Oscillations
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Question : 21
Total: 25
You are riding an automobile of mass 3000 kg. Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags 15 cm when the entire automobile is placed on it. Also, the amplitude of oscillation decreases by 50% during one complete oscillation. Estimate the values of (a) the spring 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.
Solution:
(a) Here mass supported by each wheel, M = 750 kg and x =15 cm= 0.15 m
If k is spring constant of the spring, then restoring force developed on being compressed through a distancex , F = – k x
If M is mass supported by each wheel, then
∴ k x = M g
ork =
=
= 4.9 × 10 4 N m − 1
(b) If b is damping constant for the spring and shock absorber system, then damped amplitude of oscillation is given byA = A 0 e − b T ∕ 2 M
where T is period of oscillation of the spring,A 0 A = A 0 e − b T ∕ 2 M
As the amplitude of oscillation decreases by 50% during one complete oscillation,
A 0 = A 0 e − b
M
⇒ 2 = e b
M
⇒ log e 2 =
log e e =
Now,t = 2 π √
= 2 ×
× √
=
√
∴ b =
=
= 1336.99 kg s − 1
If k is spring constant of the spring, then restoring force developed on being compressed through a distance
If M is mass supported by each wheel, then
or
(b) If b is damping constant for the spring and shock absorber system, then damped amplitude of oscillation is given by
where T is period of oscillation of the spring,
As the amplitude of oscillation decreases by 50% during one complete oscillation,
Now,
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