Spring-mass oscillate in 1-D on frictionless horizontal surface

In summary: The phase of the motion is when the mass is moving in relation to the external force, while the phase of the velocity is the angle between the velocity and the external force.
  • #1
olga11
30
0
We have a spring-mass system which oscillates in one dimension on a frictionless horizontal surface. We act an external force on the mass. F=kχo sin(ωt). Let x=Asin(ωt+φ),A>0 and ωο is the natural frequency of the system.
a) What is the phase of the motion of the mass relatively to the phase of the external force when ω<ωο and when ω>ωο?
b) What is the phase of the velocity relatively to the phase of the external force when ω<ωο and when ω>ωο?



If ω<ωο the motion of the mass is in phase φ with the external force and the amplitude of the mass oscillation is greater than the amplitude of the wiggling. As the forcing frequency approaches the natural frequency of the oscillator, the response of the mass grows in amplitude. When the forcing is at the resonant frequency, the response is technically infinite.
When the forcing frequency is greater than the natural frequency, the mass actually moves in the opposite direction of the motion, the response is out of phase with the forcing. The amplitude of the response decreases as the forcing frequency increases above the resonant frequency.

I cannot think of anything else. Could anybody please tell me if I am close to the right answer?
 
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  • #2
Nope, not quite. Have you done Fourier transforms yet? The answer to this question is easiest to see if you take the Fourier transform and calculate the transfer function of the system. Then you can graph the phase part from [itex]0 < \omega < \infty[/itex]. It helps to use a logarithmic scale on the [itex]\omega[/itex] axis.

What you should see is that the phase is bounded by [itex]-\frac{\pi}2 < \varphi < \frac{\pi}2[/itex]. That is, at the most, your mass will either lag or lead the forcing function by 90 degrees. [itex]\varphi[/itex] is only exactly zero when [itex]\omega = \omega_0[/itex].

At least, I know this to be the case when you have dissipation. But you're talking about a frictionless system, so I'm not sure.

If you're not familiar with Fourier transforms and transfer functions, you will probably just have to solve the system normally, and use some trig identities to get the answer into a form that will allow you to answer the question. I come from an engineering background, so we all study transfer functions; I don't know if they're even covered in most physics courses.
 
  • #3
Here is a link on transfer functions: http://en.wikipedia.org/wiki/Transfer_function

It focuses primarily on Laplace transforms, but Fourier transforms are a special case; scroll down the page. The Fourier transfer function is also called the "frequency response".
 
  • #4
Thank you very much. I will see what I can do and I will come back.
 
  • #5
Actually, here is a much simpler answer. We were both wrong. You are right in that the mass is in phase with the driving force for small [itex]\omega[/itex], and that it is 180 degrees out of phase for large [itex]\omega[/itex]. However, the phase lag varies continuously, and for [itex]\omega = \omega_0[/itex], the mass lags the forcing function by 90 degrees:

http://www.kettering.edu/~drussell/Demos/SHO/mass-force.html [Broken]

Edit: Whoops, I'm still slightly off. This is for a damped system. In your undamped system, you were correct at the beginning!
 
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  • #6
I think I am lost now. Does antone have a hint, please?
 
  • #7
What is the difference between the phase of the motion and the phase of the velocity?
 

1. What is a spring-mass oscillation on a frictionless surface?

A spring-mass oscillation on a frictionless surface is a type of periodic motion where a mass attached to a spring will move back and forth in a straight line. The motion is frictionless because there is no external force acting on the system to slow it down.

2. What are the factors that affect the frequency of a spring-mass oscillation?

The frequency of a spring-mass oscillation is affected by two main factors: the stiffness of the spring and the mass of the object attached to it. A stiffer spring will have a higher frequency, while a heavier mass will have a lower frequency.

3. How does the amplitude of a spring-mass oscillation change over time?

The amplitude of a spring-mass oscillation will gradually decrease over time due to the dissipation of energy through the spring and air resistance. This process is known as damping and results in the oscillation eventually coming to a stop.

4. What is the equation for the period of a spring-mass oscillation?

The period (T) of a spring-mass oscillation can be calculated using the equation T = 2π√(m/k), where m is the mass attached to the spring and k is the spring constant, a measure of the stiffness of the spring.

5. How does the presence of friction affect a spring-mass oscillation?

The presence of friction will result in the damping of a spring-mass oscillation, causing the amplitude to decrease over time. This is because friction converts the kinetic energy of the oscillation into heat, reducing the total energy of the system.

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