- #1
lightlightsup
- 95
- 9
- Homework Statement
- The suspension system of a mass ##M## automobile "sags” a distance ##x## when the chassis is placed on it. Also, the oscillation amplitude decreases by a factor ##N## each cycle. 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 each wheel supports an equal fraction of the weight. Make the simplifying assumption that the change in the period due to damping can be ignored. Express your answer in terms of the variables given and ##g##.
- Relevant Equations
- See attempt.
##x(t)=x_m e^{\frac{-bt}{2m}}cos(ωt+φ)##
Damping Factor: ##e^{\frac{-bt}{2m}}## (##b## is the damping constant)
Estimated ##ω##: ##\sqrt{\frac{k}{m}}##
More Accurate ##ω'##: ##\sqrt{\frac{k}{m}-\frac{b^2}{4m^2}}##
Also: ##T = \frac{2π}{ω}## and ##F = kx##.
So, my answers:
##k = \frac{Mg}{4x}##
Estimate: ## b = -\frac{M\ln(N)}{4π}\sqrt{\frac{g}{x}}##
More Accurate: ## b = \sqrt{\frac{gM^2\ln^2(N)}{x[16π^2+4\ln^2(N)]}}##
Apparently, my answer for ##k## is correct but my answer for ##b## is wrong.
If you need numbers to test these equations:
••60 The suspension system of a 2000 kg automobile “sags” 10 cm when the chassis is placed on it. Also, the oscillation amplitude decreases by 50% each cycle. 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 each wheel supports 500 kg.
The answers are: (a) 49,000 N/m, (b) ~1100kg/s (1092 with the estimated ##b## and 1086 with the more accurate ##b## formula).
This is from Halliday's Fundamentals of Physics, 11th Edition, Chapter 15 (Oscillations).
Damping Factor: ##e^{\frac{-bt}{2m}}## (##b## is the damping constant)
Estimated ##ω##: ##\sqrt{\frac{k}{m}}##
More Accurate ##ω'##: ##\sqrt{\frac{k}{m}-\frac{b^2}{4m^2}}##
Also: ##T = \frac{2π}{ω}## and ##F = kx##.
So, my answers:
##k = \frac{Mg}{4x}##
Estimate: ## b = -\frac{M\ln(N)}{4π}\sqrt{\frac{g}{x}}##
More Accurate: ## b = \sqrt{\frac{gM^2\ln^2(N)}{x[16π^2+4\ln^2(N)]}}##
Apparently, my answer for ##k## is correct but my answer for ##b## is wrong.
If you need numbers to test these equations:
••60 The suspension system of a 2000 kg automobile “sags” 10 cm when the chassis is placed on it. Also, the oscillation amplitude decreases by 50% each cycle. 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 each wheel supports 500 kg.
The answers are: (a) 49,000 N/m, (b) ~1100kg/s (1092 with the estimated ##b## and 1086 with the more accurate ##b## formula).
This is from Halliday's Fundamentals of Physics, 11th Edition, Chapter 15 (Oscillations).
Last edited: