Derive angular frequency for mass spring system

[so_gr_lo]

Homework Statement

Two masses m1 and m2 on the x axis are connected by a spring. The spring has stiffness s, length l and extension x. m2 is at position x2 and m1 at position x1. The equations of the motion are

m1d^2x1/dt^2 = sx and m2d^2x2/dt^2 = -sx

Combine these to show that the angular frequency is w = sqrt(s/M)

Where M = m1m2/ m1 + m2 (the reduced mass)

Relevant Equations

m1d^2x1/dt^2 = sx and m2d^2x2/dt^2 = -sx

w = sqrt(s/M)

M = m1m2/ m1 + m2

tried writing the x position as

x = Acos(wt) (ignoring the phase)

so that d²x / dt² = -w²x

Substituting that into the individual motion equations would get the required result for the individual masses, but I am not sure how to combine the equations to get the reduced mass

 

[TSny]

Can you express the extension $x$ in terms of $x_1$, $x_2$, and $l$?

 

[VVS2000]

Are the masses connected in series or to either ends of the spring?

 

[so_gr_lo]

Expresign extension in term of x1/2 gives

x2-x1 = x

Which could be substituted into each of the motion equatiosn but I'm not sure how that helps

 

[so_gr_lo]

Are the masses connected in series or to either ends of the spring?

Yes the masses are at the ends

 

[TSny]

Expresign extension in term of x1/2 gives

x2-x1 = x

Which could be substituted into each of the motion equatiosn but I'm not sure how that helps

$x_2 - x_1$ is the distance from one end of the spring to the other end of the spring. This distance will include the natural length of the spring $l$ and the extension $x$.

Once you express $x$ in terms of $x_1$ and $x_2$, you can see how $\ddot x$ is related to $\ddot x_2$ and $\ddot x_1$.

 

[so_gr_lo]

Is d²x/dt² = d²x₂/dt² - d²x₁/dt² ?

 

[TSny]

Is d²x/dt² = d²x₂/dt² - d²x₁/dt² ?

Yes. But I can't tell if you arrived at this correctly. What equation did you use for the relation between $x$ and $x_2 - x_1$?

 

[kuruman]

Here is a figure to help you out with setting the equations. When the spring is unstretched, the masses are at the dotted lines, i.e. $x_1-x_1=L$.

 

[so_gr_lo]

Yes. But I can't tell if you arrived at this correctly. What equation did you use for the relation between $x$ and $x_2 - x_1$?

I used between x = x₂-x₁

 

[TSny]

I used between x = x₂-x₁

This is not correct.
$x$ represents the amount of stretch of the spring from its unstretched length $l$. So, if $x = 0$, then $x_2 - x_1$ = $l$. Your equation doesn't satisfy this condition.

 

[so_gr_lo]

So x = x₂-x₁-l

 

[TSny]

So x = x₂-x₁-l

Yes, that's the correct relation.

 

[so_gr_lo]

Okay, but if I substitute that into the equations I get

m₁(d²x₂/dt²) - d²x₁/dt²) = sx

and similar for m₂, how does this help with combining them?

 

[TSny]

Okay, but if I substitute that into the equations I get

m₁(d²x₂/dt²) - d²x₁/dt²) = sx

This is not correct. You already had the correct equations of motion for $m_1$ and $m_2$ in the "Relevant Equations" section of your first post.

You also have $\ddot x = \ddot x_2 - \ddot x_1$. Use the equations of motion to express the right side in terms of $s$, $x$, and the masses.