A heavy particle slides under gravity down the inside of a smooth vertical tube held in vertical plane. It starts from the highest point with velocity $$2ag$$, where a is the radius of the circle. The angular position θ (as$$ shown in $$figure) at which the vertical acceleration of the particle is maximum, is given by $$cos$$$$-1pq$$. Find the value of $$p+q$$.

Question

A heavy particle slides under gravity down the inside of a smooth vertical tube held in vertical plane. It starts from the highest point with velocity $$2ag$$, where a is the radius of the circle. The angular position θ (as$$ shown in $$figure) at which the vertical acceleration of the particle is maximum, is given by $$cos$$$$-1pq$$. Find the value of $$p+q$$.

Infinity Learn · Accepted Answer

At position θ, $$v2=v02+2gh$$​ where,  $$h=a1$$−$$cos$$θ​∴ $$v2=2ag2+2ag1$$−$$cos$$θ​⇒$$v2=2ag2$$−$$cos$$θ​Also, $$N+mgcos$$θ$$=mv2a$$​⇒$$N+mgcos$$θ$$=2mg2$$−$$cos$$θ​⇒$$N=mg4$$−$$3cos$$θ​ Net vertical force, $$F=Ncos$$θ$$+mg$$​⇒$$F=mg4cos$$θ−$$3cos2$$θ$$+1$$​ This force (or$$ $$acceleration) will be maximum when dFdθ$$=0$$​⇒−$$4sin$$θ$$+6sin$$θ$$cos$$θ$$=0$$​⇒$$sin$$θ−$$4+6cos$$θ$$=0$$​⇒$$cos$$θ$$=23$$​⇒θ$$=$$$$cos$$−$$123$$​∴$$p+q=2+3=5$$

Dynamics of uniform and non uniform circular motion

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