Question

An astronaut is being tested in a centrifuge. The centrifuge has a radius of 12.0 m and, in starting, rotates according to θ = 0.370t2, where t is in seconds and θ is in radians. When t = 5.00 s, what are the magnitudes of the astronaut's

(a) angular velocity,
(b) linear velocity,
(c) tangential acceleration, and
(d) radial acceleration?

Answer 1

part a)

`\omega = 3.7 rad/s`

Part b)

v = 44.4 m/s

Part c)

`a_t = 8.88 m/s^2`

Part d)

`a_r = 164.3 m/s^2`

Step-by-step explanation:

As we know that the angle at any instant of time is given as

`\theta = 0.370 t^2`

part a)

as we know that rate of change in angle with time is angular speed

so here we have

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

`\omega = (d)/(dt)(0.370 t^2)`

`\omega = 0.74 t`

at t= 5s we will have

`\omega = (0.74)(5) = 3.7 rad/s`

Part b)

as we know the relation between linear speed and angular speed is given as

`v = R\omega`

`v = (12.0)(3.7)`

v = 44.4 m/s

Part c)

now in order to find angular acceleration we know that

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

`\alpha = (d(0.74 t))/(dt)`

`\alpha = 0.74 rad/s^2`

now tangential acceleration is given as

`a_t = R\alpha`

`a_t = 12.0(0.74)`

`a_t = 8.88 m/s^2`

Part d)

radial acceleration is given as

`a_r = (v^2)/(R)`

`a_r = (44.4^2)/(12)`

`a_r = 164.3 m/s^2`