Resonance in a Mechanical System

[amk1995]

Homework Statement

A mass m is attached to one end of a massless spring with a force constant k and an unstretched length l₀. The other end of the spring is free to turn about a nail driven into a frictionless, horizontal surface. The mass is made to revolve in a circle with an angular frequency of revolution ω.
Question:
Calculate the length l of the spring as a function of ω.

Homework Equations

F = ω²*l*m
F_Spring = k*l

The Attempt at a Solution

Fspring = Fcirc
-k*l = ω²*l*m

The issue I am having is how to get L by itself, or what I am supposed to do with l₀. Can someone guide me on where I should head in order to solve this problem? The issue is l cancels when I divide, unless I am supposed to use l₀ in place of l
Thanks!

 

[nasu]

You are using "l" to designate two different things.
If "l" in the first equation is the total length of the spring (radius of the circle), then the elastic force is not correct. The force does not depend on the total length.
What is the force when the length is lo?

 

[amk1995]

The force when length lo is just omega^2 *m*lo correct?

 

[nasu]

No. What is the elastic force ("produced" by the spring) when the spring's length is lo?
Hint: at lo the spring is unstretched.

 

[HalfLostAndSearching]

I can guide you in solving this problem by providing you with the necessary steps and equations. First, let's define the variables involved in this problem:

m = mass attached to the spring
k = force constant of the spring
l0 = unstretched length of the spring
l = length of the spring when stretched
ω = angular frequency of revolution

Now, let's look at the forces acting on the mass:

1. Centripetal force (Fc): This is the force that keeps the mass moving in a circular motion and is given by Fc = mω2l.

2. Spring force (FSpring): This is the force exerted by the spring on the mass and is given by FSpring = k(l-l0).

Since the mass is in equilibrium, these two forces must be equal in magnitude and opposite in direction. Therefore, we can set them equal to each other:

Fc = FSpring
mω2l = k(l-l0)

Now, we can solve for l:

l = (mω2l + kl0)/k

This is the length of the spring as a function of ω. We can also simplify this equation by factoring out l:

l = (mω2 + k)l0/k

This equation shows that the length of the spring is directly proportional to the angular frequency ω. As ω increases, the length of the spring also increases. We can also see that the length of the spring is inversely proportional to the force constant k. As k increases, the length of the spring decreases.

I hope this helps you in solving the problem. Remember to always define your variables and use the appropriate equations to find the solution.