You are part of a group that is designing an air filtration system Available. In

Question

You are part of a group that is designing an air filtration system Available. In order to optimize its operation you need to measure the mass of a common pollen in the air where the filter will be used. To measure the mass of pollen, you have designed a small rectangular box with a hole in one side to allow pollen between. Once inside the pollen it is given a positive electric charge and accelerated due to a force Electrostatic to acquire a speed of 1.4 m/s. The pollen then hits the end a small uniform rod which is hanging vertically from a pivot at one of its ends. Since the bar has a negative charge at its end, pollen adheres to it when this rotates (rises). By measuring the angle that the broken bar would give the mass of the particle. After the angle is measured, the load bar is inverted releasing the particle. Is a excellent design, but her friend insists that will not work. To prove it asks calculate the length of the bar giving a reasonable angle of about 10 degree for Typical particle pollen 4times10^-9 grams. His plan is to get a bar 7 times10^-4 grams with a moment of inertia 1/12 as much as if the entire mass were concentrated at its end. Is she right?

Explanation / Answer

let the length of bar L

as the body collide inelastically there is a conservation of angular momentum

let the velocity gained by centre of mass of the rod be V

4* 10-9 * 1.4 = (7*10-4 + 4*10-9) *V [ (7*10-4 + 4*10-9) = 7*10-4 approx]

=> V = 8* 10-6 m/s

now this is the velocity of the centre of mass of the rod which is located at L/2

therefore angular velocity , w = V/(L/2) = 1.6 *10-5 /L

velocity of the pollen after striking = L w = 1.6 *10-5

as the energy is conserved so initial kinetic energy of the system i converted to potential energy as it rotates by 10 degrees

let us find initial kinetic energy

initial kinetic energy ofthe bar 1/2 * I * w2 [ where I = 1/12 * 7 * 10-4* L2]

= 0.5 * 1/12 * 7 * 10-4* L2*(1.6 *10-5)2/L2

= 0.75 * 10-14

initial kinetic energy of pollen = 0.5 * 4*10-9 * (1.6 *10-5)2 = 5.12 * 10-19

total kinetic energy of the system = 0.75 * 10-14 + 5.12 * 10-19 = 0.75 * 10-14 approx

height gained by centre of mass of the rod = h =  L( 1- cos 10)/2

potential energy gained by the system =7*10-4 * g* h = 7*10-4 * 10 * L( 1- cos 10)/2

note : we neglect potential energy gained by pollen as it is very very less and it does not effect our answer

total kinetic energy of the system = potential energy gained by the system

=> 0.75 * 10-14 =   7*10-4 * 10 * L( 1- cos 10)/2

=> L = 14 * 10-11 m which is very very less

so she is right this experiment is not possible

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