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 the pollen between Once inside the pollen is given a positive electric charge and accelerated due to an electrostatic force to acquire a speed of 1.4 m / s. The pollen then hits the end of a small bar uniform, which is hanging vertically from a pivot at one end. Since the bar has a negative charge at its end, pollen adheres to it when rotating basket (sc clcva). By measuring the angle that the broken bar would give the mass of the particle. After the angle is measured, the load of the bar is inverted releasing the particle. It is an excellent design, but her friend insists that will not work. To demonstrate asks calculate the length of the bar giving a reasonable angle of about 10° for a typical pollen particle 4x10 grams. His plan is to get a bar 7x10 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 * 1.4 = (7*10 + 4*10) *V

=> V = 0.51 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) = 1.02 /L

velocity of the pollen after striking = L w = 1.02 m/s

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 * 0.07* L2*(1.02)2/L2

= 0.003

initial kinetic energy of pollen = 0.5 *0.0 4 * (1.02)2 = 0.021 J

total kinetic energy of the system = 0.0238 J

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

potential energy gained by the system =0.07 * g* h = 0.07* 10 * L( 1- cos 10)/2 = 0.0106 L

height gained by pollen = L(1-  cos 10)

potential energy gained by the pollen = 0.04*10*L(1-  cos 10) = 0.0061 L

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

=>0.0238 J = 0.0167L

=> L = 1.42 m

hence the experiment is possible

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