A spring is stretched 5 inches abobe its rest position. Itis then released and oscillates in damped harmontic motin with a frequency of 2cycles per second. After 7 seconds you measure the amplitude of the spring to be 1.5cm.

1.Find th damping constant C.
2.Find an equation that describes the postion of the spring about its rest position as afunction of time. Assum the the rest position of the spring is 6 inches above the ground and that t=0 is the moment the spring is released.

To find the damping constant C, we can use the formula for the angular frequency of a damped harmonic oscillator:

ω = √(ω₀² - (C/2m)²)

Where:
ω is the angular frequency of the damped harmonic oscillator,
ω₀ is the angular frequency of the undamped harmonic oscillator,
C is the damping constant,
m is the mass attached to the spring.

In this case, the given frequency of oscillation is 2 cycles per second. The angular frequency of a undamped harmonic oscillator can be found using the formula:

ω₀ = 2πf

Where f is the frequency in Hz.

Given that the amplitude is 1.5 cm, we need to convert it to inches:

Amplitude = 1.5 cm = 1.5/2.54 inches

The equilibrium position of the spring is 6 inches above the ground.

Let's start by finding the damping constant C.

1. Find ω₀:
ω₀ = 2πf
= 2π(2)
= 4π radians per second

2. Convert the amplitude to inches:
Amplitude = 1.5/2.54 inches = 0.59 inches

3. Find ω:
ω = √(ω₀² - (C/2m)²)
ω = 4π radians per second

4. Plug in the values:
0.59 = 0.59e^(-C * 7)
e^(-7C) = 1

5. Solve for C:
e^(-7C) = 1
Take the natural logarithm of both sides:
-7C = ln(1)
-7C = 0
C = 0

Therefore, the damping constant C is 0.

Now let's find an equation that describes the position of the spring about its rest position as a function of time.

The general equation for the damped harmonic motion is:

x(t) = A * e^(-βt) * cos(ωd * t)

Where:
x(t) is the position of the spring as a function of time,
A is the initial amplitude,
β is the damping constant (which we found to be 0),
ωd is the damped angular frequency.

In our case, the initial amplitude A is 0.59 inches, the damping constant β is 0, and the angular frequency ωd can be found using the formula:

ωd = √(ω₀² - (β/2m)²)

Since β is 0, the equation simplifies to:

ωd = ω₀

Now we can write the equation for the position of the spring:

x(t) = 0.59 * e^(0) * cos(4π * t)

Simplifying further:

x(t) = 0.59 * cos(4π * t)

Therefore, the equation that describes the position of the spring about its rest position as a function of time is x(t) = 0.59 * cos(4π * t).