Question

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

Short answer

In a critically damped shock absorber with an initial displacement of 6.41 cm and damping angular frequency of 72.4 Hz, the displacement after 0.0247 seconds is approximately 6.05 cm away from its equilibrium position.

Step-by-step solution

Understand the concept of critical damping

In a critically damped or overdamped system, the system returns to its equilibrium position without oscillating. The damping is strong enough to prevent oscillations, but not too strong that it causes the return to equilibrium to be extremely slow.

Write down the known variables

From the problem statement, we know: \(\omega_{\gamma} = 72.4 \mathrm{~Hz}\) (damping angular frequency), Initial displacement \(x_0 = 6.41 \mathrm{~cm} = 0.0641 \mathrm{~m}\), (converted to meters) Time elapsed \(t = 0.0247 \mathrm{~s}\).

Find the solution to the critically damped oscillator equation

The equation for the displacement of a critically damped oscillator is given by: \(x(t) = (x_0 + ct)e^{-\omega_{\gamma}t}\), where x(t) is the displacement after time `t` and `c` is an arbitrary constant determined by the initial conditions of the system. In our case, since the shock absorber is initially compressed, we assume the initial velocity to be \(0\). So when \(t = 0\), \(v = 0\). From the equation of displacement, \(v(t) = \frac{dx(t)}{dt} = (-x_0 \omega_{\gamma} - c\omega_{\gamma})e^{-\omega_{\gamma}t}\). When \(t = 0\), \(v(0) = 0 = (-x_0 \omega_{\gamma} - c\omega_{\gamma})e^{0}\).

Calculate the constant 'c'

From the condition \(v(0)=0\), we have: \(0 = -x_0 \omega_{\gamma} - c\omega_{\gamma}\). Rearranging, we find the constant c: \(c = -x_0 = -0.0641 \mathrm{~m}\).

Calculate the displacement after the given time

Now that we have the expression for \(x(t)\) and the value of the constant 'c', we can find the displacement after time \(0.0247 \mathrm{~s}\): \(x(t) = (x_0 + ct)e^{-\omega_{\gamma}t}\). Plugging in the values, we get: \(x(0.0247) = (0.0641 - 0.0641)e^{-72.4*0.0247} \approx 0.0605 \mathrm{~m}\).

State the final answer

The shock absorber will be approximately \(6.05 \mathrm{~cm}\) away from its equilibrium position after \(0.0247 \mathrm{~s}\).

Key concepts

Damped Oscillations

When systems capable of oscillating move back and forth around an equilibrium point, we observe what are known as oscillations. In the physical world, these oscillations don't go on forever; they gradually decrease in amplitude due to resistive forces—like friction or air resistance—which dissipate the system's energy. This process is called 'damping,' and the motion it results in is referred to as 'damped oscillations'.

Damped oscillations occur in various real-world systems, with one common example being the shock absorber in a vehicle. When pushed, the shock absorber moves back towards its original position but doesn't continue to move past it in a repetitive manner—as it would in simple harmonic motion—due to the damping force. Critical damping is a specific type of damped oscillation where the system returns to equilibrium in the shortest time without overshooting its original position. This behavior is particularly important in engineering and design, as it ensures stability and quick settling to rest, like in the case of automotive shock absorbers or in the prevention of excessive vibrations in tall buildings during strong winds or earthquakes.

To optimize understanding, visual aids such as graphs of displacement versus time for over-damped, under-damped, and critically damped systems can be very helpful. They demonstrate how the amplitude of oscillation evolves over time, helping students to visualize how critically damped systems differ from others.

Equilibrium Position

The equilibrium position in the context of oscillations is the point at which the net force acting on the system is zero. It is the position the system naturally assumes when there are no disturbances. This applies to a physical spring at rest, a pendulum hanging straight down, or, as in our exercise, a shock absorber that is neither compressed nor extended.

In the example of the shock absorber, the equilibrium position is particularly important because it's the state where the shock absorber isn't exerting any additional force to come back to its rest state. When discussing damped oscillations, understanding equilibrium is crucial as it is the final position the system aims to reach and stay at once the motion ceases. Regardless of the initial displacement or the type of damping involved, all damped oscillatory systems, in theory, aim to return to this stable point. By illustrating the concept of equilibrium with simple diagrams of forces at play, students can grasp the concept more effectively. Understanding equilibrium positions is fundamental to studying more advanced topics in mechanics and dynamics, as it forms the basis for analyzing system behavior.

Damping Angular Frequency

The damping angular frequency, often represented by the Greek letter omega with a subscript gamma \(\omega_{\gamma}\), is an important parameter in the study of damped oscillations. It is defined as the frequency at which the system would oscillate if there were no damping forces present—like the natural frequency—but adjusted for the effects of the damping. It essentially quantifies how fast the system returns to equilibrium without oscillating when there is critical damping.

Understanding damping angular frequency can be made simpler by connecting it to recognizable experiences. For example, consider striking a tuning fork and hearing its clear tone diminish over time due to damping. The rate at which it loses energy and quiets down involves the damping angular frequency. In the context of our exercise, with a damping angular frequency of 72.4 Hz, it dictates how rapidly the critically damped shock absorber reaches its equilibrium position after being disturbed. Calculations involving \(\omega_{\gamma}\) allow students to predict the future behavior of the system at specific time intervals after being displaced, as shown in the solution steps for the textbook problem. To assist learners further, practical demonstrations or simulations comparing different values of \(\omega_{\gamma}\) can vividly illustrate the effects of various damping scenarios on the time taken by the system to settle at the equilibrium position.