Question 4.SP.12: A support block is acted upon by two forces, as shown. Knowi...
A support block is acted upon by two forces, as shown. Knowing that the coefficients of friction between the block and the incline are \mu_s = 0.35 and \mu_k = 0.25, determine the force P required to (a) start the block moving up the incline, (b) keep it moving up, (c) prevent it from sliding down.
STRATEGY: This problem involves practical variations of the third type of friction problem. You can approach the solutions through the concept of the angles of friction.
MODELING:
Free-Body Diagram. For each part of the problem, draw a free-body diagram of the block and a force triangle including the 800-N vertical force, the horizontal force P, and the force R exerted on the block by the incline. You must determine the direction of R in each separate case. Note that, because P is perpendicular to the 800-N force, the force triangle is a right triangle, which easily can be solved for P. In most other problems, however, the force triangle will be an oblique triangle and should be solved by applying the law of sines.
ANALYSIS:
a. Force P to Start Block Moving Up. In this case, motion is impending up the incline, so the resultant is directed at the angle of static friction (Fig. 1). Note that the resultant is oriented to the left of the normal such that its friction component (not shown) is directed opposite the direction of impending motion.
P =(800 N) tan 44.29°
P = 780 N ←
b. Force P to Keep Block Moving Up. Motion is continuing, so the resultant is directed at the angle of kinetic friction (Fig. 2). Again, the resultant is oriented to the left of the normal such that its friction component is directed opposite the direction of motion.
P =(800 N) tan 39.04°
P = 649 N ←
c. Force P to Prevent Block from Sliding Down. Here, motion is impending down the incline, so the resultant is directed at the angle of static friction (Fig. 3). Note that the resultant is oriented to the right of the normal such that its friction component is directed opposite the direction of impending motion.
P =(800 N) tan 5.71°
P = 80.0 N ←
REFLECT and THINK: As expected, considerably more force is required to begin moving the block up the slope than is necessary to restrain it from sliding down the slope.
