An ultracentrifuge accelerates from rest to 100,000 rpm in 1.20 min. Solve c using the equation centripetal acceleration =(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 centripetal acceleration, in m/s^2, of this point at full rpm?
(d) Express this centripetal acceleration as a multiple of g.
First, we need to convert the rotation speed from rpm to rad/s:
1 rpm = 2π rad/min
So, the final angular velocity (ω) is:
ω = (100,000 rpm) * (2π rad/min) = 200,000π rad/min
Now, we convert ω to rad/s:
ω = (200,000π rad/min) * (1 min/60 s) = 10,000π rad/s
(a) To find the angular acceleration (α), we use the formula:
α = (ωf - ωi) / t
where ωi = 0 (initial angular velocity), ωf = 10,000π rad/s, and t = 1.20 min = 72 s
α = (10,000π rad/s - 0) / 72 s = 138.89 rad/s^2
(b) The tangential acceleration (at) of a point at a distance from the axis of rotation is given by:
at = r * α
where r = 13.30 cm = 0.133 m
at = (0.133 m) * (138.89 rad/s^2) = 18.59 m/s^2
(c) The centripetal acceleration (ac) of a point at full rpm is given by:
ac = (10,000π rad/s)^2 * r
Now, solving for c:
ac = (10,000π rad/s)^2 * 0.133 m = 418,879.02 m/s^2
(d) To express this centripetal acceleration as a multiple of g:
1 g = 9.81 m/s^2
So, the centripetal acceleration in terms of g is:
418,879.02 m/s^2 / 9.81 m/s^2 ≈ 42,694.94 g
Therefore, the centripetal acceleration of the point at full rpm is approximately 42,694.94 times the acceleration due to gravity.