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A cotton reel has a inner radius r and outer radius R. Mass of the reel is M and moment of inertia about longtudinal rotational axis is I. A force P is applied at the free end of thread wraped of the reel as shown in fig. If the reel moves without sliding. a. Determine the frictioal force exerted by the table on the reel and the direction in which it acts. b. I what direction does the reel begin to move? |
Answer» Solution :Method 1: As the body rolls, point `F` will be instantasneous centre of rotation (`ICR`). There is clockwise torque about `ICR`. Hecne centre of MASS will have acceleratioin in eright direction. As there is net torque about `ICR` in clockwise sense. Hence the reel will rotate in clockwise direction.  Hence the reel moves to the RIGHT. Force `P` exerts ANTICLOCKWISE torque, therefore FRICTION force must exert a larger clockwise torque to produce clockwise rolling. The equations of the motion are `SigmaF_(x)=F-f=Ma`.....i `SigmaF_(y)=N-Mg=0`............ii `Sigmatau=fxxR-Fxxr=Ialpha`..............iii As the reel rolls without slipping `a=Ralpha` From eqn i and iii we get `(F-f)/M=(R(fxxR-Fxxr))/I` `f=(F(I+MRr))/(I+MR^(2))` Force of friction comes out to be positive, thereforeour assumption about the direction of friction force was correct. On substitutingthe expression for in Eqn i we obtain `a=(FR(R-r))/(I+MR^(2))` As `Rgtr`, a positive and towards right. Similarly, from eqn iii, we obtain `alpha=(F(R-r))/(I+MR^(2))` Which is positive i.e., net torque on the reel is clockwise. Method 2: We can apply torque equation about.  ICR `implies F(R-r)=I_(p)alpha` `F(R-r)=[I+MR^2]alpha` `implies alpha=(F(R-r))/([I+MR^(2)])` |
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