Suppose that an object attached to a coiled spring is pulled down a distance of 5 inches from its rest position and then released. If the time for one oscillation is 3 seconds, write and equation that relates the displacement d of the object from its rest position after time t (in seconds). Assume no friction.
To derive the equation that relates the displacement (d) of the object from its rest position after time (t), we can use the following principles:
1. The equation for simple harmonic motion (SHM) is given by:
d = A*cos(ωt + φ)
Where,
- d is the displacement of the object from its rest position,
- A is the amplitude of the oscillation,
- ω is the angular frequency of the oscillation (ω = 2πf), and
- φ is the phase constant.
2. The period (T) of the oscillation is the time taken for one complete cycle:
T = 1/f,
where f is the frequency of oscillation.
Given that the time for one oscillation is 3 seconds, we can calculate the frequency using the formula:
f = 1/T.
Let's proceed step by step:
Step 1: Finding the Frequency (f)
Since the time for one oscillation is 3 seconds, the frequency is given by:
f = 1/T = 1/3 Hz.
Step 2: Finding the Angular Frequency (ω)
The angular frequency (ω) is related to the frequency by the equation:
ω = 2πf = 2π(1/3) = (2π/3) rad/s.
Step 3: Determining the Equation
To find the equation that relates the displacement (d) of the object to time (t), we first need to determine the amplitude (A) and phase constant (φ).
Given that the object is initially pulled down a distance of 5 inches (rest position is at equilibrium), the amplitude is 5 inches.
As there is no friction mentioned, we assume the object is released from its rest position with no initial velocity. Thus, at t = 0, the object is at its maximum displacement in the negative direction. This implies that the phase constant (φ) is 180 degrees or π radians.
Now, substituting the known values into the equation:
d = A*cos(ωt + φ)
d = 5*cos((2π/3)*t + π)
Therefore, the equation that relates the displacement (d) of the object from its rest position after time (t) is:
d = 5*cos((2π/3)*t + π).