A block having a mass of 10.0 kg is pressed against the wall bya hand exerting a force F inclined at an angle θ of 52° to the wallas shown below. The coefficient of static friction µstat betweenthe block and the wall is 0.20. We shall investigate the questionof how large the force F must be to keep the block from slidingalong the wall. There is more physics here than initially meets theeye. Think about the situation in terms of your everyday experience(or better yet, actually try it out): If you start out with a smallvalue of F, the block will tend to slide downward; as you increaseF, you reach the point at which the block will no longer slide; asyou continue increasing F, the block stays put until, at somelarger value of F, it might even begin to slide upward. This is thephysics to be investigated, both algebraically and numerically.
(a) First draw well-separated force diagrams of the block andthe region of the wall where the two are in contact (1) for thecase in which F is small enough that the block tends to slidedownward and (2) for the case in which the block tends to slideupward. Denote the various forces by appropriate algebraic symbols;do not put in numbers at this point. Describe each force in wordsand identify the third law pairs.
(b) Applying Newton’s second law, obtain algebraic expressionsfor F in terms of m, g, µstat, and θ for case 1, in which the blockis just about to start sliding downward and for case 2, in which itis just about to start sliding upward.
(c) Now put in the various numbers and calculate the value of Ffor each of the two cases. How large is the spread between the twovalues? Does your result make physical sense? What is going on atthe wall when F lies between the two extremes you have calculated?What happens to the frictional force when F lies between these twoextremes?
(d) Return to the algebraic expression for case 2 in which theblock is just about to slide upward. What does this expression sayhappens to F if you keep m and θ constant but increase the value ofµstat? What is the equation telling us happens at the point atwhich µ is large enough to make the denominator of the expressionequal to zero? Is it possible to make the block slide upward with asufficiently large F at a fixed value of θ regardless of the valueof µstat? Solve for the value of µstat at which it becomesimpossible to make the block slide upward, showing that this valuedepends only on θ and is independent of the weight of the block. Doyou find this result strange? Why or why not? Could you haveanticipated it without having made the mathematical analysis?
