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

Answer:

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)

$a_r = 164.3 m/s^2$

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

$\omega = \frac{d\theta}{dt}$

$\omega = \frac{d}{dt}(0.370 t^2)$

$\omega = 0.74 t$

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

$\alpha = \frac{d\omega}{dt}$

$\alpha = \frac{d(0.74 t)}{dt}$

$\alpha = 0.74 rad/s^2$

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

$a_r = \frac{v^2}{R}$

$a_r = \frac{44.4^2}{12}$

$a_r = 164.3 m/s^2$