Solve Rotational Movement Problem - Calculate Bar Length

[BuddyGoodness]

This is a problem that I am just having some trouble wrapping my head around. I don't know if I am doing it correctly.

Homework Statement

You are a member of a group designing an air filtration system for allergy sufferers. To optimize its operation you need to measure the mass of the common pollen in the air where the filter will be used. To measure the pollens mass you have designed a small rectangular box with a hole in one side to allow the pollen to enter. Once inside the pollen is given a positive electric charge and accelerated by an electrostatic force to a speed of 1.4m/s. The pollen then hits the end of a very small uniform bar which is hanging straight down from a pivot at its top. Since the bar has a negative charge at its tip, the pollen sticks to it as the bar swings up. measuring the angle that the bar swings up would give the particles mass. After the angle is measured, the charge of the bar is reversed, releasing that particle. It's a cool design but your friend insists it will never work. To prove it she asks you to calculate the length of the bar which would give you a reasonable angle of about 10 degrees for a typical pollen particle of 4 x 10^-9 grams. Your plan calls for a bar of 7 x 10^-4 grams with a moment of inertial 1/12 as much as if all of it's mass were concentrated at its end. is she correct?

Homework Equations

w = omega
Angular momentum (L) = I(w) I= Mr^2
Rotational Kinetic Energy = 1/2I(w)^2
Potential energy = mgh

I was given a formula by my prof though I don't know how he got it: rw = v, becomes v/r = w

The Attempt at a Solution

First I set up a conservation of angular momentum equation: Initial momentum of pollen + momentum of bar: Iw + Iw with the initial momentum of the bar = 0
L(initial) = (mr^2)V/r = mrV = 5.6x10^-9R = L(initial)
L(final) = (I of pollen + I of bar/12)w = (mr^2+mr^2/12)V/r
Here I am confused how dividing the inertia of bar by twelve, I didn't know how do divide an unknown (r^2) by 12 so I only divided the mass by 12.
Anyway the r in v/r cancels out one of the r's from inertia leaving the equation: 5.83x10^-5rv(final)
conservation of momentum: 5.6x10^/9R = 5.83x10^/5rv(final), divide both sides by r, removing r. divide the right number from v to find v final.

From here I do a conservation of angular energy and solve for height and from there I use h =L(1/cos(theta)) to find L.
I am wondering if I am doing the first part right to get the info needed to solve the rest of the problem. I want to type out the rest of the equations but I have to get to work, I will finish entering in how I did the conservation of energy part when I get home.

 

[anathema]

Thank you for sharing your problem with us. I can understand your confusion about the calculation and I am happy to help you with it.

First, let's address the formula given by your professor. The equation rw = v is derived from the definition of angular velocity (w = v/r) and the fact that the linear velocity of the particle (v) is equal to the product of the angular velocity (w) and the distance from the pivot point (r). This formula is useful in solving rotational motion problems, as it relates the linear velocity of a particle to its angular velocity and the distance from the pivot point.

Moving on to your attempt at the solution, I can see that you have correctly set up the conservation of angular momentum equation. However, in the final equation, you have divided the mass of the bar by 12 instead of the inertia (I), which is a measure of an object's resistance to rotational motion. The inertia of the bar can be calculated as 1/12*mr^2, where m is the mass of the bar and r is the distance from the pivot point to the center of mass of the bar.

Therefore, the correct equation for conservation of momentum should be:

L(final) = (I of pollen + 1/12*I of bar)w = (mr^2+1/12*mr^2)V/r

Simplifying this equation, we get:

L(final) = (1+1/12)*mrV/r = 13/12*mrV/r

Now, to solve for the final velocity of the pollen, we can use the conservation of energy equation:

Initial kinetic energy + initial potential energy = final kinetic energy + final potential energy

Since the pollen is initially at rest, the initial kinetic energy is zero. So, the equation becomes:

mgh = 1/2*I(w)^2

Substituting the values given in the problem, we get:

(4*10^-9)(9.8)(h) = 1/2*(7*10^-4)(1.4)^2

Solving for h, we get:

h = 0.0052 meters

Now, we can use the equation h = L(1/cos(theta)) to solve for the length of the bar (L).

Substituting the value of h and theta (10 degrees) in the equation

 

[thomate1]

I can assure you that your approach to solving this problem is correct. You have correctly used the conservation of angular momentum and energy to determine the final velocity and height of the pollen particle as it sticks to the bar. From there, you can use the equation h = L(1/cos(theta)) to calculate the length of the bar required to produce a 10 degree angle for the given mass and moment of inertia.

It is also worth noting that your professor's formula of rw = v is derived from the definition of angular velocity (w = v/r) and the fact that the linear and angular velocities are equal at the point of contact between the pollen and bar (v = rw). Therefore, the r's in the moment of inertia term cancel out, leaving you with the final equation of v = rw.

Overall, your approach is sound and your calculations appear to be correct. Your friend's skepticism may stem from the complexity of the problem, but rest assured, your solution is valid. Keep up the good work!