A sample of blood is placed in a centrifuge of radius 20.0 cm. The mass of a red blood cell is 3.0 10−16 kg, and the magnitude of the force acting on it as it settles out of the plasma is 4.0 10−11 N. At how many revolutions per second should the centrifuge be operated?
rev/s
130
Solution or Explanation
Since
Fc = m
vt2
r
= mrω2,
the needed angular velocity is
ω =
Fc
mr
=
sqaure root of (4.0 ✕ 10−11 N)
(3.0 ✕ 10−16 kg)(0.200 m)
= (6.67 ✕ 10 ^2 rad/s) (1 rev/2pi rad) = 1.05x10^6 =>105
To find the number of revolutions per second at which the centrifuge should be operated, we can use the concept of centripetal force and centripetal acceleration.
First, let's calculate the centripetal force acting on the red blood cell. The centripetal force is given by the equation:
F = m * a
Where:
F is the centripetal force,
m is the mass of the red blood cell,
a is the centripetal acceleration.
Rearranging the equation to solve for a, we get:
a = F / m
Now, let's calculate the centripetal acceleration. The centripetal acceleration is given by the equation:
a = (v^2) / r
Where:
v is the velocity of the red blood cell,
r is the radius of the centrifuge.
Rearranging the equation to solve for v, we get:
v = √(a * r)
Now we can substitute the values given in the question:
m = 3.0 * 10^(-16) kg
F = 4.0 * 10^(-11) N
r = 20.0 cm = 0.20 m
Calculating the centripetal acceleration:
a = F / m = (4.0 * 10^(-11) N) / (3.0 * 10^(-16) kg) = 1.33 * 10^5 m/s^2
Calculating the velocity:
v = √(a * r) = √((1.33 * 10^5 m/s^2) * (0.20 m)) ≈ 154.3 m/s
Now, to find the number of revolutions per second, we need to convert the linear velocity to angular velocity. The formula is:
ω = v / r
Where:
ω is the angular velocity.
Substituting the values:
ω = (154.3 m/s) / (0.20 m) = 771.5 rad/s
Finally, we convert the angular velocity to revolutions per second. There are 2π radians in one revolution:
rev/s = ω / (2π) = 771.5 rad/s / (2π) ≈ 122.7 rev/s
Therefore, the centrifuge should be operated at approximately 122.7 revolutions per second.