Two astronauts, each with a mass of 50 kg, are connected by a 7 m massless rope. Initially they are rotating around their center of mass with an angular velocity of 0.5 rad/s. One of the astronauts then pulls on the rope shortening the distance between the two astronauts to 4 m. What is the change in the rotational kinetic energy (in J) of this system? You may model each astronaut as a point particle.

Answer :

xero099

Answer:

[tex]\Delta K = 315.667\,J[/tex]

Explanation:

The final angular speed of the system is given by the help of the Principle of the Angular Momentum Conservation:

[tex][2\cdot (50\,kg)\cdot (3.5\,m)^{2}] \cdot (0.5\,\frac{rad}{s} ) = [2\cdot (50\,kg)\cdot (2\,m)^{2}] \cdot \omega[/tex]

The final angular speed is:

[tex]\omega = 1.531\,\frac{rad}{s}[/tex]

Finally, the change in the rotational kinetic energy is:

[tex]\Delta K = \frac{1}{2}\cdot \left[2\cdot (50\,kg)\cdot (2\,m)^{2}\cdot \left(1.531\,\frac{rad}{s} \right)^{2}-2\cdot (50\,kg)\cdot (3.5\,m)^{2}\cdot \left(0.5\,\frac{rad}{s} \right)^{2}\right][/tex]

[tex]\Delta K = 315.667\,J[/tex]

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