A gymnast was practicing an airborne movement and you filmed it in two dimension

Question

A gymnast was practicing an airborne movement and you filmed it in two dimensions. the mass of the lower body of the gymnast is 15kg, the mass of the trunk and arms is 20kg, and the mass of the head is 8kg. the displacement vectors from the whole body center of mass to the lower body, upper body, and head centers of mass are (-60 , -40) cm, (30,50)cm and (100,100)cm respectively. the moments of inertia about the centers of the mass of each segment are 2000 kg cm^2 for the lower body, 2000 kg cm^2 for the upper body and arms and 200 kg cm^2 for the head.

a) calculate the moment of inertia of the whole body about the center of mass of the whole body?

b) During the airborne movement, the gymnast's coach produced an external torque to the gymnast's body so that the gymnast could accelerate the rotation. the angular acceleration of the whole body was measured using motion analysis and it was -2rad/s/s. what is the torque produced by the coach?

am thinking of these two formulars ( T = F. D    and T = I ) but i dont know how to use them please help

thank you for any help

Explanation / Answer

a) First we need to use parallel axis theorem for each part of the body.

lower body: I=2000+md2=2000+15*(602+402)=80000

upper body: I=2000+20*(302+502)=70000

head: I=200+8*(1002+1002)=160200

So the total moment of inertia is 310200 kg cm2=31.02 kg m2

b) Now we use =I we were told that =-2 rad/s2

So =-61.04 kg m2/s2

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