Chapter 1: Problems (1.77 - 1.81)

1.77 Bones and Muscles. A patient in therapy has a forearm that weighs 20.5 N and that lifts a 112.0-N weight. These two forces have direction vertically downward. The only other significant forces on his forearm come from the biceps muscle (which acts perpendicularly to the forearm) and the force at the elbow. If the biceps produces a pull of 232 N when the forearm is raised $43^\circ$ above the horizontal, find the magnitude and direction of the force that the elbow exerts on the forearm. (The sum of the elbow force and the biceps force must balance the weight of the arm and the weight it is carrying, so their vector sum must be 132.5 N, upward)

1.78 You are hungry and decide to go to your favorite neighborhood fast-food restaurant. You leave your apartment and take the elevator 10 flights down (each flight is 3.0 m) and then go 15 m south to the apartment exit. You then proceed 0.2 km east, turn north, and go 0.1 km to the entrance of the restaurant. a) Determine the displacement from your apartment to the restaurant. Use unit-vector notation for your answer, being sure to make clear choice of coordinates. b) How far did you travel along the path you took from your apartment to the restaurant and what is the magnitude of the displacement you calculated in part (a)?

1.79 You are canoeing on a lake. Starting at your camp on the shore, you travel 240 m in the direction $32^\circ$ south of east to reach a store to purchase supplies. You know the distance because you have located both your camp and the store on a map of the lake. On the return trip you travel distance B in the direction $48^\circ$ north of west, distance C in the direction $62^\circ$ south of west, and then you are back at your camp. You measure the directions of travel with your compass, but you don't know the distances. Since you are curious to know the total distance you rowed, use vector methods to calculate the distances B and C.

1.80 You are camping with two friends, Joe and Karl. Since all three of you like your privacy, you don't pitch your tents close together. Joe's tent is 21.0 m from yours, in the direction $23.0^\circ$ south of east. Karl's tent is 32.0 m from yours, in the direction $37.0^\circ$ north of east. What is the distance between Karl's tent and Joe's tent?

1.81 Vectors $\vec{A}$ and $\vec{B}$ are drawn from a common point. Vector $\vec{A}$ has magnitude A and angle $\theta_A$ measured in the sense from the +x-axis to the +y-axis. The corresponding quantities for vector $\vec{B}$ are B and $\theta_B$. Then $\vec{A} = A\cos\theta_A\hat{i} + A\sin\theta_A\hat{j}$, $\vec{B} = B\cos\theta_B\hat{i} + B\sin\theta_B\hat{j}$, and $\phi = |\theta_B - \theta_A|$ is the angle between $\vec{A}$ and $\vec{B}$. a) Derive Eq. (1.18) from Eq. (1.21). b) Derive Eq. (1.22) from Eq. (1.27).