To solve this problem, we can use the formula for centripetal force:
Fc = (mv^2) / r
where Fc is the centripetal force, m is the mass, v is the velocity, and r is the radius.
Given:
Force (Fc) = 4.8 x 10^-11 N
Velocity (v) = 126 rev/s
First, we need to find the radius of the centrifuge. The radius can be found using the formula:
r = v/d
where d is the diameter.
Rearranging the equation, we have:
d = v/r
To convert the units, we need to multiply the value of v by 2Ï€, since 1 revolution is equivalent to 2Ï€ radians.
So, the formula becomes:
d = (2Ï€v) / r
Now, let's substitute the values into the formula:
d = (2Ï€ x 126 rev/s) / r
We know that 2Ï€ is approximately 6.28.
d = (6.28 x 126 rev/s) / r
Now, we can rearrange the formula to solve for r:
r = (6.28 x 126 rev/s) / d
Now, substitute the given force Fc = 4.8 x 10^-11 N into the centripetal force formula:
Fc = (mv^2) / r
4.8 x 10^-11 N = (m x (126 rev/s)^2) / r
Next, substitute the value of 6.28 x 126 rev/s for d:
4.8 x 10^-11 N = (m x (126 rev/s)^2) / (6.28 x 126 rev/s)
To cancel out the units of revolutions per second, divide both sides of the equation by rev/s:
4.8 x 10^-11 N / (6.28 x 126 rev/s) = m x (126 rev/s) / (6.28 x 126 rev/s)
Simplify:
4.8 x 10^-11 / 6.28 = m / 6.28
To solve for m (mass), we need to find the average density of blood. The average density of blood in humans is approximately 1.06 g/cm^3.
Now, we can solve for the mass:
m = density x volume
Let's assume the volume remains constant for different sizes of centrifuges. Therefore, the mass will be the same. We can treat the mass (m) as a constant in this equation.
So, now we can rewrite our equation:
(m / 6.28) = 4.8 x 10^-11 / 6.28
To solve for the diameter (d), we need to isolate it in the equation:
d = (6.28 x 126 rev/s) / ((m / 6.28) x 4.8 x 10^-11 / 6.28)
Now, we can substitute the value of m / 6.28 from the equation above:
d = (6.28 x 126 rev/s) / (4.8 x 10^-11 / 6.28)
Simplifying, we get:
d = (6.28 x 126 rev/s) / (4.8 x 10^-11 / 6.28)
d ≈ 3293.75 rev/s
Therefore, the diameter of the centrifuge is approximately 3293.75 cm.