Spring mass pulley system frequency equation

[Name15]

Homework Statement

A spring is attached to the ceiling, a pulley is attached to the other end of the spring, a string is suspended over the pulley, and a mass attached to the free end of this string. The system oscillates vertically.

Can someone PLEASE help me derive this equation for natural frequency:

f = (1/2pi) * SQRT(k / (m + m(s)/3))

where f = frequency
k= spring constant
m = mass attached to a spring
m(s) = mass of spring

Homework Equations

F=ma=-kx
mx''+kx=0
(Wn)^2 =k/m

The Attempt at a Solution

mx''=-kx
x''=-wn^2.x

 

[Chestermiller]

Can you provide a diagram. Your description doesn't make sense if the string is not somehow anchored on the other end.

Chet

 

[Name15]

sorry, i should have mentioned, the string is attached to say a table.

 

[Chestermiller]

Are you supposed to be considering the case where the spring has mass, or are you allowed to assume that the spring is massless? Is the pulley supposed to have mass?

I suggest that you solve this problem first for the case of a massless spring and pulley. Why? Because if you can't solve it for that case, then you'll never be able to do it for more complicated cases. Plus, when you complete that solution, you will have something to show for it, and will have developed the general technique for attacking the problem.

Have you considered drawing free body diagrams on the spring and mass, and identifying the forces acting on each? Have you written down force balance equations for the spring and mass?

Chet

 

[Name15]

i have tried many different approaches, from equation of motion to energy conservation methods, only to have been unsuccessful.

The spring and pulley masses are considered in the above. The string is massless.

 

[Chestermiller]

i have tried many different approaches, from equation of motion to energy conservation methods, only to have been unsuccessful.

The spring and pulley masses are considered in the above. The string is massless.

Well, you have to go back to basics. Can you at least determine the tensions in the string and spring if the system is in static equilibrium, without any oscillation? That would be your starting point for the dynamic analysis.

Chet

 

[Name15]

for system without mass: 2T=kx
when mass is isolated: T=mg

 

[Name15]

<< Mentor Note -- 2 threads merged >>

In an experiment to determine the natural frequency of a spring-mass-pulley system, why would the experimental natural frequency (found using 1/time) be greater than the theoretical natural frequency found using the following:

f=1/2pi∗sqrt(k/4m+ms/3)Anything other than the pulley mass and rope mass being neglected.

Also, should the experimental f be greater?