Question

An astronaut is being tested in a centrifuge. The centrifuge has a radius of 10.4 m and in starting, rotates according to ϕ = (0.326 rad/s2)t2. When t = 5.60 s, what are the astronaut's

(a) angular speed,
(b) tangential speed,
(c) tangential acceleration
(d) radial acceleration?

Answer 1

3.6512 rad/s

37.97248 m/s

6.7808 m/s²

138.645118976 m/s²

Step-by-step explanation:

r = Radius of centrifuge = 10.4 m

`\phi=0.326t^2`

Differentiating with respect to time

`\omega=(d\phi)/(dt)\\\Rightarrow \omega=(d)/(dt)0.326t^2\\\Rightarrow \omega=0.652t`

At t = 5.6 s

`\omega=0.652* 5.6\\\Rightarrow \omega=3.6512\ rad/s`

Angular speed is 3.6512 rad/s

Tangential speed is given by

`v=r\omega\\\Rightarrow v=10.4* 3.6512\\\Rightarrow v=37.97248\ m/s`

The tangential speed is 37.97248 m/s

Angular acceleration is given by

`\alpha=(d\omega)/(dt)\\\Rightarrow \alpha=0.652\ rad/s^2`

Tangential acceleration is given by

`a_t=\alpha r\\\Rightarrow a_t=0.652* 10.4\\\Rightarrow a_t=6.7808\ m/s^2`

The tangential acceleration is 6.7808 m/s²

Radial acceleration is given by

`a_r=(v^2)/(r)\\\Rightarrow a_r=(37.97248^2)/(10.4)\\\Rightarrow a_r=138.645118976\ m/s^2`

The radial acceleration is 138.645118976 m/s²