Determine the maximum horizontal force F that may be applied to the plank of mass m for which the solid sphere does not slip as it begins to roll on the plank. The sphere has a mass M and radius R. The coefficient of static and kinetic friction between the sphere and the plank are mu_(S) and mu_(k) respectively.

Determine the maximum horizontal force F that may be applied to the

1.	Determine the maximum horizontal force F that may be applied to the plank of mass m for which the solid sphere does not slip as it begins to roll on the plank. The sphere has a mass M and radius R. The coefficient of static and kinetic friction between the sphere and the plank are mu_(S) and mu_(k) respectively.
Answer» Solution : the FREE diagrams of the sphere and the plank are as shown below: Writing equation of motion For sphere EQUATIONS of motion For sphere linear acceleration `a_(1)=(mu_(s)Mg)/(M)=mu_(s)G` .. (i) Angular acceleration `alpha=((mu_(s)Mg)R)/((2)/(5)MR^(2))` `=(5)/(2)(mu_(S)g)/(R)` ...(ii) For plank linear acceleration `a_(2)=(F-mu_(S)Mg)/(m)`..(iii) For no SLIPPING `a_(2)=a_(1)+Ralpha`..(iv) Solving the above four equations we get `F=mu_(S)g(M+(7)/(2)m)` THUS, maximum value of `F` can be `mu_(S)g(M+(7)/(2)m)`