Question

| The ultracentrifuge is an important tool for separating and analyzing proteins. Because of the enormous centripetal accelerations, the centrifuge must be carefully balanced, with each sample matched by a sample of identical mass on the opposite side. Any difference in the masses of opposing samples creates a net force on the shaft of the rotor, potentially leading to a catastrophic failure of the apparatus. Suppose a scientist makes a slight error in sample preparation and one sample has a mass 10 mg larger than the opposing sample. If the samples are 12 cm from the axis of the rotor and the ultracentrifuge spins at 70,000 rpm, what is the magnitude of the net force on the rotor due to the unbalanced samples

Answer 1

The net force on the rotor due to the unbalanced samples : 64.4 N

Further explanation  

Centripetal force is a force acting on objects that move in a circle in the direction toward the center of the circle  

$\large{\boxed{\bold{F= \frac{mv^2}{R}}}$

F = centripetal force , N

m = mass , Kg

v = linear velocity , m/s

r = radius , m

The speed that is in the direction of the circle is called linear velocity  

Can be formulated:  

$\displaysyle v=2\pi.r.f$

r = circle radius  

f = rotation per second (RPS)  

The sample has a mass of 10 mg larger than the opposing sample. If the samples are 12 cm from the axis of the rotor and the ultracentrifuge spins at 70,000 rpm  

Known  

RPM = 70,000, convert to RPS = 70,000: 60 = 1166.6  

r = 12 cm = 0.12 m  

m = 10 mg = 10⁻⁵ kg  

then  

Linear velocity :

v = 2π.r.f

$\displaystyle v=2\times 3.14\times 0.12\times 1166.6\\\\v=879.15\:m/s$

Centripetal force :

$\displaystyle F=\frac{10^{-5}\times (879.15)^2}{0.12}\\\\F=\boxed{\bold{64.4\:N}}$

Learn more

the average velocity

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resultant velocity

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velocity position

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Keywords: ultracentrifuge, samples, Centripetal force, linear velocity

Answer 2

The magnitude of the net force on the rotor due to the unbalanced samples is 64.4 Newton.

How to calculate the net force magnitude?

From the information given, the velocity will be calculated as:

= 2πrf.

where, r = radius = 0.12

f = rotation per second = 70000/60 = 1166.6

Velocity will be:

= 2 × 3.14 × 0.12 × 1166.6

= 879.15 m/s

Therefore, the centripetal force will be:

= [10^-5 × (879.15)²] ) 0.12

= 64.4N

In conclusion, the magnitude of the net force is 64.4 Newton.

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