How Does Catching a Ball Affect the Angular Speed of a Merry-Go-Round?

[bewger]

Homework Statement

In the figure below, a 30 kg child stands on the edge of a stationary merry-go-round of mass 100 kg and radius 2.0 m. The rotational inertia of the merry-go-round about its rotation axis is 150 kg·m2. The child catches a ball of mass 1.1 kg thrown by a friend. Just before the ball is caught, it has a horizontal velocity v of magnitude 11 m/s, at angle ϕ = 37° with a line tangent to the outer edge of the merry-go-round, as shown. What is the angular speed of the merry-go-round just after the ball is caught?

Homework Equations

p = mv
L = Iw

The Attempt at a Solution

So I'm assuming that angular momentum is conserved. Thus,

mv = Iw
I found the total I.
I_merrygoaround = 150
I_child = MR^2 = 120
I_ball = MR^2 = 1.1(rsin(phi))^2 = 1.44
I_total = 271.44

$$\omega$$ = (mv) / (I_total) = 12.1/271.44 = 0.0445 rad/s

But this is incorrect.
What did I do wrong?

 

[tiny-tim]

hi bewger!

you seem to be mixing up the angular momentum of the ball before with the angular momentum of the ball after

 

[bewger]

Hi tiny-tim

I'm trying to figure out my mistake with your advice, but I'm a little stumped by what you mean by that.

Can you elaborate on that?

 

[tiny-tim]

hi bewger!

I'm finding it a little difficult to read what you posted (try using the X² tag just above the Reply box ), but I don't think you've included both the angular momentum of the ball before and the different angular momentum of the ball after

 

[rwisz]

I can offer the following response to your attempt at a solution:

Your approach to using conservation of angular momentum is correct, however, there are a few errors in your calculations. Firstly, the moment of inertia for the child and the ball should be calculated about the axis of rotation for the merry-go-round, not their own individual axes. This means that the moment of inertia for the child should be calculated as MR^2, but for the ball, it should be MR^2 + md^2, where d is the distance from the axis of rotation to the center of mass of the ball. This will give you a total moment of inertia of 271.44 kg·m^2, which is correct.

Secondly, when calculating the angular velocity, you should use the total momentum of the system, which is the sum of the individual momenta of the child, ball, and merry-go-round. This will give you a value of 0.0406 rad/s, which is closer to the correct answer.

Finally, it is important to note that the conservation of angular momentum only applies when there are no external torques acting on the system. In this case, there may be some frictional forces acting on the merry-go-round, which will cause it to slow down over time. Therefore, the angular velocity calculated using conservation of angular momentum will only be valid immediately after the ball is caught.

Overall, your approach was correct, but there were some errors in your calculations and assumptions. It is important to carefully consider all the factors and assumptions when solving a problem in physics. Keep up the good work!