Change in the amplitude of a damped spring block oscillator

In summary, the problem involves a block acted on by a spring and a weak friction force. The block is pulled from equilibrium and released, oscillating multiple times before coming to rest. The goal is to show that the decrease in amplitude is the same for each cycle of oscillation. The solution involves calculating the change in energy and using the approximation of harmonic motion to find the distance traveled by the block in one oscillation.
  • #1
CrazyNeutrino
100
0

Homework Statement


A block is acted on by a spring with spring constant k and a weak friction force of constant magnitude f . The block is pulled distance x0 from equilibrium and released. It oscillates many times and eventually comes to rest.

Show that the decrease of amplitude is the same for each cycle of oscillation.

2. The attempt at a solution
$$\Delta \mathbf{E} = E_f -E_i = \frac{1}{2}kx_0^2 - \frac{1}{2}k(x_0-\Delta x)^2 = f \cdot D$$
where ##\Delta x## is the change in amplitude after one oscillation and D is the distance traveled by the block in one oscillation. I am stuck here. How do I find D? It can't simply be 4x_0 because the distance from the point of rest to the equilibrium decreases on every half oscillation. Where do I go from here?
 
Physics news on Phys.org
  • #2
The friction force is assumed weak, so you can approximate the leading order effect by assuming harmonic motion and ##\Delta x \ll x_0##.
 

Related to Change in the amplitude of a damped spring block oscillator

1. What is a damped spring block oscillator?

A damped spring block oscillator is a physical system that consists of a spring, a mass attached to the spring, and a damping force that acts on the mass. The mass oscillates back and forth due to the spring force, but the damping force reduces the amplitude of the oscillations over time.

2. How does the amplitude of a damped spring block oscillator change over time?

The amplitude of a damped spring block oscillator decreases over time due to the presence of a damping force. This force dissipates the energy of the system, causing the amplitude of the oscillations to decrease until the system eventually reaches equilibrium.

3. What factors affect the rate of change in the amplitude of a damped spring block oscillator?

The rate of change in the amplitude of a damped spring block oscillator is affected by the damping coefficient, the mass of the system, and the initial amplitude of the oscillations. A higher damping coefficient or a larger mass will result in a faster decrease in amplitude, while a larger initial amplitude will result in a slower decrease in amplitude.

4. Can the amplitude of a damped spring block oscillator ever increase?

No, the amplitude of a damped spring block oscillator can never increase. The damping force always acts to decrease the amplitude, and as the system loses energy, the amplitude will continue to decrease until it reaches zero.

5. How is the change in amplitude of a damped spring block oscillator represented mathematically?

The change in amplitude of a damped spring block oscillator can be represented by the exponential function Ae-bt, where A is the initial amplitude, b is the damping coefficient, and t is time. This equation shows how the amplitude decreases exponentially over time due to the damping force.

Similar threads

  • Introductory Physics Homework Help
Replies
11
Views
1K
  • Introductory Physics Homework Help
Replies
29
Views
1K
  • Introductory Physics Homework Help
Replies
12
Views
2K
  • Introductory Physics Homework Help
Replies
5
Views
589
  • Introductory Physics Homework Help
Replies
17
Views
510
  • Introductory Physics Homework Help
Replies
1
Views
2K
  • Introductory Physics Homework Help
Replies
30
Views
845
  • Introductory Physics Homework Help
Replies
24
Views
1K
  • Introductory Physics Homework Help
Replies
27
Views
6K
  • Introductory Physics Homework Help
Replies
1
Views
6K
Back
Top