Sheet Set Natural

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Sheet Set Natural

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How to find the natural length of a spring with given, constant force, radius, mass and revolutions?

A ball with mass of 0.1kg rests on horizontal sheet of essentially frictionless ice. It is attached by a spring with constant force of 80 N/m to a post set in the ice. Once given push, the ball revolves uniformly in circle with radius 0.5 m around the post. If the ball revolves makes 2 complete revolutions in 1 s, find the natural length for the spring.

Notice that the ball is moving in a circle.

_Anything_ that moves in a circle (at constant speed) has an acceleration equal to this amount:

a = ω²r

(where ω = its angular velocity, and r = its distance from the center of the circle).

Therefore, the net force on the ball is:

F_net = ma = mω²r

But this force is supplied entirely by the spring. That is, the spring must be stretched just enough to supply a force of mω²r.

F_spring = mω²r

But F_spring is also equal to the spring constant (k) times the amount of stretch (x). That means:

F_spring = kx = mω²r

Now, x and r and the “natural length” of the spring (call it L) are related as follows:

r = (natural length) + (amount of stretch)
r = x + L

So that means you can get rid of “x” in the previous equation by replacing it with “r – L”:

k(r-L) = mω²r

The problem gives you “k” and “r” and “m” directly. You can calculate “ω” by converting (2 revolutions/sec) into radians per second.

Then all you need to do is solve for “L”.

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