1-) A 210g , 43.0-cm-diameter turntable rotates on frictionless bearings at 68.0

Question

1-) A 210g , 43.0-cm-diameter turntable rotates on frictionless bearings at 68.0rpm . A 24.0g block sits at the center of the turntable. A compressed spring shoots the block radically outward along a frictionless groove in the surface of the turntable. What is the turntable's rotation angular velocity when the block reaches the outer edge?

2-) A 45kg figure skater is spinning on the toes of her skates at 0.50rev/s . Her arms are outstretched as far as they will go. In this orientation, the skater can be modeled as a cylindrical torso (40kg , 20 cm average diameter, 160 cm tall) plus two rod-like arms (2.5 kg each, 71cm long) attached to the outside of the torso. The skater then raises her arms straight above her head, where she appears to be a 45 kg, 20-cm-diameter, 200-cm-tall cylinder. What is her new rotation frequency, in revolutions per second?

Explanation / Answer

1.

W1=68 rpm =68*2pi =427.26 rev/s

By Conservation of angular momentum

I1*W1 =I2*W2

[(1/2)*M*R2]*W1=[(1/2)*M*R2+mR2]*W2

=>(1/2)*0.21*68 =[(1/2)*0.21+0.024]*W2

W2=55.35 rpm

b)

R=D/2 =0.1 m

Initial Moment of inertia

I1=(1/2)*M*R2 +2*[(1/12)*m*L2+m*(R+L/2)^2]

I1=(1/2)*40*0.12+2[(1/12)*2.5*0.712 +2.5*(0.1+0.71/2)2]

I1=1.445 Kg-m2

Final Moment of inertai

I2=(1/2)*M*R2

I2=(1/2)*45*0.1^2 =0.225 Kg-m^2

By conservation of angular momentum

I1*W1=I2*W2

1.445*0.5=0.225*W2

W2=3.2 rev/s

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