A 139 kg horizontal platform is a uniform disk of radius 1.85 m and can rotate about the vertical axis through its center. A 65.9 kg person stands on the platform at a distance of 1.03 m from the center, and a 27.3 kg dog sits on the platform near the person at 1.43 m from the center. Find the moment of inertia of this system, consisting of the platform and its occupants, with respect to the axis.

Moment of inertia:

To find the moment of inertia of the system consisting of the platform, person, and dog, we need to consider the individual moments of inertia and then sum them up. The moment of inertia of an object depends on its mass and distribution of mass around the axis of rotation.

Given information:

- Mass of the platform (M): 139 kg

- Radius of the platform (R): 1.85 m

- Mass of the person (m1): 65.9 kg

- Distance of the person from the center (r1): 1.03 m

- Mass of the dog (m2): 27.3 kg

- Distance of the dog from the center (r2): 1.43 m

First, let's calculate the moment of inertia of the platform alone. A uniform disk has a known formula for its moment of inertia:

I_platform = (1/2) * M * R^2

I_platform = (1/2) * 139 kg * (1.85 m)^2

I_platform = 301.957 kg·m²

Next, let's calculate the moment of inertia contributed by the person:

I_person = m1 * r1^2

I_person = 65.9 kg * (1.03 m)^2

I_person = 71.351 kg·m²

Similarly, let's calculate the moment of inertia contributed by the dog:

I_dog = m2 * r2^2

I_dog = 27.3 kg * (1.43 m)^2

I_dog = 55.759 kg·m²

Finally, we can find the total moment of inertia of the system by summing up the individual contributions:

Total moment of inertia (I_total) = I_platform + I_person + I_dog

I_total = 301.957 kg·m² + 71.351 kg·m² + 55.759 kg·m²

I_total = 429.067 kg·m²

Therefore, the moment of inertia of the system, consisting of the platform, person, and dog, with respect to the given vertical axis, is approximately 429.067 kg·m².

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