A student sits on a rotating stool holding two 2.1-kg objects. When his arms are

Question

A student sits on a rotating stool holding two 2.1-kg objects. When his arms are extended horizontally, the objects are 1.0 m from the axis of rotation and he rotates with an angular speed of 0.75 rad/s. The moment of inertia of the student plus stool is 3.0 kg · m2 and is assumed to be constant. The student then pulls in the objects horizontally to 0.21 m from the rotation axis. (a) Find the new angular speed of the student. rad/s (b) Find the kinetic energy of the student before and after the objects are pulled in. before J after J

Explanation / Answer

Given

   mass of objects m1=m2 = 2.1 kg, r1 = 1m (stretched condition)

   pulled in condition , r2 = 0.21 m

moment of inertia is constat I = 3.0 kg m2

conservation of momentum

   I1W1 = I2W2

I1 = 3+ m1r1^2+m2r1^2 , given m1=m2 ==> I1 = 3+2mr1^2 = 3+2*2.1*1^2 kg m2 = 7.2 kg m2

and I2 = 3+ 2m*r2^2 = 3 + 2*2.1*0.21^2 kg m2 = 3.18522 kg m2

  
   I1W1 = I2W2

   W2 = I1W1/I2
   W2 = 7.2*0.75 / 3.18522 rad/s

   w2 = 1.695330 rad/s

a) new angular speed of the student is W2 = 1.695330 rad/s

b) kinetic energy of the student before the objects pulled in is

   k.e = 0.5*I1*W1^2 = 0.5*7.2*0.75^2J = 2.025 J

after pulled in

   k.e = 0.5*I2*w2^2 = 0.5*3.18522*1.695330^2 J = 4.57739 J

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