Solving Oscillating Mass-Spring System w/ Non-Resonant Force

In summary, the homework statement is that you need to find the position or velocity of a mass attached to a spring when subjected to an oscillating force. The attempt at a solution is to use a model and solve for the position or velocity. However, the problem is that you need to solve for the position or velocity when the force has a frequency different from the natural frequency of the oscillating system.
  • #1
Yann
48
0
1. Homework Statement and 2. Homework Equations

Find position/velocity of a mass m attached to a spring of constant k when subjected to an oscillatinf roce

[tex]
F(t) = F sin(Bt)
[/tex]

With [tex]B\not = \sqrt{k/m}[/tex]

The Attempt at a Solution



Model;

[tex]
mx'' + kx = F \sin(Bt)
[/tex]

I have no idea if/how it can be solved (without a computer, of course). Because;

[tex]
mr^2 + k = 0
[/tex]

Gives

[tex]
r = ±\sqrt{-k/m}
[/tex]

As [tex]B\not = \sqrt{k/m}[/tex] it can't be an answer.
 
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  • #3
I'm not sure I understand your point, it's more a problem of math than a problem of physics, I must solve;

[tex]
mx'' + kx = F \sin(Bt)
[/tex]

But I don't know how
 
  • #4
Yann said:
I'm not sure I understand your point, it's more a problem of math than a problem of physics, I must solve;

[tex]
mx'' + kx = F \sin(Bt)
[/tex]

But I don't know how

First solve the homogenous equation first:

mx'' + kx = 0

Then you need a particular solution for
[tex]
mx'' + kx = F \sin(Bt)
[/tex]

just plug in x = Asin(Bt) into the differential equation and solve for A...

Then your general solution is the solution for the homogenous equation + the particular solution Asin(Bt)...

And finally you need to deal with initial conditions...
 
  • #5
Thx for the help, I solved the diff. equation. But will the solution to the differential equation give me the position or the velocity at time t ? And there's no initial condition, only [tex]B\not = \sqrt{k/m}[/tex], I don't know what to do with it.
 
  • #6
You actually need two boundary conditions, but since you don't have them you can probably just leave the two constants unsolved for.
 

What is an oscillating mass-spring system?

An oscillating mass-spring system is a physical system that consists of a mass attached to a spring. The mass is able to move back and forth due to the force exerted by the spring. This system is commonly used in physics experiments to study the behavior of simple harmonic motion.

How do you solve an oscillating mass-spring system with a non-resonant force?

To solve an oscillating mass-spring system with a non-resonant force, you can use the equation of motion for simple harmonic motion: x = A*sin(ωt + φ), where x is the displacement of the mass, A is the amplitude, ω is the angular frequency, and φ is the phase angle. You can also use the equation for the total energy of the system: E = 1/2*k*A^2, where k is the spring constant. By solving these equations, you can determine the displacement, velocity, and acceleration of the mass at any given time.

What is the difference between a resonant and non-resonant force in an oscillating mass-spring system?

A resonant force in an oscillating mass-spring system is a force that has the same frequency as the natural frequency of the system. This can cause the system to oscillate with a larger amplitude, leading to resonance. On the other hand, a non-resonant force has a different frequency than the natural frequency of the system, resulting in smaller amplitudes and no resonance.

What factors can affect the behavior of an oscillating mass-spring system with a non-resonant force?

The behavior of an oscillating mass-spring system with a non-resonant force can be affected by several factors, including the amplitude and frequency of the force, the mass of the object, and the stiffness of the spring. Additionally, factors such as air resistance and friction can also affect the behavior of the system.

What are some real-world applications of an oscillating mass-spring system with a non-resonant force?

An oscillating mass-spring system with a non-resonant force has various real-world applications, such as in the construction of bridges and buildings, designing shock absorbers for vehicles, and creating musical instruments. It is also commonly used in studies of earthquake seismology and in the development of suspension systems for cars and trains.

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