Finding a change in angular velocity, with out I equation

[Shepherd7]

Homework Statement

A playground carousel is rotating about its center. The radius of the carousel is 1.5m, its initial angular speed is 3.14 rad/s and its moment of inertia is 125 kg m^2. A 40kg person climbs onto the carousel near its outer edge. What is the angular speed of the carousel after the person climbs aboard?

Homework Equations

Krot = 1/2 I W^2
L=IW
I = intergral r^2 dm
I = Icm + Md^2
Here's the kicker we can't use the equations for the moment of inertia about a rotating disk. Can't use I = 1/2MR^2

The Attempt at a Solution

I'm just lost I've tried solving this problem multiple ways with different equations.
This is my last attempt

Li=Lf
IiWi = IfWf
IiWi/Wf = If

1/2 Ii Wi^2 = 1/2 If Wf^2

1/2 Ii Wi^2 = 1/2 IiWi/Wf * Wf^2

Ii Wi^2 / Ii Wi = Wf
(125 * 3.14^2) / (125*3.14) = Wf = 3.14

Am I any where close!

 

[Troels]

Am I any where close!

It seems that you are trying to apply conservation of angular momentum as well as conservation of rotational energy. you could do both, but not at the same time, as they are redundant; the former is simpler:

[tex]L_\mathrm{before}=L_\mathrm{after}=L[/tex]

so

[tex]I_\mathrm{before}\omega_\mathrm{before}=I_\mathrm{after}\omega_\mathrm{after}[/tex]

as you stated, yourself. To complete the argument however, you need to know the moment of inertial after the person gets on. Fortunately, that is a simple matter if you assume the person to be a point mass:

[tex]I_\mathrm{after}=I_\mathrm{before}+m_\mathrm{person}r^2[/tex]

putting is together:

[tex]I_\mathrm{before}\omega_\mathrm{before}=\left(I_\mathrm{before}+m_\mathrm{person}r^2\right)\omega_\mathrm{after}[/tex]

or

[tex]\omega_\mathrm{after}=\frac{I_\mathrm{before}}{I_\mathrm{before}+m_\mathrm{person}r^2}\omega_\mathrm{before}[/tex]

As this qoutient is less than one, the angular velocity will decrease to conserve the angular momentum.

 

[kerimek]

I would like to first address the statement that equations for the moment of inertia of a rotating disk cannot be used. This is incorrect, as the moment of inertia for a rotating disk is given by I = 1/2MR^2. However, I understand that this may be a restriction given by the homework assignment, so I will provide a solution without using this equation.

To solve this problem, we can use the conservation of angular momentum. This states that the initial angular momentum of the system is equal to the final angular momentum of the system. In this case, the initial angular momentum of the carousel is given by IiWi, where Ii is the moment of inertia and Wi is the initial angular speed. The final angular momentum of the system will be equal to the sum of the angular momentum of the carousel and the person, which can be expressed as IfWf + mpRp, where If is the final moment of inertia, Wf is the final angular speed, mp is the mass of the person, and Rp is the distance from the center of rotation to the person.

Setting these two equal, we can solve for the final angular speed:

IiWi = IfWf + mpRp

Solving for Wf:

Wf = (IiWi - mpRp) / If

Plugging in the given values, we get:

Wf = (125*3.14*3.14 - 40*1.5) / (125 + 40*1.5^2)

Wf = 2.44 rad/s

Therefore, the angular speed of the carousel after the person climbs aboard is 2.44 rad/s.