What is the value of the damping constant?

In summary, the conversation discusses a damped harmonic oscillator with a mass of 2.0kg and a spring with a stiffness of 10 N/m. The damping force is proportional to the velocity, and after 4 complete cycles, the amplitude of the oscillator reduces to three-fourths of its initial value. The questions asked involve the value of the damping constant and the amount of energy dissipated during the 4 cycles. The conversation also presents an equation to calculate the amplitude at different time intervals, where a mistake was made in the initial amplitude calculation. The individual asks for help and feedback on their progress.
  • #1
hemetite
50
0
Qn4.

A damped harmonic oscillator involves a block of mass 2.0kg and a spring with a stiffness 10 N/m. The damping force is proportioanla to the velocity of the oscillator. Initially it osicillates with an amplitude of 25cm. Due to the damping, the amplitude fallsto three-fourth of this initial value after 4 complete cycles..

(a) What is the value of the damping constant?
(b) How much energy has been "dissipated" during the 4 cycles?

Here are my thoughts..

Initial A=0.25m
after 4 cycle it reduced to = 3/4 * 0.25 = 0.1875m

w=sqrt [ k/m - (b/2m)sq]
= sqrt [ 10/2 - (bsq)/16
= sqrt [(80-bsq) / 16 ]

i will be using

x(t) = A exp (-b/2m)t sin (wt + teta) -----------> equation 1

at t= 0 for the first cycle A= 0.25m

1 cycle = 2pi, after 4 complete cycle. it will be at 8pi

therefore putting the values A= 0.25, t=8pi, m=2kg and w= sqrt [(80-bsq) / 16 ]
into equation 1

0.1875= 0.25 exp (-b/4) 8pi sin (w8pi) * sqrt [(80-bsq) / 16 ]

here i solve for b...to get answer the first answer


am i on the right track?
 
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  • #2
hemetite said:
Qn4.


x(t) = A exp (-b/2m)t sin (wt + teta) -----------> equation 1

at t= 0 for the first cycle A= 0.25m

mistake...for first Amplitude to occur...t= pi/2

what i have done so far...correct?...

help? hint?..critics?
 
  • #3

1. What is the damping constant?

The damping constant is a parameter that describes the rate at which a system loses energy. It is commonly denoted by the letter "b" and is often used in the context of oscillatory or vibrating systems.

2. How is the damping constant calculated?

The damping constant is typically calculated using experimental data or mathematical models. In experimental settings, it can be determined by measuring the amplitude and frequency of oscillations and using equations such as the damped harmonic oscillator equation. In mathematical models, it is often included as a variable that can be adjusted to best fit the observed behavior of the system.

3. What is the role of the damping constant in a system?

The damping constant plays a crucial role in determining the behavior of a system. It affects the rate at which the system loses energy, the amplitude of oscillations, and the system's response to external forces. A higher damping constant results in a faster loss of energy and a shorter period of oscillation, while a lower damping constant leads to a slower loss of energy and a longer period of oscillation.

4. How does the value of the damping constant impact a system's stability?

The value of the damping constant can greatly impact a system's stability. If the damping constant is too low, the system may become unstable and exhibit large, uncontrollable oscillations. On the other hand, if the damping constant is too high, the system may become overdamped, meaning it will not oscillate at all. Finding the optimal value of the damping constant is essential for maintaining stability in a system.

5. Can the damping constant be changed or controlled?

Yes, the damping constant can be changed or controlled in some systems. In experimental settings, it can be manipulated by adjusting the physical properties of the system, such as the material used or the shape of the components. In mathematical models, it can be adjusted by changing the value of the variable representing the damping constant. However, in some systems, the damping constant is fixed and cannot be altered.

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