An ultracentrifuge accelerates from rest to 100,000 rpm in 1.20 min. Solve c using the equation a=(v^2)/r
(a) What is its angular acceleration in rad/s^2?
(b) What is the tangential acceleration, in m/s^2, of a point 13.30 cm from the axis of rotation?
(c) what is the velocity?
(d) Express this centripetal acceleration as a multiple of g.
(a) First, convert the final angular velocity from rpm to rad/s:
100,000 rpm = 100,000 revolutions/minute * 2π radians/revolution * 1 minute/60 seconds = 10^5 * 2π / 60 rad/s ≈ 10472.39 rad/s
Now, use the formula for angular acceleration:
a = (v^2)/r
Angular acceleration = (10472.39 rad/s) / (1.20 minutes * 60 seconds/minute) = 1456.02 rad/s^2
(b) Now, calculate the tangential acceleration at a point 13.30 cm from the axis of rotation:
r = 13.30 cm = 0.1330 m
Tangential acceleration = a_t = r * α = 0.1330 m * 1456.02 rad/s^2 ≈ 193.90 m/s^2
(c) To find the velocity, use the formula v = r * ω:
v = 0.1330 m * 10472.39 rad/s ≈ 1391.10 m/s
(d) Finally, express the centripetal acceleration as a multiple of g:
g = 9.81 m/s^2
Centripetal acceleration = r * ω^2 = (0.1330 m) * (10472.39 rad/s)^2 ≈ 14711.70 m/s^2
Multiple of g = 14711.70 m/s^2 / 9.81 m/s^2 ≈ 1500.27g
Therefore, the centripetal acceleration at a point 13.30 cm from the axis of rotation is approximately 1500.27 times the acceleration due to gravity.