Question

Luc, who is 1.80 m tall and weighs 950 N, is standing at the center of a playground merry-go-round with his arms extended, holding a 4.0 kg dumbbell in each hand. The merry-go-round can be modeled as a 4.0-m-diameter disk with a weight of 1500 N. Luc's body can be modeled as a uniform 40-cm-diameter cylinder with massless arms extending to hands that are 85 cm from his center. The merry-go-round is coasting at a steady 35 rpm when Luc brings his hands in to his chest. Afterward, what is the angular velocity, in rpm, of the merry-go-round?

Answer 1

given

Luc weight is W = 950 N ==> m = 950/10 = 95 kg

mass of dumbell md = 4 kg , mass of merry go round is m_mgr = 1500 /10 = 150 kg  

radius of merry go roung r_mgr = 2 m, luc body modeled as cylinder with radius isr_l = 20 cm =0.2 m

diameter of the cylinder is d_l = 40 cm radius r_l = 20 cm

when arms extended the distance from the axis of rotation isl_arms 0.85 m

initial angular velocity is W1 = 35 rpm

we know that the moent of inertia of merro go round is I_mgr = m*r^2 /2

By conservation of angular momentum I1*W1 = I2*W2

W2 = I1*W1 /I2

here I1 = (1/2)m_mgr*r_mgr^2 + m*r_l^2 / 2 + 2*md*L_arms^2

and I2 = (1/2)m_mgr*r_mgr^2 + m*r_l^2 / 2 + 2*md*r_l^2

substituting the values  

with stretched arms

I1 = (0.5)(150*9.8*2^2+0.5*95*0.2^2+0.5*2*4*0.85^2 kg m^2 = 2942.395 kg m^2

after pulled in

I2 = (0.5)(150*9.8*2^2+0.5*95*0.2^2+0.5*2*4*0.2^2 kg m^2 = 2941.03 kg m^2

I1*W1 = I2*W2

W2 = I1*W1 /I2

W2 = (2942.395)(35*2pi/60)/(2941.03 ) rad/s

W2 = 3.6668925292452 rad/s

w2 = 3.6668925292452*60/2pi

W2 = 35.016244 rpm

so the answer is 35.016244 rpm