Question

A shock absorber that provides critical damping with \(\omega_{\gamma}=72.4 \mathrm{rad} / \mathrm{s}\) is compressed by \(6.41 \mathrm{~cm} .\) How far from the equilibrium position is it after \(0.0247 \mathrm{~s} ?\)

Short answer

Answer: The position of the shock absorber at \(t = 0.0247\, \text{s}\) is approximately \(2.52\,\text{cm}\) away from the equilibrium position.

Step-by-step solution

Write down the equation for critical damping

The equation for critical damping is given by: \( x(t) = e^{-\omega_{\gamma} t} \left( A + B \cdot t \right) \) where \(x(t)\) is the distance from the equilibrium position at time \(t\), \(\omega_{\gamma}\) is the damping coefficient, \(A\) represents the initial position, and \(B\) is the initial velocity divided by \(\omega_{\gamma}\).

Determine the initial conditions

We have the initial position, which is the shock absorber compressed by \(6.41\,\text{cm}\). Therefore, we have: \(A = 6.41\, \text{cm}\) However, we don't have the initial velocity, but we know that it is at rest and thus the initial velocity is \(0 \,\text{cm} / \text{s}\). Therefore, \(B = \frac{0\,\text{cm} / \text{s}}{\omega_{\gamma}}\)

Calculate the position at the given time

We have \(\omega_{\gamma} = 72.4\,\text{rad} / \text{s}\), \(t = 0.0247\, \text{s}\), and \(B = 0\). Now we can substitute the values into the equation from Step 1 to find the position of the shock absorber at time \(t\): \(x(0.0247) = e^{-72.4 \cdot 0.0247} \left( 6.41 + 0\right) \approx 2.52\,\text{cm}\) Now we have the position of the shock absorber at \(t = 0.0247\, \text{s}\), and the shock absorber is approximately \(2.52\,\text{cm}\) away from the equilibrium position.

Key concepts

Damping Coefficient

The damping coefficient, often denoted by symbols such as \(\omega_\gamma\) or \(\zeta\), is a crucial parameter in physics and engineering that characterizes the amount of damping in a system. In the context of mechanical vibrations, it relates to the resistance that a shock absorber or any kind of damper provides to motion. A larger damping coefficient signifies a system that will slow down more quickly.

In the case of critical damping, which serves as the threshold between oscillatory and non-oscillatory systems, the damping coefficient is finely tuned to prevent the system from oscillating after being disturbed. Instead, the system returns to the equilibrium position without overshooting it. As seen in the given exercise, critical damping enables us to predict how an object such as a compressed shock absorber will return to rest over time.

Equilibrium Position

The equilibrium position is the state where the forces on an object are balanced, resulting in no net force acting on the object. When a system is undisturbed, it resides at the equilibrium position. If a system is moved from this position, the forces will aim to return the object back to equilibrium. It is the position which we regard as the 'natural' or 'resting' state.

In our exercise, the initial compression of the shock absorber places it away from its equilibrium position by 6.41 cm. The formula used to calculate the shock absorber's return involves this displacement from equilibrium, and through critical damping, ensures a smooth and swift return to equilibrium without oscillation. Understanding the concept of equilibrium is vital, as it allows us to comprehend how systems react to perturbations and the forces at play to restore balance.

Exponential Decay

Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value, and it is often modeled with an exponential function. In physics, this concept is commonly applied to processes such as radioactive decay and the discharge of a capacitor, as well as damping in mechanical systems.

The equation used in our exercise involving a damping scenario is a perfect example of exponential decay. The expression \( e^{-\omega_{\gamma} t}\) represents how the displacement from equilibrium decreases over time. The influence of exponential decay in the given exercise shows that as time progresses, the compression of the shock absorber diminishes rapidly, meaning the shock absorber returns closer to its equilibrium position as time elapses, without any oscillations, due to the critical damping condition.