Question

An astronaut is being tested in a centrifuge. The centrifuge has a radius of 7.40 m and, in starting, rotates according to θ = 0.190t², where t is in seconds and θ is in radians. When t = 3.60 s, what are the magnitudes of the astronaut's (a) angular velocity, (b) linear velocity, (c) tangential acceleration, and (d) radial acceleration?

Answer 1

Angular displacement of the astronaut is given by,

$\theta$(t) = 0.19t²

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

If we differentiate the equation of angular displacement with respect to time we will get the equation for angular velocity,

$\omega$(t) = 0.38t

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

If we differentiate the equation of angular velocity with respect to time we will get the equation for angular acceleration,

$\alpha$(t) = 0.38

Radius of the centrifuge = R = 7.4 m

Angular velocity at t=3.6 sec = $\omega$₁ = (0.38)(3.6) = 1.368 rad/s

Angular acceleration at t=3.6 sec = $\alpha$₁ = 0.38 rad/s²

Linear velocity of the astronaut at t=3.6 sec = V₁

V₁ = $\omega$₁R

V₁ = (1.368)(7.4)

V₁ = 10.12 m/s

Tangential acceleration of the astronaut at t=3.6 sec = a_t1

a_t1 = $\alpha$₁R

a_t1 = (0.38)(7.4)

a_t1 = 2.812 m/s²

Radial acceleration of the astronaut at t=3.6 sec = a_r1

a_r1 = $\omega$₁²R

a_r1 = (1.368)²(7.4)

a_r1 = 13.85 m/s²

a) Magnitude of astronaut's angular velocity at t=3.6 sec = 1.368 rad/s

b) Magnitude of astronaut's linear velocity at t=3.6 sec = 10.12 m/s

c) Magnitude of astronaut's tangential acceleration at t=3.6 sec = 2.812 m/s²

d) Magnitude of astronaut's radial acceleration at t=3.6 sec = 13.85 m/s²