A uniform rod of mass m and length l which can rotate freely in vertical plane without friction, is hinged at its lower end on a table. If a sphere of mass m and radius $$R=L3$$ is placed in contact with the vertical rod and a horizontal force F $$=$$ $$80$$ N is applied at the upper end of the rod. Find the horizontal component of hinge reaction (in$$ $$N) acting on the rod just after force F starts acting.

Question

A uniform rod of mass m and length l which can rotate freely in vertical plane without friction, is hinged at its lower end on a table. If a sphere of mass m and radius $$R=L3$$ is placed in contact with the vertical rod and a horizontal force F $$=$$ $$80$$ N is applied at the upper end of the rod. Find the horizontal component of hinge reaction (in$$ $$N) acting on the rod just after force F starts acting.

Infinity Learn · Accepted Answer

Torque equation for rod about O, $$I0$$α$$=Fl$$−$$Nl3$$∴ $$ml23$$⋅α$$=Fl$$−$$Nl3ml$$α$$3=F$$−$$N3$$             …(i)The$$ motion of the sphere is translational only acceleration of the sphere $$(a) $$=$$ acceleration of point Plα$$3=a$$                        …(ii)From$$ $$(i) and (ii) $$ma=F$$−$$N3$$            …(iii)For$$ sphere, $$N=ma$$                          …$$(iv)Solving$$ $$(iii) and (iv) we get, $$a=3F4m$$∴$$N=3F4Acceleration$$ of the COM of the rod $$isacm=$$α$$l2=3a2=9F8mLet$$ horizontal component of hinge reaction is Nx. Force equation for rod, F−$$3F4+Fx=mac$$⇒F−$$3F4+Fx=m9F8mFx=9F8$$−$$F4=7F8=7$$×$$808=70$$ N

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