A hanging spring is compressed 3 inches from its rest position and released at t = 0 seconds. It returns to the same position after 0.8 seconds.
I need help Finding:
a) the amplitude of the motion
b) the period of the motion
c) the frequency of the motion
d) a function that models the displacement, y, of the end of the spring from the rest position at time, t.
e) the displacement from the rest position at t= 3 min rounded to the tenths place
(a)
compressed 3 inches from its rest position
the amplitude is 3
(b)
returns to the same position after 0.8 seconds
the period is 0.8 seconds
If it had not been at its maximum compression when released, then things would have been a bit more complicated. But it only returns that far after a full period.
(c) frequency = 1/period
(d) cos(t) has its maximum at t=0. So, if y is the height of the end of the spring, which is hanging down, its maximum is when t=0. So,
y = cos(kt)
cos(kt) has period 2π/k. Since our period is 0.8 (or 4/5), we need 2π/k = 4/5. k=5π/2
y = cos(5π/2 t)
(e) now just plug in t=180 (3 minutes is 180 seconds)
To find the answers to these questions, we can use the properties and formulas of simple harmonic motion provided by the problem.
a) The amplitude of the motion is the maximum distance that the spring travels from its rest position. In this case, it is compressed 3 inches, so the amplitude is 3 inches.
b) The period of the motion is the time taken for the spring to complete one full cycle or oscillation. From the problem, we know that it returns to the same position after 0.8 seconds, so the period is 0.8 seconds.
c) The frequency of the motion is the number of cycles or oscillations per unit of time. It is the reciprocal of the period. So, the frequency can be calculated by taking the reciprocal of the period: frequency = 1 / period = 1 / 0.8 = 1.25 Hz.
d) A function that models the displacement, y, of the end of the spring from the rest position at time, t, can be represented by a sinusoidal function. For simple harmonic motion, it can be written as:
y(t) = A * sin(2πft + φ)
where:
A represents the amplitude,
f represents the frequency,
t represents the time, and
φ represents the phase angle.
In this case, we already know the amplitude (A = 3 inches) and frequency (f = 1.25 Hz). The phase angle (φ) is not provided in the problem, so we'll assume it to be zero.
Therefore, the function that models the displacement of the spring would be:
y(t) = 3 * sin(2π * 1.25t + 0)
e) To find the displacement from the rest position at t = 3 minutes (which is 3 * 60 seconds = 180 seconds), we can substitute t = 180 into the displacement function and round the result to the tenths place:
y(180) = 3 * sin(2π * 1.25 * 180 + 0)
Calculate the value of y(180) using a calculator or software that can compute trigonometric functions.
Please note that the calculation of y(180) will provide the displacement from the rest position, which means that if the spring has returned to its rest position at that time, the displacement would be 0. Otherwise, it will give the displacement at that time relative to the rest position.