Forced Oscillator where Damping is Negligible

[Becca93]

Homework Statement

Damping is negligible for a 0.139 kg mass hanging from a light 7.00 N/m spring. The system is driven by a force oscillating with an amplitude of 1.88 N. At what frequency will the force make the mass vibrate with an amplitude of 0.430 m? There are two possible solutions, enter one of them.

Homework Equations

A = (Fo/m) / (√((ω^2 - ωo^2)^2 + (bω/m)^2)
Damping is negligible, therefore b = 0, therefore
A = (Fo/m) / (√((ω^2 - ωo^2)^2)

ωo = √(k/m)

The Attempt at a Solution

So,
m = 0.139 kg
k = 7.00 N/m
Fo = 1.88 N
A = 0.430 m

A = (Fo/m) / (√((ω^2 - ωo^2)^2)
Rearranged to find ω, is

(ω^2 - √(k/m)^2)^2 = (Fo/m) / A
ω^2 = √((Fo/m)/A) + (k/m)
ω = √( √((Fo/m)/A) + (k/m) )

So,
ω = √( √((1.88/0.139)/0.430) + (7/0.139) )
ω = √( 1.77 + 50.3597)
ω = 7.22 rad/s

ω = 2(pi)f
7.22 /2(pi) = f
f = 1.49 Hz

This is not the correct answer and I have no idea where I'm going wrong.

Am I using the wrong equations? Are my calculations incorrect? Any assistance would be much appreciated.

 

[vela]

You simplified $\sqrt{(\omega^2-\omega_0^2)^2}$ incorrectly.

 

[Becca93]

You simplified $\sqrt{(\omega^2-\omega_0^2)^2}$ incorrectly.

How? I did the following:

√((ω^2 - ωo^2)^2)

Shouldn't the square root and the square cancel, leaving

(ω^2 - ωo^2)

ω^2 - (√k/m)^2

ω^2 - (k/m)

Resulting in:

ω^2 = √((Fo/m)/A) + (k/m)

Where did I go wrong?

 

[vela]

After setting b=0 and moving some stuff around, you should have
$$A = \frac{F_0/m}{\sqrt{(\omega^2-\omega_0^2)^2}} = \frac{F_0/m}{\omega^2-\omega_0^2}$$My take was that you ended up with
$$A = \frac{F_0/m}{\sqrt{(\omega^2-\omega_0^2)^2}} = \frac{F_0/m}{(\omega^2-\omega_0^2)^2}$$ (compare to "(ω^2 - √(k/m)^2)^2 = (Fo/m) / A" which is what you wrote in your first post) which leads to the incorrect answer.

If you check the units on your result, you'll see they don't work out. That means you messed up the algebra somewhere.

 

[Becca93]

After setting b=0 and moving some stuff around, you should have
$$A = \frac{F_0/m}{\sqrt{(\omega^2-\omega_0^2)^2}} = \frac{F_0/m}{\omega^2-\omega_0^2}$$My take was that you ended up with
$$A = \frac{F_0/m}{\sqrt{(\omega^2-\omega_0^2)^2}} = \frac{F_0/m}{(\omega^2-\omega_0^2)^2}$$ (compare to "(ω^2 - √(k/m)^2)^2 = (Fo/m) / A" which is what you wrote in your first post) which leads to the incorrect answer.

If you check the units on your result, you'll see they don't work out. That means you messed up the algebra somewhere.

Ahh, yes, I see what I did. I did mess up the algebra without noticing. I had it straight in my head but not on paper.

Thank you very much for pointing that out for me. I've got the correct answer now.