A sphere of mass M and radius R moves with velocity v through a region of space that contains small and identical particles of mass “m” each. All particles are at rest. There are n of these particles per unit volume (nm=$$ρ$$=1000$$$$π$$kgm−$$3). Assume m

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A sphere of mass M  and radius R moves with velocity v  through a region of space that contains small and identical particles of mass “m”  each. All particles are at rest. There are n  of these particles per unit volume (nm=$$ρ$$=1000$$$$π$$kgm−$$3).  Assume m<

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Consider a particle that makes contact with the sphere at an angle θ  with  respect to the line of motion. In the frame of the heave sphere (see$$ fig.$$). The particle comes in with the velocity $$-V$$ and then bounces off with a horizontal velocity component of  $$Vcos2$$θ.  So in this frame (and$$ hence also in the lab $$frame), the particle increases its horizontal momentum by $$mV(1+cos2$$θ$$)$$.The sphere must therefore loss this momentum.The area on the sphere that lies between  θ and  θ$$+d$$θ  (which$$ is a circle of radius  Rsinθ$$) sweeps out volume at rate  $$V(2$$$$π$$Rsinθ$$)(Rd$$θ$$)$$$$cos$$θ. Then the force experienced by sphere (that$$ is, the rate of  change in $$momentum) is $$thereforeF=$$∫$$0$$π$$/2n(2V$$$$π$$$$R2sin$$θ$$cos$$θ$$)$$.$$mV(1+cos2$$θ$$)d$$θ $$=2$$$$π$$$$nmR2V2$$∫$$0$$π$$/2sin$$θ$$cos$$θ(1+cos2$$θ$$)d$$θ$$=2$$$$π$$$$nmR2V2$$∫$$0$$π$$/2(sin2$$θ$$2+sin4$$θ$$4)d$$θ $$=2$$$$π$$$$nmR2V2$$∫$$0$$π$$/2($$−$$cos2$$θ$$4-cos4$$θ$$16) $$=2$$$$π$$$$nmR2V212$$ $$=$$πρ$$R2V2$$ $$=$$π×$$1000$$π×$$1100$$×$$64$$ $$=640$$ N

A sphere of mass M and radius R moves with velocity v through a region of space that contains small and identical particles of mass “m” each. All particles are at rest. There are n of these particles per unit volume (nm=ρ=1000πkgm−3). Assume m<

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