Angular momentum of merry go round turntable

[MAins]

Hi, I got this question wrong on a test and am really quite lost about it. I'd appreciate somebody running me through it.

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"Suppose a 55 kg person stands at the edge of a 6.5 m diameter merry go round turntable that is mounted on a frictionless bearing and has a moment of inertia of 1700 kg m^2. The turntable is at rest initially, but when the person begins running at a speed of 3.8 m/s (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Caculate the angular velocity of the table."

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So I set up conservation of angular momentum
L_f = L_i
and
(Iw)_f = (Iw)_i

Can somebody please help me from there?

 

[JaWiB]

I guess you'd treat the person as a point mass, so the person has moment of inertia mr^2. Can you calculate his angular velocity?

 

[Moetasim]

Hi, it seems like you are on the right track with using conservation of angular momentum and the equation (Iw)_f = (Iw)_i. Let's break down the problem step by step to find the solution.

First, let's define our variables:
m = mass of person (55 kg)
r = radius of merry go round (6.5 m)
I = moment of inertia of the turntable (1700 kg m^2)
w = angular velocity of the turntable (unknown)
v = linear velocity of the person (3.8 m/s)

Next, let's consider the initial and final states of the system. Initially, the turntable is at rest and the person is running at a speed of 3.8 m/s around its edge. This means that the person has a linear velocity of 3.8 m/s with respect to the turntable, but they are also rotating with the turntable at the same angular velocity. So we can say that the initial angular velocity of the turntable is also 3.8 m/s.

Now, let's use the equation (Iw)_f = (Iw)_i to find the final angular velocity (w) of the turntable. Plugging in our values, we get:
(1700 kg m^2)(w) = (55 kg)(6.5 m)(3.8 m/s)
Solving for w, we get:
w = (55 kg)(6.5 m)(3.8 m/s) / (1700 kg m^2)
w = 0.75 rad/s

Therefore, the angular velocity of the turntable is 0.75 rad/s in the opposite direction of the person's motion. This means that the turntable is rotating at a slower speed than the person is running, but in the opposite direction.

I hope this explanation helps you understand the problem better. Remember to always define your variables and consider the initial and final states of the system when solving problems involving conservation of angular momentum. Best of luck in your studies!