Chapter 1: Problems (1.72 - 1.76)

1.72 On a training flight, a student pilot flies from Lincoln, Nebraska to Clarinda, Iowa, and then to St. Joseph, Missouri, and then to Manhattan, Kansas (Fig. 1.33). The directions are shown relative to north: $0^\circ$ is north, $90^\circ$ is east, $180^\circ$ is south, and $270^\circ$ is west. Use the method of components to find a) the distance she has to fly from Manhattan to get back to Lincoln; b) the direction (relative to north) she must fly to get there. Illustrate your solution with a vector diagram.

1.73 A graphic artist is creating a new logo for her company's Web site. In the graphics program she is using, each pixel in an image file has coordinates (x, y), where the origin (0, 0) is at the upper-left corner of the image, the +x-axis points to the right, and the +y-axis points down. Distances are measured in pixels. a) The artist draws a line from the pixel location (10, 20) to the location (210, 200). She wishes to draw a second line that starts at (10, 20), is 250 pixels long, and is at an angle of $30^\circ$ measured clockwise from the first line. At which pixel location should this second line end? Give your answer to the nearest pixel. b) The artist now draws an arrow that connects the lower right end of the first line to the lower right end of the second line. Find the length and direction of this arrow. Draw a diagram showing all three lines.

1.74 Getting Back. An explorer in the dense jungles of equatorial Africa leaves his hut. He takes 40 steps northeast, then 80 steps $60^\circ$ north of west, then 50 steps due south. Assume his steps all have equal length. a) Sketch, roughly to scale, the three vectors and their resultant. b) Save him from becoming hopelessly lost in the jungle by giving him the displacement, calculated using the method of components, that will return him to his hut.

1.75 A ship leaves the island of Guam and sails 285 km at $40.0^\circ$ north of west. In which direction must it now head and how far must it sail so that its resultant displacement will be 115 km directly east of Guam?

1.76 A boulder of weight w rests on a hillside that rises at a constant angle $\alpha$ above the horizontal, as shown in Fig. 1.34. Its weight is a force on the boulder that has direction vertically downward. a) In terms of $\alpha$ and w, what is the component of the weight of the boulder in the direction parallel to the surface of the hill? b) What is the component of the weight in the direction perpendicular to the surface of the hill? c) An air conditioner unit is fastened to a roof that slopes upward at an angle of $35.0^\circ$. In order that the unit not slide down the rood, the component of the unit's weight parallel to the roof cannot exceed 550 N. What is the maximum allowed weight of
the unit?