Question

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) \(4690 \mathrm{~kg} / \mathrm{s}\)

Short answer

Question: Determine the damping constant required for the shock absorber on each wheel to produce critical damping for a car with a mass of 1640 kg and a suspension spring total elongation of 30.0 cm. Answer: \(234\mathrm{~kg/s}\)

Step-by-step solution

Determine the mass acting on each suspension spring

As there are 4 wheels, we can evenly distribute the mass of the car on each wheel. So, each wheel has to support a mass of \(1640\mathrm{~kg}/4 = 410\mathrm{~kg}\).

Calculate the spring constant of the suspension spring on each wheel

The total elongation of the suspension spring on each wheel is \(30.0\mathrm{~cm}\). We can use Hooke's law to calculate the spring constant, which is given by \(F = kx\). The force acting on each spring is the gravitational force on the mass, which is \(F = mg\), where \(m = 410\mathrm{~kg}\) and \(g = 9.81\mathrm{~m/s^2}\). So, \(F = 410\mathrm{~kg} \times 9.81\mathrm{~m/s^2} = 4021.4\mathrm{~N}\). Since \(k = F/x\), we have \(k = 4021.4\mathrm{~N} / (30.0\mathrm{~cm} \times 0.01\mathrm{~m/cm}) = 13405\mathrm{~N/m}\).

Calculate the critical damping constant using the formula c=2*sqrt(mk)

Now, we can use the formula for the critical damping constant, which is \(c = 2\sqrt{mk}\). Plugging in the values of m and k we calculated in the previous steps, we get \(c = 2\sqrt{410\mathrm{~kg} \times 13405\mathrm{~N/m}} \approx 234\mathrm{~kg/s}\).

Select the correct answer from the options given

Comparing our calculated value with the options given in the exercise, the closest option is b) \(234\mathrm{~kg/s}\). Therefore, the damping constant required for the shock absorber on each wheel to produce critical damping is \(234\mathrm{~kg/s}\).

Key concepts

Damping Constant

When a car travels over bumpy roads, its shock absorbers play a crucial role in smoothing out the ride. The shock absorbers are there to control the motion of the springs and thus the car. A fundamental aspect of this control mechanism is the damping constant. The damping constant measures the resistance force the shock absorber applies to control the spring's motion. This resistance force is directly proportional to the velocity of the spring's motion and acts in the opposite direction, helping to dissipate the energy of the system.

In the context of the automobile lift exercise, finding the appropriate damping constant involves ensuring that the motion of the car upon being released from the lift is critically damped. This means that the car would return to its rest position as quickly as possible without oscillating. For critical damping, the damping constant 'c' must be precisely calculated to match the conditions given by the formula for critical damping, which is \[ c = 2\sqrt{mk} \], where 'm' is the mass supported by each wheel, and 'k' is the spring constant of the suspension spring. In essence, the damping constant is essential for determining how quickly the automobile stabilizes after being displaced.

Hooke's Law

To understand the behavior of springs and elastic materials, one must delve into Hooke's Law. This principle is fundamental in physics and engineering, describing how the force needed to extend or compress a spring by some distance is proportional to that distance. Mathematically, Hooke's Law is expressed as \[ F = kx \], where 'F' stands for the force applied to the spring, 'k' is the spring constant representing the stiffness of the spring, and 'x' is the displacement of the spring from its equilibrium position.

In our automobile lift scenario, we apply Hooke's Law to determine the spring constant of the car's suspension system. By knowing the weight of the car (the force exerted on the spring due to gravity) and the distance the suspension spring is compressed (the displacement), we can calculate the spring constant, 'k'. This spring constant is crucial as it not only determines the spring's responsiveness to weight changes but also is a key component in calculating the damping constant necessary for critical damping.

Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that displacement. It is the idealized motion of systems that can oscillate back and forth, such as a mass on a spring or a pendulum swinging with small displacements. The main feature of SHM is its sine or cosine waveform when graphing displacement over time.

In SHM, the system has a natural frequency at which it oscillates if not affected by external forces or damping. This natural frequency is determined by both the mass of the oscillating object and the spring constant. For the automobile lifted off the ground, if it were to be released and behave according to SHM, it would oscillate up and down. However, with the addition of damping (particularly critical damping), we prevent the oscillation, instead inducing a return to the equilibrium position in the shortest possible time without oscillating. This is desired in many physical systems, such as car suspensions, to avoid prolonged bouncing after encountering a road bump.