Angular Frequency of a two-object system

[hidemi]

Homework Statement

A 2.00 m-long 6.00 kg ladder pivoted at the top hangs down from a platform at the circus. A 42.0 kg trapeze artist climbs to a point where her center of mass is at the center of the ladder and swings at the system's natural frequency. The angular frequency (in s −1) of the system of ladder and woman is
A) 1.01
B) 3.07
C) 4.03
D) 8.05
E) 16.2

The answer is B.

Relevant Equations

w= √(mgd/I)

I calculate as follow and get a correct answer, but I wonder why the weight of the ladder 6 kg is not included in the mass (m) in the numerator.

w= √(mgd/I)
= √ { （42*10*1)/ [(1/12)(6)(2^2)+42*1] }

= √ (420/44)​

= 3.06

 

[Doc Al]

but I wonder why the weight of the ladder 6 kg is not included in the mass (m) in the numerator

It should be.

 

[hidemi]

It should be.

If the 6kg is included into the numerator, the answer wouldn't be correct.
How did you calculate to get the given answer? Is there something wrong in my original calculation?
Thanks!

 

[Doc Al]

Is there something wrong in my original calculation?

Your original calculation -- as you point out yourself -- uses incorrect physics. (The mass of the ladder should be included.) So that cannot be the answer unless the problem is mistaken (which does happen).

But there's another error in your calculation, besides the missing mass in the numerator.

 

[TSny]

Is there something wrong in my original calculation?

What is the moment of inertia of the ladder if it’s rotating about one end?

 

[hidemi]

What is the moment of inertia of the ladder if it’s rotating about one end?

I = √（mgd/I）= √[(42+6)*9.8*1 / (1/3*6*2^2 + 42*1^2)] = 3.067
Is this correct? Should the 'd' be half of the ladder length for both the ladder and woman?

 

[Doc Al]

Is this correct?

Yes, looks good now.

Should the 'd' be half of the ladder length for both the ladder and woman?

That 'd' is the distance from the pivot point to the center of mass of the entire system. (In this case, the center of mass of the system is at the midpoint of the ladder, but only because the woman happens to be at that point. What if she were at some other point?)