To determine the period of the simple harmonic motion described by the equation [tex]\( d = 5 \sin \left( \frac{\pi}{4} t \right) \)[/tex], we need to follow these steps:
1. Identify the form of the equation:
The equation given is [tex]\( d = 5 \sin \left( \frac{\pi}{4} t \right) \)[/tex]. Here, [tex]\( d \)[/tex] represents the displacement as a function of time [tex]\( t \)[/tex], and it conforms to the general form of simple harmonic motion, which is [tex]\( d = A \sin(Bt) \)[/tex], where:
- [tex]\( A \)[/tex] is the amplitude.
- [tex]\( B \)[/tex] is the angular frequency.
2. Determine the angular frequency ([tex]\( B \)[/tex]):
From the given equation [tex]\( d = 5 \sin \left( \frac{\pi}{4} t \right) \)[/tex], we can see that the angular frequency [tex]\( B \)[/tex] is [tex]\( \frac{\pi}{4} \)[/tex].
3. Calculate the period (T):
The period [tex]\( T \)[/tex] of simple harmonic motion is the time it takes for the motion to complete one full cycle. The formula to calculate the period is given by:
[tex]\[
T = \frac{2\pi}{B}
\][/tex]
where [tex]\( B \)[/tex] is the angular frequency.
4. Substitute the angular frequency into the formula:
Substituting [tex]\( B = \frac{\pi}{4} \)[/tex] into the formula, we get:
[tex]\[
T = \frac{2\pi}{\frac{\pi}{4}}
\][/tex]
5. Simplify the expression:
Simplify the fraction:
[tex]\[
T = 2\pi \div \frac{\pi}{4} = 2\pi \times \frac{4}{\pi}
\][/tex]
When you multiply, the [tex]\( \pi \)[/tex] terms cancel out:
[tex]\[
T = 2 \times 4 = 8
\][/tex]
Therefore, the period [tex]\( T \)[/tex] of the simple harmonic motion described by the given equation is [tex]\( 8 \)[/tex] units of time.
