An astronaut is being tested in a centrifuge. The centrifuge has a radius of 7 m

Question

An astronaut is being tested in a centrifuge. The centrifuge has a radius of 7 m and, in starting, rotates according to ? = 0.30t2, where t is in seconds and ? is in radians.
(a) When t = 9.0 s, what is the magnitude of the astronaut's angular velocity?
rad/s

(b) When t = 9.0 s, what is the magnitude of the astronaut's linear speed?
m/s

(c) When t = 9.0 s, what is the magnitude of the astronaut's tangential acceleration?
m/s2

(d) When t = 9.0 s, what is the magnitude of the astronaut's radial acceleration?
m/s2

Explanation / Answer

Given

theta = 0.30 t^2

Radius of the centrifuge is , r = 10 m

a) The angular velocity , = d theta/ dt = d(0.3 t^2) / dt

     = 0.6 t

   when t = 9 s,

= 0.6 *9 s =5.4 rad/s

b)when t=9.0 s

linear speed w=d theta/dt

w=d(0.30 t^2)/dt

=0.6t=0.6*9=5.4 rad/s

v=wr

=5.4 rad/s*12m=64.8 m/s

c)

when t=9.0 s

astronaut's tangential acceleration a_t=alpha*r

=dw/dt*12

=12*d(0.6t/dt)

   =12*0.6=7.2 m/s^2

d)

astronaut's radial acceleration

a_r=w^2*r

=(5.4 rad/s)^2*12=349.92 m/s^2

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