First, we need to convert the rotation speed to rad/s. Given that 1 rotation is equal to 2π radians, and 1 minute has 60 seconds, we have:
(7.1 × 10^4 rotations/min) × (2π radians/rotation) × (1 min/60 s) = 7.47 × 10^3 radians/s
Next, we can compute the centripetal acceleration for the unbalanced mass:
a = ω^2 × r
a = (7.47 × 10^3)^2 × 0.11
a ≈ 3.71 × 10^7 m/s^2
Now, we can calculate the net force acting on the unbalanced mass:
F = m × a
F = (11 × 10^-6 kg) × (3.71 × 10^7 m/s^2)
F ≈ 0.408 N
The magnitude of the net force on the rotor due to the unbalanced samples is approximately 0.408 N.
The ultracentrifuge is an important tool for separating and analyzing proteins in biological research. Because of the enormous centripetal accelerations that can be achieved, the apparatus must be carefully balanced so that each sample is matched by another on the opposite side of the rotor shaft. Failure to do so is a costly mistake. Any difference in mass of the opposing samples will cause a net force in the horizontal plane on the shaft of the rotor. Suppose that a scientist makes a slight error in sample preparation, and one sample has a mass 11mg greater than the opposing sample.
If the samples are 11cm from the axis if the rotor and the ultracentrifuge spins at 7.1×10^4 rpm, what is the magnitude of the net force on the rotor due to the unbalanced samples?
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