Question

An astronaut is being tested in a centrifuge. The centrifuge has a radius of 9 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?

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

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

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

Answer 1

5.4 rad/s

48.6 m/s

5.4 m/s²

262.44 m/s²

Step-by-step explanation:

r = Radius of centrifuge = 9 m

`\theta=0.3t^2`

Differentiating with respect to time

`(d\theta)/(dt)=0.6t=\omega`

At t = 9 s

`\omega=0.6* 9=5.4\ rad/s`

Angular velocity is 5.4 rad/s

Linear speed

`v=r\omega\\\Rightarrow v=9* 5.4=48.6\ m/s`

The linear speed of the astronaut is 48.6 m/s

Differentiating
`\omega` with respect to time

`(d\omega)/(dt)=0.6=\alpha`

Tangential acceleration is given by

`a_t=\alpha r\\\Rightarrow a_t=0.6* 9=5.4\ m/s^2`

Tangential acceleration of the astronaut is 5.4 m/s²

Radial acceleration is given by

`a_r=(v^2)/(r)\\\Rightarrow a_r=(48.6^2)/(9)\\\Rightarrow a_r=262.44\ m/s^2`

The radial acceleration of the astronaut is 262.44 m/s²