The Vertical Force of a Spring: A Logical Argument

[hutchphd]

Alas I miss the grocery store...

This colloquy has seemed to me a prime example of the pitfalls of doing physics without formal maths
In particular, as I have mentioned,

surmises regarding the tension and the effects of gravity and of opposition thereto thereon.

I still do not know what the surmises are or even their number. So much the pity, because I feel that some of the results you to which you allude may not be correct; the formal approach in this case is actually simpler and my intuition alone would not necessarily get me to the correct result. It is a case study in method.
I have been mildly frustrated because it is impossible to provide an answer (even when known) if you cannot ascertain the specific question!
But a good time was had by all...

 

[sysprog]

Alas I miss the grocery store...

This colloquy has seemed to me a prime example of the pitfalls of doing physics without formal maths
In particular, as I have mentioned,

I still do not know what the surmises are or even their number. So much the pity, because I feel that some of the results you to which you allude may not be correct; the formal approach in this case is actually simpler and my intuition alone would not necessarily get me to the correct result. It is a case study in method.
I have been mildly frustrated because it is impossible to provide an answer (even when known) if you cannot ascertain the specific question!
But a good time was had by all...

Whence the rotation?

 

[hutchphd]

Whence the rotation?

initial conditions: For instance horizontal impulse followed by vertical impulse a quarter period π/2ω iater...or pull it off equilibrium sideways and give it the appropriate vertical speed...or give it a tangent spiral in an initial interval.
The circle is certainly an allowed solution whatever the size of gravity.

 

[sysprog]

initial conditions: For instance horizontal impulse followed by vertical impulse a quarter period π/2ω iater...or pull it off equilibrium sideways and give it the appropriate vertical speed...or give it a tangent spiral in an initial interval.
The circle is certainly an allowed solution whatever the size of gravity.

Why would both the radius of the circle and also the velocity of the weight be invariant within the cycle whatever the varying direction of the weight within the cycle with respect to the invariant direction of the gravity?

 

[haruspex]

Why would both the radius of the circle and also the velocity of the weight be invariant within the cycle whatever the varying direction of the weight within the cycle with respect to the invariant direction of the gravity?

Because when you do the math, which you have conspicuously failed to do, you find that, given the right initial conditions, the resultant of the tension in the string and gravity is of constant magnitude and is always normal to the velocity.

 

[hutchphd]

Why would both the radius of the circle and also the velocity of the weight be invariant within the cycle whatever the varying direction of the weight within the cycle with respect to the invariant direction of the gravity?

I think it is intuitive that this is possible in the absence of gravity (yes?). It certainly is easy to show also.
It has been shown almost trivially (see #83 ) the solution with gravity is identical in shape and speed and acceleration to the no gravity one.
Therefore it is absolutely true. One need not look farther.

 

[haruspex]

conducting a few crude experiments with the produce scale,

The hard part about demonstrating the result experimentally is the zero relaxed length condition. One way would be with a torsion spring on a horizontal axis connected to the axle of a drum. A string is wound around the drum. Just where the string leaves the drum, at the same height as its axis, it passes through a small smooth aperture, to prevent sideways movement, then, with the torsion spring in its relaxed state, connects immediately to the mass.
Thus, the distance of the mass from the aperture is proportional to the angle of torsion on the spring.

 

[hutchphd]

Its tricky to get an exact circle even if the springs have a nonzero outstretched length. Why is it easier for the nonzero case?
Seems to me the method usually used is to displace the object radially outward and then supply just the right amount of tangential velocity (pull it out and shove it sideways).

The astronauts get to make corrective burns to circularize their orbits...

 

[kuruman]

I am a latecomer to this thread, but there is a simple heuristic way to look at this by bringing two considerations together.

First, a vertical spring-mass system in gravity behaves as a horizontal spring-mass system with the equilibrium position lowered to $\dfrac{mg}{k}$ from the relaxed position.
In this case the position of the mass relative to the relaxed position can be written as $$y(t) =A\cos(\omega t)-\frac{mg}{k}.$$ The initial conditions are $y(0)=A-\dfrac{mg}{k}~;~~\dot y(0)=0.$

Second, a mass, undergoing uniform circular motion in the $xy$-plane of radius $A$ with angular speed $\omega$, has coordinates $$x(t)=A\sin(\omega t)~;~~y(t)=A\cos(\omega t).$$ That's for a line of sight along the $z$-axis. If viewed along the $x$-axis the motion is the same (mathematically) as that of a spring-mass system oscillating along the $y$-axis such that $\sqrt{\frac{k}{m}}$ matches the angular speed of the circular motion.

Putting the two considerations together, we can immediately write an expression for the position of the mass$$\vec r(t)=A\sin(\omega t)~\hat x+\left(A\cos(\omega t)-\frac{mg}{k}\right)~\hat y.$$We can verify that this works with a parametric plot ($A=1;~mg/k=A/3$)

Clearly, $$\ddot x =-\omega^2 x~;~~\ddot y=-\omega^2 y$$which are the equations of motion for a spring of zero length placed at the origin of Cartesian coordinates. Thus, we have a solution that is consistent with Newton's second law. Initial conditions and anything else one cares to find can be obtained from $\vec r(t)$. Finally, I see no reason why this cannot be extended to a sphere in three dimensions.