The top of a flagpole sways back and forth in the wind. The top sways 10cm to the right (+10) and 10cm to the left (-10) of its resting position and moves back and forth 4 times every second. At t=0, the pole was at its resting position and then it started moving to the right.
a) Determine the equation of a sinusoidal function that describes the distance the top of the pole is from its resting position in terms of time.
b) if the wind speed decreases such that the sway of the top of the pole is reduced by 20%, determine the new equation of the sinusoidal function. Assume the period remains the same.
a) To determine the equation of a sinusoidal function that describes the distance the top of the pole is from its resting position in terms of time, we need to first determine the amplitude, period, and phase shift.
Amplitude:
The amplitude is the distance between the maximum and minimum points of the oscillation. In this case, the top of the pole sways 10 cm to the right (+10) and 10 cm to the left (-10) of its resting position. Since the maximum displacement from the resting position is 10 cm, the amplitude is 10 cm.
Period:
The period is the time it takes for one complete cycle of oscillation. In this case, the top of the pole moves back and forth 4 times every second. Therefore, the period is 1/4 second.
Phase Shift:
At t=0, the pole was at its resting position and then it started moving to the right. This means there is no phase shift.
The equation of a sinusoidal function that describes the distance the top of the pole is from its resting position in terms of time can be written as:
f(t) = A * sin(2π / T * t)
where A is the amplitude, T is the period, and t represents time.
Substituting the values we found earlier, the equation becomes:
f(t) = 10 * sin(2π / (1/4) * t)
Simplifying further:
f(t) = 10 * sin(8πt)
Therefore, the equation of the sinusoidal function that describes the distance the top of the pole is from its resting position in terms of time is f(t) = 10 * sin(8πt).
b) If the wind speed decreases such that the sway of the top of the pole is reduced by 20%, the new amplitude will be 80% of the original amplitude.
New Amplitude:
New Amplitude = 0.8 * 10 cm = 8 cm
The period remains the same, so the new equation of the sinusoidal function becomes:
f(t) = 8 * sin(2π / (1/4) * t)
Simplifying further:
f(t) = 8 * sin(8πt)
Therefore, the new equation of the sinusoidal function that describes the distance the top of the pole is from its resting position in terms of time is f(t) = 8 * sin(8πt).