Question

A block having a mass of 10.0 kg is pressed against the wall by a hand exerting a force F inclined at an angle θ of 52° to the wall as shown below. The coefficient of static friction µstat between the block and the wall is 0.20. We shall investigate the question of how large the force F must be to keep the block from sliding along the wall. There is more physics here than initially meets the eye. Think about the situation in terms of your everyday experience (or better yet, actually try it out): If you start out with a small value of F, the block will tend to slide downward; as you increase F, you reach the point at which the block will no longer slide; as you continue increasing F, the block stays put until, at some larger value of F, it might even begin to slide upward. This is the physics to be investigated, both algebraically and numerically.

(a) First draw well-separated force diagrams of the block and the region of the wall where the two are in contact (1) for the case in which F is small enough that the block tends to slide downward and (2) for the case in which the block tends to slide upward. Denote the various forces by appropriate algebraic symbols; do not put in numbers at this point. Describe each force in words and identify the third law pairs.

(b) Applying Newton’s second law, obtain algebraic expressions for F in terms of m, g, µstat, and θ for case 1, in which the block is just about to start sliding downward and for case 2, in which it is just about to start sliding upward.

(c) Now put in the various numbers and calculate the value of F for each of the two cases. How large is the spread between the two values? Does your result make physical sense? What is going on at the wall when F lies between the two extremes you have calculated? What happens to the frictional force when F lies between these two extremes?

(d) Return to the algebraic expression for case 2 in which the block is just about to slide upward. What does this expression say happens to F if you keep m and θ constant but increase the value of µstat? What is the equation telling us happens at the point at which µ is large enough to make the denominator of the expression equal to zero? Is it possible to make the block slide upward with a sufficiently large F at a fixed value of θ regardless of the value of µstat? Solve for the value of µstat at which it becomes impossible to make the block slide upward, showing that this value depends only on θ and is independent of the weight of the block. Do you find this result strange? Why or why not? Could you have anticipated it without having made the mathematical analysis?

Answer 1