A student sits on a rotating stool holding two 3.09-kg masses. When his arms are extended horizontally, the masses are 1.08 m from the axis of rotation, and he rotates with an angular speed of 0.770 rad/s. The moment of inertia of the student plus stool is 3.25 kg⋅m2 and is assumed to be constant. (Note that this moment of inertia does not include the two 3.09-kg masses.) The student then pulls the masses horizontally to 0.34 m from the rotation axis. (a) Find the new angular speed of the student. Find the kinetic energy of the rotating system (student, stool, and masses) (b) before and (c) after the masses are pulled in. Caution: There are two weights.

a

The New angular speed is  

b

The Kinetic energy before the masses are pulled in is  

c

The Kinetic energy after the masses are pulled in is  

Explanation:

From the we are told that masses are 1.08 m from the axis of rotation, this means that

             The radius

              The  mass is

              The  angular speed

  The moment of inertia of the system excluding the two mass

           New radius  

Generally the conservation of angular momentum can be mathematical represented as

Where is the final angular speed

            is the initial  angular speed

          is the initial moment of inertia

            is the final moment of inertia

Moment of inertia is mathematically represented as

Where I is the moment of inertia

          m is the mass

           r is the radius

So the Initial moment of inertia is given as  

The multiplication by is because we are considering two masses

So the final  moment of inertia is given as  

Substituting these values into equation 1

Generally Kinetic energy is mathematically represented in term of moment of inertia as

Now considering the kinetic energy before the masses are pulled in,

The Moment of inertia would be  

  The Angular speed would be  

Now substituting these value into the equation above

Now considering the kinetic energy after the masses are pulled in,                      

The Moment of inertia would be  

The Angular speed would be  

Now substituting these value into the equation above

the body's situation won't change the forces are equal opposit and on the same body

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