An object is attached to a coiled spring. The object is pulled down (from its rest position) and then released. Given that the distance from its rest position at time t = 0 is 3 inches, its amplitude is 3 inches, and its period is 1.5 seconds, write an equation for the distance d as a function of time t in seconds of the object from its rest position.
To write the equation for the distance of the object from its rest position as a function of time, we can use the equation of simple harmonic motion:
d(t) = A * cos(ωt + φ)
Where:
- d(t) represents the distance of the object from its rest position at time t
- A represents the amplitude (maximum displacement) of the motion
- ω represents the angular frequency of the motion
- t represents time
- φ represents the phase constant (initial phase)
Given that the amplitude is 3 inches, we have A = 3 inches.
To find the angular frequency (ω), we can use the formula:
ω = 2π / T
Where:
- T represents the period of the motion
Given that the period is 1.5 seconds, we have T = 1.5 seconds.
Plugging the values into the formula, we have:
ω = 2π / 1.5 ≈ 4.18879 rad/s
To determine the phase constant (φ), we need to consider the initial conditions. In this case, we are given that the distance from the rest position at t = 0 is 3 inches. Since the cosine function has its maximum value at t = 0, we can say that φ = π/2 to match the given initial condition.
Substituting the known values into the equation, we get:
d(t) = 3 * cos(4.18879t + π/2)
So, the equation for the distance of the object from its rest position as a function of time is:
d(t) = 3 * cos(4.18879t + π/2)