a. The magnitude of the angular momentum of the system is [tex]3750 \;Kgm^2/s[/tex]
b. The rotational energy of the system is 1875 Joules.
c. The new angular momentum of the system is [tex]3750 \;Kgm^2/s[/tex]
d. To calculate the astronauts’ new speeds is [tex]10 \;m/s^2[/tex]
e. The new rotational energy of the system is 7500 joules.
f. The amount of work the astronaut did in shortening the rope is 5625 Joules.
Given the following data:
a. To calculate the magnitude of the angular momentum of the system:
First of all, we would determine the distance of each astronaut from the center of mass of the rope.
[tex]D_c = \frac{1}{2} \times distance\\\\D_c = \frac{1}{2} \times 10\\\\D_c = 5\;meter[/tex]
Mathematically, angular momentum of the system is given by:
[tex]W_m = 2mvD_c\\\\W_m = 2 \times 75 \times 5 \times 5\\\\W_m = 3750 \;Kgm^2/s[/tex]
b. To calculate the rotational energy of the system:
[tex]R_E = 2 \times \frac{1}{2} MV^2\\\\R_E = 2 \times \frac{1}{2} \times 75 \times 5^2\\\\R_E = 75 \times 25\\\\R_E = 1875 \;Joules[/tex]
c. To calculate the new angular momentum of the system:
Based on the law of conservation of momentum, the initial angular momentum is equal to the final or new angular momentum.
New angular momentum = [tex]3750 \;Kgm^2/s[/tex]
d. To calculate the astronauts’ new speeds:
Note: By pulling on the rope, one of the astronauts shortens the distance between them to 5.00 m.
[tex]D_n = \frac{1}{2} \times distance\\\\D_n = \frac{1}{2} \times 5\\\\D_n = 2.5\;meter[/tex]
From the new angular momentum:
[tex]W_n = 2mV_2D_n[/tex]
[tex]3750 = 2 \times 75 \times V_2 \times 2.5\\\\3750 = 375V_2\\\\V_2 = \frac{3750}{375} \\\\V_2 = 10 \; m/s^2[/tex]
e. To calculate the new rotational energy of the system:
[tex]R_N = 2 \times \frac{1}{2} MV_2^2\\\\R_N = 2 \times \frac{1}{2} \times 75 \times 10^2\\\\R_N = 75 \times 100\\\\R_N = 7500 \;Joules[/tex]
f. To find how much work the astronaut did in shortening the rope:
[tex]Work\;done = R_N - R_E\\\\Work\;done = 7500 - 1875[/tex]
Work done = 5625 Joules
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