Sinusoidal functions
A pendulum is swinging back and forth. After t seconds, the horizontal distance from the bob to the place where it was released is given by H(t)=7−7cos( 2π(t−2)/20). How often does the bob cross its midline?
it crosses the midline twice each period.
H(t)=7−7cos( 2π(t−2)/20)
cos(2π(t-2)/20) = cos(π/10 (t-2))
So the period is 2π/(π/10) = 20 seconds
It crosses the midline every 10 seconds
If it were Sin, how many times would it cross midline? 3?
To determine how often the bob crosses its midline, we need to find the period of the sinusoidal function H(t).
The general form of a sinusoidal function is H(t) = A + B * cos(C(t - D)). In this case, we have H(t) = 7 - 7 * cos((2π(t - 2))/20).
Comparing this to the general form, we can see that A = 7, B = 7, C = 2π/20, and D = 2.
The period T of a sinusoidal function is given by T = 2π/C.
Therefore, the period of H(t) is T = 2π/(2π/20) = 20.
The bob crosses its midline once per complete oscillation, which is equal to the period. Therefore, the bob crosses its midline every 20 seconds.
To determine how often the pendulum bob crosses its midline, we need to find the number of complete cycles the function completes. In sinusoidal functions, a complete cycle occurs when the function goes through one period of oscillation.
To determine the period of the given sinusoidal function, we need to identify the coefficient of the "t" term inside the cosine function. In this case, the coefficient is (2π/20). The period is then given by the formula:
Period = (2π) / coefficient
Plugging in the coefficient, we have:
Period = (2π) / (2π/20)
= 20
Therefore, the period of the function is 20 seconds.
Since the pendulum bob crosses its midline twice in one complete cycle (once going down and once going up), we divide the period by two to find how often the bob crosses its midline:
Crossings = Period / 2
= 20 / 2
= 10
Therefore, the pendulum bob crosses its midline 10 times in 20 seconds, or once every 2 seconds.