A buoy floating in the ocean is bobbing in simple harmonic motion with amplitude 7ft and period 8 seconds. Its displacement d from sea level at time t=0 seconds is 0 ft, and initially it moves upward. (Note that upward is the positive direction.)

Question

A buoy floating in the ocean is bobbing in simple harmonic motion with amplitude 7ft and period 8 seconds. Its displacement d from sea level at time t=0 seconds is 0 ft, and initially it moves upward. (Note that upward is the positive direction.)

Give the equation modeling the displacement d as a function of time t

Answer 1

The equation modeling the displacement d as a function of time t for simple harmonic motion is:

d(t) = A * cos(ωt + ϕ)

Where:
- d(t) represents the displacement from the equilibrium position at time t.
- A represents the amplitude of the motion.
- ω represents the angular frequency of the motion.
- ϕ represents the phase angle.

In this case, we are given that the amplitude A is 7ft. The period T is 8 seconds, and we know that the angular frequency ω is related to the period by ω = 2π / T.

Let's calculate ω first:

ω = 2π / T
ω = 2π / 8
ω = π / 4

Now, since the buoy starts at the equilibrium position at t = 0 and moves upward, we can deduce that the phase angle ϕ is 0.

Hence, the equation modeling the displacement d as a function of time t is:

d(t) = 7 * cos((π / 4)t + 0)
OR
d(t) = 7 * cos((π/4)t)

Answer 2

To determine the equation that models the displacement (d) as a function of time (t), we need to understand the key components of simple harmonic motion.

Simple harmonic motion is characterized by three variables: amplitude (A), period (T), and initial phase (ϕ). In this scenario, the given values are:

Amplitude (A) = 7 ft
Period (T) = 8 seconds

The equation representing the displacement as a function of time in simple harmonic motion is:

d(t) = A * sin(2πt / T + ϕ)

Let's go through the process of finding the equation step by step.

Step 1: Calculate the angular frequency (ω).
The angular frequency (ω) is given by the formula ω = 2π / T, where T is the period. In this case, the period is 8 seconds.

ω = 2π / 8
ω = π / 4

Step 2: Determine the initial phase (ϕ).
The initial phase (ϕ) represents the phase at t = 0. Since the buoy initially moves upward, the initial phase should be 0.

ϕ = 0

Now we have all the necessary values to finalize the equation.

d(t) = A * sin(ωt + ϕ)
d(t) = 7 * sin((π / 4) * t + 0)
d(t) = 7 * sin((π/4)t)

Thus, the equation modeling the displacement (d) as a function of time (t) is:

d(t) = 7 * sin((π/4)t)

Answer 3

As before,

d(t) = A + Bsin(kt)
has
amplitude = B
period = 2π/k
d(0) = A