# Damped Oscillator Conceptual Problem and Differential Equation Solution

• ggilvar99
In summary, the conversation discusses a conceptual problem involving a block connected to a spring and a viscous damping mechanism. The observations made include the static compression of the spring when the block is pushed horizontally with a force equal to its weight and the viscous resistive force when the block moves with a certain speed. The task is to write the differential equation for horizontal oscillations of the mass and find the angular frequency for a specific case. The conversation also mentions relevant equations and attempts at a solution. The suggestion is made to obtain the values for k and lambda from the given conditions in order to solve the problem.
ggilvar99
Hey guys I'm new to the forum and having a little trouble with this conceptual problem.

1. A block of mass m is connected to a spring, the other end of which is fixed. There is also a viscous damping mechanism. The following observations have been made of this system:

i) If the block is pushed horizontally with a force equal to mg, the static compression of the spring is equal to h

ii) The viscous resistive force is equal to mg as the block moves with a speed u.

a) Write the differential equation governing horizontal oscillations of the mass in terms of m, g, h and u.

b) for the particular case of u = 3√gh, what is the angular frequency of the damped oscillations?

2. Homework Equations :

mx'' + λx' + kx = 0

3. The Attempt at a Solution :

F = mg = -kh (x = h)

F = mg = -λu for x' = u

At this point I'm somewhat lost and not sure what they're looking for. If the viscous force = mg at velocity u, how can you translate that into a differential equation that covers all velocities of the mass? Any help would be greatly appreciated

Can you obtain express ##k## and ##\lambda## from the conditions given?

Oh, you're saying to set k = -(mg)/h and lambda = -(mg)/u and plug that into the diffeq? I don't know why that never occurred to me, thanks a lot for the suggestion!

## 1. What is a damped oscillator?

A damped oscillator is a system or object that oscillates or moves back and forth, but with each oscillation, the amplitude or energy of the oscillation decreases due to an external force or resistance. This is in contrast to an undamped oscillator, where the oscillations would continue indefinitely without any external forces.

## 2. What types of forces can cause damping in an oscillator?

There are several types of forces that can cause damping in an oscillator, including frictional forces, air resistance, and electromagnetic forces. These forces can act to decrease the amplitude of the oscillations and eventually bring the oscillator to rest.

## 3. How is damping measured in an oscillator?

Damping is typically measured by the damping ratio or damping coefficient, which represents the amount of energy lost per oscillation in the system. It can also be measured by the decay constant, which is the rate at which the amplitude of the oscillations decreases over time.

## 4. What are some real-world examples of damped oscillators?

Damped oscillators can be found in many everyday objects, such as a swinging pendulum, a car's suspension system, or a guitar string. They are also commonly found in mechanical and electrical systems, such as in shock absorbers, electronic circuits, and musical instruments.

## 5. How is the behavior of a damped oscillator different from an undamped oscillator?

The main difference between a damped and undamped oscillator is that the amplitude of the oscillations in a damped oscillator decreases over time, while an undamped oscillator maintains a constant amplitude. Additionally, the frequency of oscillation may also be affected by damping, causing the oscillator to have a different period or time for one complete oscillation.

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