The centrifuge is a device in which a small container of material is rotated at a high speed on a circular path. Such a device is used in medical laboratories, for instance to cause the more less dense blood serum and collect at the bottom. The question is suppose the centripetal Acceleration is 6.5*10^3 times as large as the acceleration due to gravity. How many revolutions per minute is the sample making, if it is located at a radius of 5.00 cm from axis of rotation?
To find the number of revolutions per minute (RPM) that the sample is making, we need to first determine the centripetal acceleration.
The centripetal acceleration (ac) of an object moving in a circular path is given by the formula:
ac = (v^2) / r
Where:
- ac is the centripetal acceleration
- v is the velocity of the object
- r is the radius of the circular path
In this case, we are told that the centripetal acceleration is 6.5 * 10^3 times as large as the acceleration due to gravity (g). The acceleration due to gravity is approximately 9.8 m/s^2.
So, we can write the equation as:
ac = 6.5 * 10^3 * g
Next, we need to find the velocity of the sample. The velocity (v) can be calculated using the formula:
v = ω * r
Where:
- v is the velocity
- ω is the angular velocity (in radians)
- r is the radius of the circular path
Since we want to find the RPM, we need to convert the angular velocity from radians to revolutions per minute (RPM):
ω (in RPM) = ω (in radians) * (1 min / 2π radians) * (1 rev / 1 min)
Now, let's rearrange the formulas and solve the problem.
Given:
Radius (r) = 5.00 cm = 0.05 m
Acceleration due to gravity (g) = 9.8 m/s^2
Centripetal acceleration (ac) = 6.5 * 10^3 * g
First, calculate the velocity (v):
v = √(ac * r)
v = √(6.5 * 10^3 * 9.8 * 0.05)
Next, calculate the angular velocity (ω):
ω = v / r
Finally, convert the angular velocity to RPM:
ω (in RPM) = ω (in radians) * (1 min / 2π radians) * (1 rev / 1 min)
Substitute the calculated values into the equations to find the RPM:
ω (in RPM) = (ω * 60) / (2π)
RPM = (ω * 60) / (2π)
Solve the equations to find the RPM.
To find the number of revolutions per minute (rpm) the sample is making, we need to use the relationship between centripetal acceleration, acceleration due to gravity, and the radius of rotation.
The centripetal acceleration (ac) is given as 6.5 * 10^3 times the acceleration due to gravity (g). We can express this as:
ac = 6.5 * 10^3 * g
The centripetal acceleration is also related to the angular velocity (ω) and the radius (r) of rotation by the formula:
ac = ω^2 * r
Since we want to find the sample's rpm, we need to convert ω to revolutions per minute.
1 revolution is equal to 2π radians, and 1 minute is equal to 60 seconds. So, the angular velocity ω in radians per minute is given by:
ω = 2π * (rpm/60)
Plugging this into the formula for centripetal acceleration, we have:
6.5 * 10^3 * g = (2π * (rpm/60))^2 * r
Simplifying, we get:
rpm = √[(6.5 * 10^3 * g * 60^2)/(2π * r)]
Now, substituting the given values:
g = 9.8 m/s^2 (acceleration due to gravity)
r = 5.00 cm = 0.05 m (radius of rotation)
rpm = √[(6.5 * 10^3 * 9.8 * 60^2)/(2π * 0.05)]
Calculating this expression gives us the number of revolutions per minute the sample is making.