\)[/tex] is given by:
[tex]\[ T = \frac{2\pi}{B} \][/tex]
4. Calculate the period using the identified [tex]\(B\)[/tex]:
Substitute [tex]\( B = \frac{\pi}{2} \)[/tex] into the formula for the period:
[tex]\[ T = \frac{2\pi}{\frac{\pi}{2}} \][/tex]
Simplifying this expression, we get:
[tex]\[ T = \frac{2\pi}{\frac{\pi}{2}} = \frac{2\pi \times 2}{\pi} = \frac{4\pi}{\pi} = 4 \][/tex]
So, the period [tex]\(T\)[/tex] of the simple harmonic motion given by [tex]\( d = 9 \cos \left(\frac{\pi}{2} t\right) \)[/tex] is [tex]\(4\)[/tex].
Answer:
[tex]\[ \boxed{4} \][/tex]
