An astronaut is being tested in a centrifuge. The centrifuge has a radius of 5.20 m and, in starting, rotates according to θ = 0.170t2, where t is in seconds and θ is in radians. When t = 4.70 s, what are the magnitudes of the astronaut's (a) angular velocity, (b) linear velocity, (c) tangential acceleration, and (d) radial acceleration?

A) angular velocity; ω = 1.598 rad/s

B) linear velocity; V = 8.31 m/s

C) Tangential Acceleration; a_t = 1.768 m/s²

D) Radial Acceleration; a_r = 13.28 m/s²

Explanation:

We are given that;

Radius; r = 5.2m

Time; t = 4.7 sec

θ = 0.170t²

Thus, angular acceleration would be the second derivative of θ which is d²θ/dt²

Thus,α = d²θ/dt² = 0.34 rad/s²

A) Formula for angular velocity is;

ω = αt

Where α is angular acceleration and t is time.

Thus; ω = 0.34 x 4.7

ω = 1.598 rad/s

b) formula for linear velocity is given by; V = ωr

We have ω = 1.598 rad/s and r = 5.2m

Thus; V = 1.598 x 5.2

V = 8.31 m/s

c) formula for tangential acceleration is;

a_t = αr

a_t = 0.34 x 5.2

a_t = 1.768 m/s²

d) formula for radial acceleration is;

a_r = rω²

a_r = 5.2 x 1.598²

a_r = 13.28 m/s²