You are riding in an automobile of mass 3000 kg. Assuming that you are examiningthe oscillation characteristics of its suspension system. The suspension sags15 cm when the entire automobile is placed on it. Also, the amplitude of oscillationdecreases by 50% during one complete oscillation. Estimate the values of (a) thespring constant k and (b) the damping constant b for the spring and shock absorber system of one wheel, assuming that each wheel supports 750 kg.
Given: The mass of automobile is 3000 kg , the displacement is 15 cm , each wheel support a mass of 750 kg and the amplitude of oscillations decreased by 50 % during each compete oscillation.
a)
If k is the spring constant of one spring, then the total spring constant of four springs in parallel is 4 k .
The restoring force on the system is given as,
F = 4 k x M g = 4 k x k = M g 4 x
Where, the mass is M , the displacement is x .
By substituting the given values in the above formula, we get
k = 3000 × 10 4 × 0 ⋅ 15 k = 5 × 10 4 N
Thus, the spring constant is 5 × 10 4 N .
b)
The time period is given as,
T = 2 π m k
Where, the mass supported by one wheel is m .
By substituting the given values in the above equation, we get
T = 2 π × 750 5 × 10 4 = 0 ⋅ 77 sec
The damping equation is given as,
x = x 0 e − b T 2 m
Where, the damping constant is b .
By substituting the given values in the above equation, we get
50 100 x 0 = x 0 e − b × 0.77 2 × 750 ( 5.13 × 10 − 4 ) b = ln 2 b ≈ 1351 kg/s
Thus, the value of damping constant b is 1351 kg/s .
Given: The mass of automobile is
a)
If
The restoring force on the system is given as,
Where, the mass is
By substituting the given values in the above formula, we get
Thus, the spring constant is
b)
The time period is given as,
Where, the mass supported by one wheel is
By substituting the given values in the above equation, we get
The damping equation is given as,
Where, the damping constant is
By substituting the given values in the above equation, we get
Thus, the value of damping constant
