Question

If you kick a harmonic oscillator sharply, you impart to it an initial velocity but no initial displacement. For a weakly damped oscillator with mass \(m\), spring constant \(k\). and damping force \(F_{y}=-b v,\) find \(x(t),\) if the total impulse delivered by the kick is \(J_{0}\).

Short answer

Answer: The position function \(x(t)\) for a weakly damped harmonic oscillator under these conditions is given by: \(x(t) = \frac{J_{0}}{(r_2-r_1)(r_2e^{r_2t_1}-r_1e^{r_1t_1})}(e^{r_1t}-e^{r_2t})\), where \(J_0\) is the initial impulse provided, \(r_1\) and \(r_2\) are the roots of the characteristic equation, and \(t_1\) is the time at which the initial impulse is delivered.

Step-by-step solution

Equation of Motion for a Damped Harmonic Oscillator

The equation of motion for a damped harmonic oscillator is given by Newton's second law: \(m\ddot{x}(t)=-kx(t)-bv(t)\), where: - \(x(t)\) is the displacement of the oscillator at time \(t\), - \(v(t)=\dot{x}(t)\) is the velocity of the oscillator at time \(t\), - \(\ddot{x}(t)\) is the acceleration of the oscillator at time \(t\), - \(m\) is the mass of the oscillator, - \(k\) is the spring constant, - \(b\) is the damping coefficient, and - \(F_{y}=-bv\) is the damping force.

Introducing the Impulse Delivered by the Kick

The total impulse delivered by the kick is defined as the change in momentum, which can be integrated over time: \(J_{0}=\int_0^{t_1} F_{k}(t) dt = m(v(t_1)-v(0))\). Since the kick delivers an initial velocity but no initial displacement, the initial conditions are: 1. \(x(0)=0\), and 2. \(v(t_1)-v(0)=\frac{J_{0}}{m}\).

Solving the Equation of Motion for the Position Function \(x(t)\)

Now we want to find the position function x(t) that satisfies the equation of motion and both initial conditions. Assuming a solution in the form x(t) = e^(rt), the equation of motion becomes: \(r^2 me^{rt} = -ke^{rt} - bre^{rt}\) Canceling the exponentials (since they are nonzero) and solving for r, we get: \(r^2 m + br + k = 0\). This is a quadratic equation for r, with the following solutions: \(r = \frac{-b\pm\sqrt{b^2 - 4mk}}{2m}\). Since the oscillator is weakly damped, let's assume a real, distinct pair of roots, which corresponds to \(b^2 > 4mk\). The general solution for x(t) is: \(x(t) = C_1e^{r_1 t} + C_2e^{r_2 t}\).

Applying the Initial Conditions and Solving for the Constants

To find the constants \(C_1\) and \(C_2\), we apply the initial conditions: 1. \(x(0)=0 \Rightarrow C_1+C_2=0 \Rightarrow C_2=-C_1\) 2. \(v(t_1)-v(0)=\frac{J_{0}}{m}\) We know that \(v(t)=\dot{x}(t)=C_1 r_1e^{r_1 t} + C_2 r_2e^{r_2 t}\), so the second initial condition gives: \(\frac{J_{0}}{m} = v(t_1)-v(0) = C_1(r_1 e^{r_1 t_1} -r_1) + C_2(r_2 e^{r_2 t_1} -r_2)\) But since \(C_2=-C_1\), we can substitute and solve for \(C_1\): \(C_1 = \frac{J_{0}}{(r_2-r_1)(r_2e^{r_2t_1}-r_1e^{r_1t_1})}\) Now we can find the particular solution: \(x(t) = \frac{J_{0}}{(r_2-r_1)(r_2e^{r_2t_1}-r_1e^{r_1t_1})}(e^{r_1t}-e^{r_2t})\). This is the position function \(x(t)\) for a weakly damped harmonic oscillator with an initial velocity and no initial displacement, and it satisfies both the equation of motion and the initial conditions.

Key concepts

Damped Harmonic Motion

Damped harmonic motion describes the behavior of a system that experiences a restoring force proportional to displacement, as well as a resistive force that opposes the motion, commonly referred to as damping. In the context of a damped harmonic oscillator, such as a mass attached to a spring moving in a viscous medium, this damping force is usually modeled as being proportional to the velocity of the mass.

Mathematically, the damping force can be expressed as \(F_y = -bv\), where \(b\) is the damping coefficient, and \(v\) is the velocity of the oscillator. The presence of damping reduces the amplitude of oscillations over time, leading to attenuated motion. This is a realistic scenario in most physical systems, where friction or air resistance dissipates energy, making perfect harmonic motion (no damping) an idealization.

In the exercise, you're asked to find the resulting motion of a weakly damped harmonic oscillator after it receives an impulse. Weak damping implies that the resistive force is not strong enough to prevent oscillatory behavior immediately, but it will gradually reduce the amplitude of the motion.

Equation of Motion

The equation of motion for any mechanical system describes how the position of the system changes over time. For a damped harmonic oscillator, Newton's second law is used to relate the motion of the system to the forces acting on it. Specifically, the second law (force equals mass times acceleration) can be extended to include damping and is written in the form of a differential equation.

\[m\ddot{x}(t) = -kx(t) - bv(t)\]

This differential equation models the dynamics of a simple harmonic oscillator with mass \(m\), spring constant \(k\), and damping coefficient \(b\), with \(x(t)\) signifying the displacement from equilibrium, \(v(t)=\dot{x}(t)\) representing the velocity, and \(\ddot{x}(t)\) denoting the acceleration of the oscillator. Here, the negative sign indicates that both the restoring and damping forces oppose the displacement and velocity of the mass, respectively.

Impulse and Momentum

Impulse is defined as the change in momentum of a system, which can result from collisions or the application of an external force over a period of time. It is a crucial concept in understanding how forces affect the motion of objects.

The impulse delivered to a system can be calculated as the integral of force over the time interval that the force is applied:

\[J = \int F dt\]

In the given exercise, an initial impulse \(J_0\) is imparted to the harmonic oscillator, which causes an immediate change in its velocity. Since momentum \(p\) is the product of mass \(m\) and velocity \(v\), the impulse-momentum theorem can be written as:

\[J_0 = m(v(t_1) - v(0))\]

This expresses the total impulse as the difference between the final and initial momentum, helping us find the initial conditions for the velocity after the kick.

Differential Equations in Physics

Differential equations are fundamental to physics as they provide a mathematical framework for modeling and analyzing dynamic systems. These equations relate the rates of change of a system to the state of the system itself.

In physics, second-order differential equations are often used to describe systems with acceleration. The harmonic oscillator problem exemplifies this, where acceleration \(\ddot{x}(t)\) depends on displacement \(x(t)\) and velocity \(v(t)\). Analytical solutions to these equations reveal how physical quantities evolve over time.

For our damped harmonic oscillator, the analytical solution involves proposing a particular form for \(x(t)\) and solving a characteristic equation to find constants that satisfy both the equation of motion and the initial conditions. The resulting position function for \(x(t)\) reveals not just the position of the oscillator at any given time but also contains information about the amplitude, frequency, and damping of the motion.