# * *Damped Harmonic Motion (Differential Equations)

• acedeno
In summary, the conversation is about solving a damped harmonic oscillator equation using a trial function and determining the numerical values of two constants with given initial conditions. The solution involves complex numbers, so if the person missed the class on complex numbers, they may not be able to solve the problem.

#### acedeno

*URGENT*Damped Harmonic Motion (Differential Equations)

A damped harmonic oscillator satisfies the following equation: d2x/dt2 = − 5x − 2dx/dt

(a) By assuming a trial function of the form x = A e^qt, determine the solution of this
equation "from scratch." Express your final answer as a real function, that is, there
should be no i’s in your final answer (where i = (−1)^½).

(b) Your solution to part (a) should have two constants (of integration).
If at t = 0, x = 0.0100 m and dx/dt = 0, determine the numerical values of these two
constants, correct to 3 significant digits.

I have no idea how to do this question. Could somebody(if they can) show me the solution? I have a midterm tomorrow and these are the types of questions I will need answer. I missed this class so I'm completely lost :S

You should really keep to the physics forums rules and show your own attempt. If you missed the class on complex numbers, then you might not be able to do this question, because q must be a complex number.

## 1. What is damped harmonic motion in the context of differential equations?

Damped harmonic motion refers to a type of motion where a system or object experiences a restoring force that is proportional to its displacement from its equilibrium position, as well as a damping force that is proportional to its velocity. This type of motion can be described using a second-order differential equation.

## 2. What is the difference between damped harmonic motion and simple harmonic motion?

The main difference between damped harmonic motion and simple harmonic motion is the presence of a damping force in damped harmonic motion. This force causes the amplitude of the motion to decrease over time, while in simple harmonic motion, the amplitude remains constant.

## 3. How is damped harmonic motion modeled using differential equations?

Damped harmonic motion can be modeled using a second-order differential equation of the form mx'' + bx' + kx = 0, where m is the mass of the object, b is the damping coefficient, and k is the spring constant. This equation can be solved using various methods, such as the method of undetermined coefficients or the Laplace transform.

## 4. What are some real-life examples of damped harmonic motion?

Damped harmonic motion can be observed in many real-life systems, such as a mass-spring system with air resistance, a pendulum with friction, or an RLC circuit with resistance. It can also be seen in everyday objects, such as a door closing on its own or a car's suspension system.

## 5. How does the damping coefficient affect damped harmonic motion?

The damping coefficient, denoted by b, determines the amount of damping in a system and therefore affects the behavior of damped harmonic motion. A higher damping coefficient leads to faster decay of the amplitude, while a lower damping coefficient results in slower decay. A critical damping coefficient results in the fastest return to equilibrium without overshooting.