An automobile with a mass of 1640 kg is lifted into the air. During the lift, the suspension sp

step 1 This question cover the concept of critical damping and in critical damping the damping constant

An automobile with a mass of 1640 kg is lifted into the...

An automobile with a mass of $1640 \mathrm{~kg}$ is lifted into the air. During the lift, the suspension spring on each wheel lengthens by $30.0 \mathrm{~cm} .$ What damping constant is required for the shock absorber on each wheel to produce critical damping? a) $101 \mathrm{~kg} / \mathrm{s}$ b) $234 \mathrm{~kg} / \mathrm{s}$ c) $1230 \mathrm{~kg} / \mathrm{s}$ d) $2310 \mathrm{~kg} / \mathrm{s}$ e) $4690 \mathrm{~kg} / \mathrm{s}$

An automobile with a mass of $1640 \mathrm{~kg}$ is lifted into the air. During the lift, the suspension spring on each wheel lengthens by $30.0 \mathrm{~cm} .$ What damping constant is required for the shock absorber on each wheel to produce critical damping?
a) $101 \mathrm{~kg} / \mathrm{s}$
b) $234 \mathrm{~kg} / \mathrm{s}$
c) $1230 \mathrm{~kg} / \mathrm{s}$
d) $2310 \mathrm{~kg} / \mathrm{s}$
e) $4690 \mathrm{~kg} / \mathrm{s}$

Key Concepts

Spring Constant

The spring constant, denoted as k, quantifies the stiffness of a spring and follows from Hooke's Law, which states that the force exerted by a spring is proportional to its displacement (F = kx) from equilibrium. In static equilibrium scenarios, such as a spring supporting a weight, the spring constant is determined by dividing the applied force by the displacement it causes, providing insight into how much the spring resists deformation.

Critical Damping

Critical damping is a condition in a damped harmonic oscillator where the system returns to its equilibrium state as quickly as possible without oscillating. This occurs when the damping constant c is exactly equal to 2 multiplied by the square root of the product of the mass and the spring constant (c = 2?(mk)). This balance ensures that the energy is dissipated efficiently, preventing any overshoot or oscillatory behavior.

Mass Distribution in Multi-Support Systems

When a system’s total mass is supported by multiple elements, the mass associated with each support is effectively reduced by evenly distributing the overall load. This concept is crucial for analyzing systems such as automobile suspensions, where the overall mass is divided among the individual springs and shock absorbers. The effective mass per component influences both the static deflection of the spring and the dynamic damping requirements necessary for critical damping.

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