Mass Oscillation: Conditions for Simple Harmonic Motion

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In summary, the conversation discusses the conditions for a mass (m) located on the +x axis, at a distance of X from the origin, to be approximated as simple harmonic motion. This is determined by the equation of motion, which involves the gravitational forces acting on the mass from two fixed masses (M) located on the +y and -y axes. The relevant condition is that x is much smaller than L, and the first term after the linear term in the Taylor expansion can be ignored. This leads to an equation with a linear term and a restoring force, allowing for simple harmonic motion to occur.
  • #1
esradw
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Hello,
In my question,I have two masses ( M ) ,one fixed at +y and the other at -y axis and both have a distance of L from the origine. The third mass (m) is located on the +x axis at the distance of X.

I know that the gravitational forces are acting on the (m) by both masses (M), The net force is on the x-axis toward (-x) and magnitude of 2Fgrav and this force will accelerate the (m) toward equilibrium (Origine) and once it is there the Fgrav =0 but because it has a velocity it will continue until its velocity=0 , So this Fgrav force on the X axis is Restoring force .Therefore,the mass will oscilate. My question is since my equation is
(x:+2GMmx/(L^2+X^2)^3/2=0) , I can't say this is Simple harmonic oscillation because my equation of motion doesn't just consist of x but x/(...+x^2)^3/2,
So under what condition for x ,it is possible to say that the motion of the mass (m) can be approximated as Simple harmonic motion ?

Any idea?

thanks
 
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  • #2
if you know that the mass will oscillate back and forth on the x-axis then you don't need the y components and you can use trig to isolate the x component of the force. this will give you an equation in x
 
  • #3
Do you mean that since my equation of motion x (2dot)+2GMm(x/(L^2+x^2)^3/2=0, ( for SHM, the equation of motion x(2dot)+W^2x=0 ) I can just say that (L^2+x^2)^3/2 = 1 so x=(1-L^2)^1/2 )

Is this right ?

thanks
 
  • #4
esradw said:
So under what condition for x ,it is possible to say that the motion of the mass (m) can be approximated as Simple harmonic motion ?
As with most other problems where the approximation to a harmonic oscillator is made, the relevant regime is one of small oscillations, ie: x << L (so that, to first order in L2 + x2 ~ L2 )

Typically, you write the taylor expansion, and will see that the first term after the linear term is of order 3 in x/L. You can throw away this and smaller terms.
 
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  • #5
thank you very much,I understand now
 

1. What is mass oscillation?

Mass oscillation, also known as simple harmonic motion, is a type of repetitive motion in which an object oscillates back and forth around its equilibrium position. The motion is periodic, meaning that it repeats itself at regular time intervals.

2. What are the conditions for simple harmonic motion?

The conditions for simple harmonic motion are: 1) the restoring force must be directly proportional to the displacement of the object from its equilibrium position, 2) the motion must be periodic, and 3) there must be no energy loss or friction present in the system.

3. What is the role of mass in simple harmonic motion?

The mass of an object affects the period of its oscillation. A greater mass will result in a longer period, while a lighter mass will have a shorter period. However, the mass does not affect the frequency of the oscillation.

4. How is simple harmonic motion different from other types of motion?

Simple harmonic motion is different from other types of motion in that it follows a specific pattern of oscillation, in which the displacement of the object is directly proportional to the restoring force and the motion is periodic. Other types of motion, such as linear or circular motion, do not necessarily follow these patterns.

5. What are some real-life examples of mass oscillation?

Some real-life examples of mass oscillation include the motion of a pendulum, a mass on a spring, and the vibration of a guitar string. These all follow the conditions for simple harmonic motion and can be observed in various forms in daily life.

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