Force on Body Attached to Spring at Displacement x - A.P. French

In summary, the first equation is a Taylor series for the restoring force, and it reduces to the linear restoring force when the displacement is small. The second equation is the derivative of the first equation, and it shows that the mass is trying to resist displacement by pushing against the spring.
  • #1
Slimy0233
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Source: A.P. French's Vibrations and Waves

I do not recognize the first equation, can someone explain how it came to be? The reasoning behind it.

How can force on a body attached to a spring at small displacement x be represented as

1685899566771.png


? I know recognize F = - kx (restoring force)

I realize that the mass is at equilibrium and not rest, thus there were/are multiple forces acting on the spring, thus, I guess my question simplifies, what is the nature of the forces

1685899663451.png


if -kx is restoring force, what are the rest of the forces, can someone please state an example for better understanding?

edit: Good God, creating a post is no joke :')
edit 2: The math which was visible at first is not visible now.
 
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  • #2
The first equation is just a Taylor series. Whatever form the restoring force takes (not just Hooke's law), you can Taylor expand it into a series, a polynomial of some high (possibly infinite) order. But if we agree to restrict ourselves to the region where ##x## is small then ##x^2##, ##x^3##, etcetera, must be really small and we can neglect all terms except the ##x## one. Then the force reduces to the linear restoring force you are familiar with.

(That's a paraphrase of the paragraph between the two marked equations, by the way.)
 
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Ibix said:
The first equation is just a Taylor series. Whatever form the restoring force takes (not just Hooke's law), you can Taylor expand it into a series, a polynomial of some high (possibly infinite) order.
I am sorry, can you please explain this more. Especially the "whatever form the restoring force takes" part.
 
  • #4
Well, a restoring force is just how strongly a system resists deformation or displacement. It doesn't have to be directly proportional to the displacement. An obvious example is a pendulum, where the restoring force is proportional to the sine of the displacement angle, ##\theta##. But you can expand that sine as a Taylor series, and as long as you keep the angle small then you can neglect the ##\theta^3## and higher terms (the even power terms are zero in this case). This is the formal justification for writing "##\sin\theta\approx\theta## for small ##\theta##". Once you have done that, you have justified modelling a small-amplitude pendulum as a simple harmonic oscillator.

(And, although French doesn't mention it above, you can find out how big ##\theta## has to be for the ##\theta^3## term to matter, and hence how small a "small" amplitude actually must be.)
 
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  • #5
So, in the case of the pendulum, the first equation you have marked would be $$\begin{eqnarray*}
F(\theta)&=&-mg\sin\theta\\
&=&-mg\left(\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-\ldots\right)
\end{eqnarray*}$$As long as ##\theta## is small this is approximately ##F(\theta)\approx-mg\theta##. This is the right hand side of the second equation you have marked, which would therefore be$$ml\frac{d^2\theta}{dt^2}=-mg\theta$$
 
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  • #6
@Ibix These are one of the best answers I have ever received. Thank you very much!!

Beautifully explained!
 
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  • #7
You're very welcome. I don't think the explanation is quite as unparalleled as you say, but I'm glad it helped you.
 

FAQ: Force on Body Attached to Spring at Displacement x - A.P. French

What is the formula for the force on a body attached to a spring at displacement x?

The force on a body attached to a spring at displacement x is given by Hooke's Law, which states F = -kx, where F is the force, k is the spring constant, and x is the displacement from the equilibrium position.

How is the spring constant (k) determined?

The spring constant (k) is determined by the stiffness of the spring. It can be found experimentally by measuring the force applied to the spring and the resulting displacement, and then using the formula k = F/x.

What is the significance of the negative sign in Hooke's Law?

The negative sign in Hooke's Law indicates that the force exerted by the spring is always in the opposite direction to the displacement. This means the spring always tries to return to its equilibrium position, providing a restoring force.

How does the mass of the body affect the motion of the spring system?

The mass of the body affects the motion of the spring system by influencing its natural frequency of oscillation. The angular frequency (ω) of the system is given by ω = sqrt(k/m), where m is the mass of the body. A larger mass results in a lower frequency of oscillation.

What are the conditions for simple harmonic motion in a spring-mass system?

The conditions for simple harmonic motion in a spring-mass system include a linear restoring force proportional to displacement (Hooke's Law), negligible damping, and no external driving forces. Under these conditions, the system will oscillate with a sinusoidal motion.

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