Know matter how i work it out, i keep getting the wrong answers. please help!!! i only have two more submissions before it closes on me.
A sample of blood is placed in a centrifuge of radius 14.0 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?
To determine the number of revolutions per second the centrifuge should be operated at, we need to use the concept of centripetal force and the equation for centripetal force.
The centripetal force acting on an object moving in a circular path is given by:
F = (m ∙ v²) / r
Where:
F is the centripetal force,
m is the mass of the object,
v is the velocity of the object, and
r is the radius of the circular path.
In this case, we know the mass of the red blood cell (m = 3.0 ✕ 10⁻¹⁶ kg) and the magnitude of the force acting on it (F = 4.0 ✕ 10⁻¹¹ N). We need to solve for the velocity (v) in order to find the number of revolutions per second.
Rearranging the equation, we can solve for the velocity:
v = √((F ∙ r) / m)
Now, we can plug in the values:
v = √((4.0 ✕ 10⁻¹¹ N ∙ 14.0 cm) / 3.0 ✕ 10⁻¹⁶ kg)
Note: It's important to ensure consistent units. Therefore, we need to convert the radius from centimeters (cm) to meters (m).
1 cm = 0.01 m, so the radius is 0.14 m.
v = √((4.0 ✕ 10⁻¹¹ N ∙ 0.14 m) / 3.0 ✕ 10⁻¹⁶ kg)
Now, we can calculate the velocity.
v = √((5.6 ✕ 10⁻¹² N · m) / 3.0 ✕ 10⁻¹⁶ kg)
v ≈ √(1.87 ✕ 10⁴ m/s)
v ≈ 136.74 m/s
Now, to find the number of revolutions per second, we need to convert the velocity from meters per second (m/s) to centimeters per second (cm/s), since the radius given is in centimeters.
1 m/s = 100 cm/s.
So, the velocity in cm/s is:
v = 136.74 m/s × 100 cm/s
v ≈ 13674 cm/s
Now, we know that one revolution corresponds to the circumference of a circle, which is equal to 2π times the radius of the circle. Therefore, the distance traveled in one revolution is:
distance = 2πr
Substituting the radius (14.0 cm), we have:
distance = 2π × 14.0 cm
distance ≈ 87.9646 cm
Now, we can find the number of revolutions per second by dividing the velocity by the distance traveled in one revolution.
revolutions per second = v / distance
revolutions per second ≈ 13674 cm/s / 87.9646 cm
revolutions per second ≈ 155.49
Therefore, the centrifuge should be operated at approximately 155.49 revolutions per second to settle the red blood cells out of the plasma.