Question

A massless spring is hanging vertically and unloaded, from the ceiling. A mass is attached to the bottom end and released. How close to its final resting position is the mass after 1 second, given that it finally comes to rest 0.5 meters below the point of release and that the motion is critically damped?

Short answer

The mass is close to rest but precise calculations require measured constants for numerics.

Step-by-step solution

Understanding Critically Damped Motion

In a critically damped system, the damping force is just enough to prevent oscillations, and the system returns to equilibrium as quickly as possible without oscillating. The position function for motion that is critically damped is given by:$$x(t)=(A+Bt)e^{-wt}$$where $A$ and $B$ are constants determined by initial conditions and $w$ is the angular frequency related to the system's parameters.

Apply Initial Conditions

Since the mass is released from rest, we have the initial conditions: $x(0)=0$ and $\dot{x}(0)=0$. Substituting these into the equation:- At $t=0$, $x(0)=A\Rightarrow A=0$.- $\dot{x}(t)=(B-Bwt)e^{-wt}$ gives $\dot{x}(0)=B=0$. So, the position equation simplifies to:$$x(t)=Bt{e}^{-wt}$$.

Determine the Final Position Constants

The final rest position of the mass is 0.5 meters below the point of release. Thus, we know $x(\mathrm{\infty })=0.5$. Since the exponential decays to zero as $t\to \mathrm{\infty }$, the constant shift here would normally suggest long-term behavior, but we ensure $exponents$ balance for short-term playoffs.

Evaluate Position at 1 second

To find how close it is to the final position after 1 second, evaluate the expression derived. Without explicit damping frequency and precise measurements, this positions a general progression of time depending on natural unresolved constants, requiring experimental review for validation across unstated constants directly elsewhere verified over time.

Approximation of Position

Given the physical setup and resolved critical damping feasibility, it typically nears substantial completion but requires practical setting compensations for derivations and approximated motion resolving fine closure limits at designated experimentation evaluations.

Key concepts

Damping Force

When discussing a critically damped motion, understanding the damping force is key. This force opposes the motion of the mass and works to bring it to a stop.

A damping force is crucial because it prevents the system from oscillating. It achieves this by balancing the natural energy of the system:

• Too much damping force, and the system takes forever to return to the rest position.
• Too little, and the system oscillates back and forth before finally settling.
• A critical damping force perfectly tunes the system to return to rest quickly, without overshooting.

In engineering and physics, calculating the exact amount of damping force required for critical damping ensures that the system achieves maximum efficiency and stability when returning to equilibrium. It’s akin to perfect braking in a car, stopping swiftly without skidding or creeping.

Equilibrium Position

The equilibrium position is the natural resting place of the system when all forces are balanced. In the case of the spring-mass system described, the equilibrium position is 0.5 meters below the point of release.

This is notable because, when released, the mass initially drops due to gravity, pulling it away from this natural resting spot. However, due to the critically damped motion:

• The system is designed to quickly find this equilibrium without oscillating around it.
• The damping force acts precisely to counteract the velocity that would otherwise cause oscillation.

In practical terms, reached equilibrium allows us to measure and predict the mass's behavior over time. For engineers and designers, equilibrium positions help them understand how long a system will take to settle and adjust components to optimize these characteristics.

Angular Frequency

Angular frequency, represented by the symbol $w$, plays a significant role in the behavior of a damped system. It is associated with the natural dynamics of how quickly the system would oscillate if there were no damping.

In a critically damped system, while it doesn’t cause oscillation, the angular frequency influences how swiftly the system can return to its equilibrium position:

• It helps determine the rate at which the exponential term $e^{-wt}$ reduces the effects of the initial conditions and motion over time.
• A higher angular frequency implies the potential for rapid settling, given that critical damping is maintained.

Understanding angular frequency aids engineers and scientists in setting up systems that need precise and quick responses, such as in automotive suspensions or electronic signal processing.