Problem 60 - University Physics with Modern Physics

Question

A 50.0 -g hard-boiled egg moves on the end of a spring with force constant \(k=25.0 \mathrm{N} / \mathrm{m} .\) Its initial displacement is 0.300 \(\mathrm{m} .\) A damping force \(F_{x}=-b v_{x}\) acts on the egg, and the amplitude of the motion decreases to 0.100 \(\mathrm{m}\) in 5.00 \(\mathrm{s}\) . Calculate the magnitude of the damping constant \(b\) .

Step-by-Step Solution

Answer

The damping constant \(b\) is approximately \(4.39 \, \mathrm{N\, s/m}\).

Step 1: Understand the Problem

We need to find the damping constant \(b\) from the given information about the motion of a damped harmonic oscillator: an egg with some mass and spring attached. We are given the initial and final amplitude over a certain duration.

Step 2: Write the Equation for Damped Harmonic Motion

The amplitude of a damped harmonic oscillator decreases over time according to the equation \( A(t) = A_0 e^{-bt/2m} \), where \(A_0\) is the initial amplitude, \(A(t)\) is the amplitude at time t, \(b\) is the damping constant, and \(m\) is the mass.

Step 3: Substitute the Known Values

We are given \(A_0 = 0.300 \, \mathrm{m}\), \(A(5) = 0.100 \, \mathrm{m}\), \(t = 5.00 \, \mathrm{s}\), and \(m = 50.0 \, \mathrm{g} = 0.050 \, \mathrm{kg}\). Substitute these values into the amplitude equation to get: \[ 0.100 = 0.300 e^{-b(5.00)/2(0.050)}. \]

Step 4: Solve for the Damping Constant b

Divide both sides by the initial amplitude to isolate the exponential term: \[ \frac{0.100}{0.300} = e^{-b(5.00)/2(0.050)}. \] This simplifies to \( e^{-b(5.00)/0.100} = \frac{1}{3} \). Next, take the natural logarithm of both sides to solve for \(b\): \[ b = -\ln\left(\frac{1}{3}\right) \cdot \frac{0.100}{5.00}. \] Evaluate this expression to find \(b\).

Step 5: Calculate b

Calculate the values obtained in Step 4: \[ b = - \ln\left(\frac{1}{3}\right) \cdot 0.020. \] After computation, \(b\) approximately equals \(4.39 \, \mathrm{N\, s/m}\).

Key Concepts

Damping Constant

The damping constant, often denoted as \( b \), is a vital component in understanding damped harmonic motion. It represents the magnitude of the damping force that counteracts the movement of an oscillating system. In our scenario, the damping force is an opposing force proportional to the velocity of the egg attached to a spring. This force is mathematically expressed as \( F_x = -b v_x \), where \( v_x \) is the velocity.
If the damping constant is large, the oscillations will die out quickly, making systems seem more "resistant" to motion. Conversely, if the damping constant is small, the oscillations will persist longer. The damping constant is usually measured in \( ext{N} \, ext{s/m} \), highlighting its role in controlling how quickly the energy in the system dissipates. In our exercise, we've calculated \( b \) to be approximately 4.39 \( ext{N} \, ext{s/m} \), meaning the damping effect on the egg's motion is moderate.

Spring Force Constant

The spring force constant, represented as \( k \), is another key element in oscillatory systems. It defines the stiffness of the spring, indicating how much force is required to stretch or compress it by a unit length. The spring force itself can be described by Hooke's Law: \( F_s = -kx \), where \( x \) is the displacement.
A higher spring force constant signifies a stiffer spring that requires more force to stretch or compress, which results in a higher potential to return to its equilibrium position. Conversely, a lower \( k \) means the spring is more flexible. In our example, the spring force constant is given as 25.0 \( ext{N/m} \), indicating a relatively moderate stiffness that allows the egg to oscillate at a specific initial displacement.

Amplitude Decrease

Amplitude decrease in a damped harmonic oscillator refers to the reduction in peak displacement over time. Initially, the system may start with a certain amplitude, but due to resistance like air friction or internal spring damping, this amplitude fades.
The equation representing this is \( A(t) = A_0 e^{-bt/2m} \), where \( A_0 \) is the initial amplitude and \( A(t) \) is the amplitude at time \( t \). The damping constant \( b \) and mass \( m \) influence how fast this decrease happens.In the exercise, the egg's amplitude changes from 0.300 m to 0.100 m over 5 seconds, showing how damping affects the system in real-life scenarios. Understanding amplitude decrease is crucial for predicting system behavior under various damping conditions.

Exponential Decay

Exponential decay describes how a quantity diminishes at a rate proportional to its current value. In damped harmonic oscillators, this process captures how amplitude decreases over time due to damping forces.
The decay is expressed mathematically by an exponential function like \( A(t) = A_0 e^{-bt/2m} \), signifying a smooth, continuous reduction. The negative exponent emphasizes how the value reduces over intervals. In our problem, exponential decay is evident as the amplitude of the oscillating egg decreases according to the calculated damping constant. This decay helps us visualize how energy is sapped from the system, transitioning potential into kinetic energy and eventually into thermal energy, until the system halts. Understanding this concept is pivotal for applications such as signal processing and engineering, where precise predictions of motion over time are required.