Question

In a Beams ultracentrifuge, the rotor is suspended magnetically in a vacuum. Since there is no mechanical connection to the rotor, the only friction is the air resistance due to the few air molecules in the vacuum. If the rotor is spinning with an angular speed of 5.00 × 10⁵ rad/s and the driving force is turned off, its spinning slows down at an angular rate of magnitude 0.400 rad/s². The rotor of the Beams ultracentrifuge is a rod 26.4 cm long, turning about a perpendicular axis through its center.

For a point at the end of the rotor, find the initial speed.

For a point at the end of the rotor, find the magnitude of the tangential acceleration component.

For a point at the end of the rotor, find the maximum radial acceleration component.

Answer 1

w_o = initial angular speed = 5 x 10⁵ rad/s

r = radius = length of the rod = 26.4 cm = 0.264 m

v_o = initial linear speed

initial linear speed is given as

v_o = r w_o

v_o = (0.264) (5 x 10⁵ )

v_o = 1.32 x 10⁵ m/s

= angular acceleration = 0.4 rad/s²

a = linear acceleration

Linear acceleration is given as

a = r

a = (0.264) (0.4)

a = 0.1056 m/s²

radial acceleration is given as

a_r = v_o²/r

a_r = (1.32 x 10⁵ )²/(0.264)

a_r = 6.6 x 10¹⁰ m/s²