Solve Centrifuge Problem with Calculus & Angular Velocity

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In summary, the conversation discusses a physics problem involving a centrifuge tube filled with a homogenous liquid and a rotating arm. The problem requires knowledge of calculus and the goal is to find the resultant pressure of the liquid at a distance from the axis of rotation. The suggestion is to start by writing the net force on a small slice of the liquid and using Newton's 2nd law to find the force, which is centripetal acceleration.
  • #1
Omar.Castillo
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I have a problem with a small physics problem. It requires some knowledge of calculus which I don't yet posses I am looking here for ideas. The problem states that there is a centrifuge tube with a length 12cm that is caused to spin by a rotating arm with lengt 8 cm. The tube is filled with 10 cm of a homogenous liquid with density p. It is rotating at a full angular velocity w and its traveling fast enough to be almost horizontal.

The question requires me to show that the resultant pressure of the liquied at a distance r from the axis of rotation is delta p/ delta r= prw^2

any suggetions on how to get this result?
 
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  • #2
Start by writing the net force on a small slice (thickness delta r) of the liquid. Use Newton's 2nd law to find that force. The acceleration is centripetal.
 
  • #3


Dear colleague,

Thank you for reaching out for ideas on how to solve this centrifuge problem using calculus and angular velocity. I can offer some suggestions on how to approach this problem.

Firstly, it is important to understand the basic principles of centrifugal force and how it relates to angular velocity. Centrifugal force is the outward force experienced by an object in circular motion, and it increases with the square of the angular velocity. This means that as the angular velocity increases, so does the centrifugal force.

In the given problem, the centrifuge tube is rotating at a full angular velocity w, and it is filled with a homogenous liquid with density p. We can use the equation for centrifugal force, F = mrw^2, where m is the mass of the liquid and r is the distance from the axis of rotation, to determine the force acting on the liquid at a distance r.

Next, we can use the concept of pressure to determine the resultant pressure of the liquid at a distance r from the axis of rotation. Pressure is defined as the force per unit area, and in this case, the force acting on the liquid is due to centrifugal force. Therefore, we can write the equation for pressure as P = F/A, where A is the cross-sectional area of the liquid.

To find the change in pressure, we can use the concept of differentiation from calculus. The change in pressure, delta P, can be represented as the derivative of pressure with respect to the distance r, which is written as delta P/delta r. This gives us the equation delta P/delta r = dP/dr.

Now, we can substitute the values for force and area in the equation for pressure to get P = mrw^2/A. Taking the derivative of this equation with respect to r, we get dP/dr = mrw^2/A.

Finally, we can substitute the given values of length and density to get the final result of delta P/delta r = prw^2. This shows that the change in pressure with respect to distance is directly proportional to the density, angular velocity, and distance from the axis of rotation.

I hope this explanation helps you understand the problem better and provides you with a clear approach to solving it. If you have any further questions, please feel free to reach out.

Best regards,

 

1. What is a centrifuge and how does it work?

A centrifuge is a laboratory instrument that spins samples at high speeds in order to separate different components based on their density. This is achieved by applying angular velocity, which causes the samples to move outward and form layers based on their density. The faster the centrifuge spins, the more separation occurs.

2. How can calculus be used to solve a centrifuge problem?

Calculus is used to analyze the rate of change in a system, which is crucial in understanding the behavior of a centrifuge. By using calculus concepts such as derivatives and integrals, we can determine the angular velocity needed to achieve a desired separation and how long the centrifuge should run for optimal results.

3. What are the key factors that affect the centrifugation process?

The key factors that affect the centrifugation process include the speed or angular velocity of the centrifuge, the size and shape of the sample, the type of rotor being used, and the density and viscosity of the sample components. These factors can also be adjusted to optimize the separation process.

4. How does changing the angular velocity affect the separation of samples in a centrifuge?

The angular velocity directly affects the separation of samples in a centrifuge. A higher angular velocity will cause the samples to move outward faster, resulting in a greater separation. However, if the velocity is too high, it can cause samples to mix back together, so it is important to determine the optimal velocity for the specific sample being used.

5. Can calculus also be used to determine the optimal angle for the centrifuge rotor?

Yes, calculus can also be used to determine the optimal angle for the centrifuge rotor. By analyzing the forces acting on the samples and the rotor, we can determine the optimal angle that will result in the most efficient separation process.

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