a) The angular velocity can be calculated using the equation v = √(a/r), where a is the centripetal acceleration (10 gs) and r is the radius (10 m). This gives an angular velocity of 31.62 rad/s.
b) The angle θ can be calculated using the equation θ = tan-1(Fc/Fg), where Fc is the centripetal force and Fg is the force of gravity. The centripetal force can be calculated using the equation Fc = mv2/r, where m is the mass of the cage, v is the angular velocity (31.62 rad/s), and r is the radius (10 m). The force of gravity can be calculated using the equation Fg = mg, where m is the mass of the cage and g is the acceleration due to gravity (9.8 m/s2). This gives an angle θ of 45.2° below the horizontal.
A large centrifuge is used to expose aspiring astronauts to accelerations similar to those experienced in rocket launches and atmospheric reentries.
a) At what angular velocity is the centripetal acceleration 10 gs if the rider is 10.0 m from the center of rotation?
(b) The rider's cage hangs on a pivot at the end of the arm, allowing it to swing outward during rotation as shown in the figure. At what angle θ below the horizontal will the cage hang when the centripetal acceleration is 10 gs? (Hint: The arm supplies centripetal force and supports the weight of the cage. Draw a free-body diagram of the forces to see what the angle θ should be.)
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