Finding the dampening constant

[bebop721]

The suspension (four springs) of a mass 2,603 kg car sags a distance of 23.2cm when the entire weight of the car is placed on it The effective amplitude decays to 51.6% of its initial value in 4 oscillations (exact). What is the value of the damping constant b of one of the springs (in kg/s)? (If necessary use g=9.84m/s2


any help at all would be much appreciated thank-you

 

[cristo]

What thoughts do you have on the question? Please note that you must show your work before we can help you.

 

[m2003]

Based on the given information, we can use the formula for the damping ratio to find the value of the damping constant b:

ζ = (b/2mω) = (ln(A/A_n))/N

Where:
ζ = damping ratio
b = damping constant
m = mass of the car (2,603 kg)
ω = angular frequency (2πf)
A = initial amplitude (23.2 cm)
A_n = amplitude after 4 oscillations (51.6% of initial value)
N = number of oscillations (4)

First, we need to calculate the angular frequency:

ω = 2πf = 2π/T = 2π/(4T) = π/T

Where T is the time for 4 oscillations. We can calculate T by multiplying the time for one oscillation (T_1) by 4:

T = 4T_1 = 4(4s) = 16s

Therefore, ω = π/16s ≈ 0.1963 rad/s

Now, we can plug in all the values into the formula for the damping ratio:

ζ = (b/2mω) = (ln(A/A_n))/N
ζ = (b/2(2,603 kg)(0.1963 rad/s)) = (ln(23.2 cm/(0.516)(23.2 cm)))/4
ζ ≈ 0.0051

Solving for b, we get:

b = 2ζmω = 2(0.0051)(2,603 kg)(0.1963 rad/s) ≈ 2.537 kg/s

Therefore, the value of the damping constant b for one of the springs is approximately 2.537 kg/s. This value may vary depending on the accuracy of the given information and the assumptions made in the calculation.