FMAX $($N$)=19\mathrm{.}66\backslash$theta $($rad$)=67$ M $($g$)=590$ H$1($cm$)=57$ H$2($cm$)=68$ R $($cm$)=74$

You are hired to design a robot arm that can catch spherical objects when they fall. In Figure $1$ the robot arm is connected to the ceiling above the inclined plane with a purpose to catch the uniform

spherical object when it starts to move downwards. The friction coefficient between the sphere and the inclined plane is enough to provide a smooth rolling without slipping. Robot arm always

grabs the sphere from its equatorial plane parallel to the inclined plane. There is no friction between the pincer of the robot and the sphere, so$,$ when the sphere is grabbed, it can be pulled up by rolling on the inclined surface.when the sphere is released without an initial velocity;

a$)$ Draw a free body diagram and find the force FMIN that should be applied by the robot arm to keep the sphere in equilibrium.

b$)$ If the robot arm force FMAXis larger than the requiredvalue for the equilibrium condition $($FMIN$),$ what is the acceleration of the sphere rolling upwards on the inclined plane.

c$)$ Find the minimum value of the coefficient of static friction $$s between the inclined plane and the sphere so that the sphere can roll without slipping.

d$)$ Calculate the work done by the robot arm to pull the spherical object from H$1$ to H$2.$

e$)$ Does the friction between the sphere and inclined plane $($rolling friction, fs$)$ do work? Explain why or why not using dynamics$$ rules. What are the main sources of rolling friction, and which physical properties does it depend?

f$)$ What physical properties and methods would you use to minimize the power consumption and increase the mechanical efficiency of the robot arm? Express your answers by implementing the concepts that we have covered in the class and mathematically show how

your newly assigned values would increase or decrease the desired properties.