How fast (in rpm) must a centrifuge rotate if a particle 69-cm from the axis of rotation is to experience an acceleration of 200438 g's?
(can you show me how you get the answer) be thorough
Well, let's start by finding the acceleration in terms of g's. 1 g is equivalent to the acceleration due to gravity on Earth, which is approximately 9.8 m/s^2.
So, the acceleration in terms of g's is 200438 g's * 9.8 m/s^2/g = 1964308 m/s^2.
The formula for centripetal acceleration is given by a = rω^2, where a is the centripetal acceleration, r is the distance from the axis of rotation, and ω is the angular velocity.
We have the acceleration (a = 1964308 m/s^2) and the distance from the axis of rotation (r = 69 cm = 0.69 m), and we need to find ω.
Substituting the given values into the equation, we get:
1964308 m/s^2 = (0.69 m) ω^2
To solve for ω, we divide both sides of the equation by (0.69 m):
ω^2 = 1964308 m/s^2 / 0.69 m
ω^2 ≈ 2849344.93 s^-2
Now, we take the square root of both sides to find ω:
ω ≈ √(2849344.93 s^-2)
ω ≈ 1688.94 s^-1
Finally, we convert the angular velocity from s^-1 to rpm:
1 rpm = 1/60 s^-1
So, we have:
ω ≈ 1688.94 s^-1 * (1/60) rpm/s
ω ≈ 28.15 rpm
Therefore, the centrifuge must rotate at approximately 28.15 rpm for a particle 69 cm from the axis of rotation to experience an acceleration of 200438 g's.
To calculate the required rotational speed (in rpm) of a centrifuge for a given acceleration, we will use the following formula:
g-force = (r * ω^2) / g
Where:
- g-force is the acceleration experienced by the particle in units of g (1 g = 9.8 m/s^2)
- r is the distance of the particle from the axis of rotation in meters (converted from centimeters)
- ω is the angular speed (in radians per second)
- g is the acceleration due to gravity (9.8 m/s^2)
Let's break down the steps to find the answer:
Step 1: Convert the distance from cm to meters
Given that r = 69 cm, we need to convert it to meters by dividing it by 100:
r = 69 cm / 100 = 0.69 meters
Step 2: Convert the acceleration from g's to m/s^2
The acceleration is provided as 200438 g's. To convert to m/s^2, we multiply it by 9.8:
g-force = 200438 * 9.8 = 1,966,912 m/s^2
Step 3: Rearrange the formula to solve for ω
Rearranging the formula, we can solve for the angular speed ω:
ω = sqrt((g * g-force) / r)
Step 4: Substitute the values and calculate ω
Substituting the known values into the formula:
ω = sqrt((9.8 * 1,966,912) / 0.69)
Calculating this gives us:
ω ≈ 2,467.38 radians per second
Step 5: Convert the angular speed from radians per second to rpm
To convert from radians per second to revolutions per minute (rpm), we use the following conversion factor:
1 revolution = 2π radians
1 minute = 60 seconds
To find the angular speed in rpm, we divide ω by 2π and then multiply by 60:
angular speed (rpm) = (ω * 60) / (2π)
Substituting the known value, we get:
angular speed (rpm) ≈ (2,467.38 * 60) / (2π)
Calculating this gives us approximately:
angular speed ≈ 23,530.31 rpm
Therefore, the centrifuge must rotate at approximately 23,530.31 rpm for the particle located 69 cm from the axis of rotation to experience an acceleration of 200438 g's.
centrifuge means centripetal force
v^2 / r = 200438 g ... v in m/s
rpm = v * 60 / (2 π r)
a = 200438 g's
=200438 * 9.81 m/s^2
You have got to be kidding
r = 0.69 meters
a = omega^2 r
solve for omega
which is in radians/second
divide by 2 pi to get revolutions per second
multiply by 60 to get revolutions per minute