Question

An astronaut is being tested in a centrifuge. The centrifuge has a radius of 11.0 m and, in starting, rotates according to θ = 0.260t2, where t is in seconds and θ is in radians. When t = 2.40 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:

a) 1.248 rad/s

b) 13.728 m/s

c) 0.52 rad/s^2

d) 17.132m/s^2

Explanation:

You have that the angles described by a astronaut is given by:

$\theta=0.260t^2$

(a) To find the angular velocity of the astronaut you use the derivative og the angle respect to time:

$\omega=\frac{d\theta}{dt}=\frac{d}{dt}[0.260t^2]=0.52t$

Then, you evaluate for t=2.40 s:

$\omega=0.52(2.40)=1.248\frac{rad}{s}$

(b) The linear velocity is calculated by using the following formula:

$v=\omega r$

r: radius if the trajectory of the astronaut = 11.0m

You replace r and w and obtain:

$v=(1.248\frac{rad}{s})(11.0m)=13.728\frac{m}{s}$

(c) The tangential acceleration is:

$a_T=\alpha r\\\\\alpha=\frac{\omega^2}{2\theta}=\frac{(1.248rad/s)^2}{2(0.260(2.40s)^2)}=0.52\frac{rad}{s^2}$

(d) The radial acceleration is:

$a_r=\frac{v^2}{r}=\frac{(13.728m/s)^2}{11.0m}=17.132\frac{m}{s^2}$