91Ó°ÊÓ

In a damped harmonic oscillator, the damping constant, denoted as \( b \), plays a crucial role in determining how quickly the motion diminishes over time. It is a measure of how strong the damping force is in resisting the motion of the system. To find \( b \), we use the exponential decay relation of amplitude: \[ A(t) = A_0 e^{-\frac{b t}{2m}} \] Here, \( A_0 \) is the initial amplitude, \( A(t) \) is the amplitude at time \( t \), and \( m \) is the mass of the oscillating object. Given that the amplitude decreases from 0.300 m to 0.100 m in 5 seconds, we can set up the equation as follows:\[ \frac{0.100}{0.300} = e^{-\frac{5b}{0.1}} \] Simplifying, we get:\[ \frac{1}{3} = e^{-50b} \] Taking the logarithm of both sides allows us to solve for \( b \):\[ \ln\left(\frac{1}{3}\right) = -50b \] Finally, we find:\[ b = -\frac{\ln\left(\frac{1}{3}\right)}{50} \approx 0.0220\, \text{kg/s} \] This damping constant indicates how the energy is dissipated in the system and how it affects the rate of amplitude decay.

The natural frequency of a harmonic oscillator is the frequency at which it would oscillate without any damping or external forces. It is represented by the angular frequency \( \omega_0 \) and is calculated using the formula:\[ \omega_0 = \sqrt{\frac{k}{m}} \] where- \( k \) is the spring force constant, and- \( m \) is the mass of the oscillating object.In the given exercise,- \( k = 25.0 \) N/m, and- \( m = 0.05 \) kg.Plugging these values into the formula, we find:\[ \omega_0 = \sqrt{\frac{25.0}{0.05}} = \sqrt{500} \] which simplifies to:\[ \omega_0 = 22.36 \text{ rad/s} \] This natural frequency tells us how fast the system would oscillate if it were ideal, with no damping or external interruptions. It is an intrinsic property of the oscillator and can help predict its behavior in various conditions.

In a damped harmonic oscillator, the amplitude of oscillation decreases over time, following an exponential decay pattern. This pattern can be expressed by:\[ A(t) = A_0 e^{-\frac{b t}{2m}} \] Here:- \( A_0 \) is the initial amplitude,- \( A(t) \) is the amplitude at time \( t \), - \( b \) is the damping constant, and - \( m \) is the mass.The exponential term \( e^{-\frac{b t}{2m}} \) is responsible for the decay, reducing the amplitude over time. As \( t \) increases, this term becomes smaller, indicating a decrease in amplitude.In the problem,- the initial amplitude \( A_0 = 0.300 \) m,- the amplitude at time \( t = 5 \) s is \( 0.100 \) m.The exponential decay reflects the energy loss of the system due to the damping force. It illustrates how, over a specific timeframe, a once larger oscillation diminishes, showcasing the effects of the damping force on the motion.