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

Answer:

3.6512 rad/s

37.97248 m/s

6.7808 m/s²

138.645118976 m/s²

Explanation:

r = Radius of centrifuge = 10.4 m

$\phi=0.326t^2$

Differentiating with respect to time

$\omega=\dfrac{d\phi}{dt}\\\Rightarrow \omega=\dfrac{d}{dt}0.326t^2\\\Rightarrow \omega=0.652t$

At t = 5.6 s

$\omega=0.652\times 5.6\\\Rightarrow \omega=3.6512\ rad/s$

Angular speed is 3.6512 rad/s

Tangential speed is given by

$v=r\omega\\\Rightarrow v=10.4\times 3.6512\\\Rightarrow v=37.97248\ m/s$

The tangential speed is 37.97248 m/s

Angular acceleration is given by

$\alpha=\dfrac{d\omega}{dt}\\\Rightarrow \alpha=0.652\ rad/s^2$

Tangential acceleration is given by

$a_t=\alpha r\\\Rightarrow a_t=0.652\times 10.4\\\Rightarrow a_t=6.7808\ m/s^2$

The tangential acceleration is 6.7808 m/s²

Radial acceleration is given by

$a_r=\dfrac{v^2}{r}\\\Rightarrow a_r=\dfrac{37.97248^2}{10.4}\\\Rightarrow a_r=138.645118976\ m/s^2$

The radial acceleration is 138.645118976 m/s²