A pendulum is swinging next to a wall. The distance from the bob of the swinging pendulum to the wall varies in a periodic way that can be modeled by a trigonometric function. The function has period 0.8 seconds, amplitude 6cm, and midline H=15cm. At time t=0.5 seconds, the bob is at its midline, moving towards the wall. Find the formula of the trigonometric function that models the distance H from the pendulum's bob to the wall after t seconds. Define the function using radians. H(t)=◻

Question

A pendulum is swinging next to a wall. The distance from the bob of the swinging pendulum to the wall varies in a periodic way that can be modeled by a trigonometric function. The function has period 0.8 seconds, amplitude  6cm, and midline  H=15cm. At time  t=0.5 seconds, the bob is at its midline, moving towards the wall. Find the formula of the trigonometric function that models the distance  H from the pendulum's bob to the wall after  t seconds. Define the function using radians.  H(t)=◻

Bytelearn · Accepted Answer

Amplitude Explanation: The amplitude of the trigonometric function is given as 6 cm. This means the maximum deviation from the midline is 6 cm.Midline Definition: The midline of the function is given as H = 15 cm. This is the average value around which the function oscillates.Period Calculation: The period of the function is given as 0.8 seconds. The period T of a trigonometric function is related to the angular frequency ω by the formula ω = 2π/T. Let's calculate ω. ω = 2π / 0.8 = 2π / (4/5) = (2π * 5) / 4 = (5π) / 2Phase Shift Calculation: Since the pendulum is at its midline and moving towards the wall at t = 0.5 seconds, we can use the cosine function to model the motion, as the cosine function starts at its maximum value and decreases (which corresponds to the pendulum moving towards the wall from the midline). However, we need to adjust the phase shift to account for the fact that at t = 0.5 seconds, the pendulum is at the midline, not at an extreme. The phase shift φ can be found by setting the cosine function to 0 at t = 0.5 seconds, since the midline corresponds to the zero point of the cosine function's oscillation. cos(ωt + φ) = 0 Let's find φ using the angular frequency ω we calculated. cos((5π/2) * 0.5 + φ) = 0 cos((5π/4) + φ) = 0 Since cos(π/2) = 0, we need φ such that (5π/4) + φ = π/2. φ = π/2 - (5π/4) φ = -3π/4Function Modeling: Now we have all the components to write the function. The trigonometric function that models the distance H from the pendulum's bob to the wall after t seconds is: H(t) = amplitude * cos(ωt + φ) + midline Substituting the values we have: H(t) = 6 * cos((5π/2)t - 3π/4) + 15

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