1) Can two different forces, acting through the same point, produce the same torque on an object? Answer: Yes, as long as the component of the force perpendicular to the line joining the axis to the force is the same for both forces. 2) If you stand with your back towards a wall and your heels touching the wall, you cannot lean over to touch your toes. Why? Answer: As you bend over your center of gravity moves forward and eventually is beyond the area of the floor in touch with your feet. This does not happen when you do it away from the wall because part of your body moves back and the center of mass remains over your feet. 3) Two equal forces are applied to a door at the doorknob. The first force is applied perpendicular to the door; the second force is applied at 30° to the plane of the door. Which force exerts the greater torque? A) the first applied perpendicular to the door B) the second applied at an angle C) both exert equal non-zero torques D) both exert zero torques E) Additional information is needed. 4) A heavy boy and a lightweight girl are balanced on a massless seesaw. If they both move forward so that they are one-half their original distance from the pivot point, what will happen to the seesaw? A) It is impossible to say without knowing the masses. B) It is impossible to say without knowing the distances. C) The side the boy is sitting on will tilt downward. D) Nothing, the seesaw will still be balanced. E) The side the girl is sitting on will tilt downward. 5) A figure skater is spinning slowly with arms outstretched. She brings her arms in close to her body and her angular speed increases dramatically. The speed increase is a demonstration of A) conservation of energy: her moment of inertia is decreased, and so her angular speed must increase to conserve energy. B) conservation of angular momentum: her moment of inertia is decreased, and so her angular speed must increase to conserve angular momentum. C) Newton’s second law for rotational motion: she exerts a torque and so her angular speed increases. D) This has nothing to do with mechanics, it is simply a result of her natural ability to perform. 6) A girl weighing 450. N sits on one end of a seesaw that is 3.0 m long and is pivoted 1.3 m from the girl. If the seesaw is just balanced when a boy sits at the opposite end, what is his weight? Neglect the weight of the seesaw. 7) An 82.0 kg painter stands on a long horizontal board 1.55 m from one end. The 15.5 kg board is 5.50 m long. The board is supported at each end. (a) What is the total force provided by both supports? (b) With what force does the support, closest to the painter, push upward? FIGURE 11-4 8) The mobile shown in Figure 11-4 is perfectly balanced. What must be the masses of m1, m2, and m3?

## 1) Can two different forces, acting through the same point, produce the same torque on an object? Answer: Yes, as long as the component of the force perpendicular to the line joining the axis to the force is the same for both forces. 2) If you stand with your back towards a wall and your heels touching the wall, you cannot lean over to touch your toes. Why? Answer: As you bend over your center of gravity moves forward and eventually is beyond the area of the floor in touch with your feet. This does not happen when you do it away from the wall because part of your body moves back and the center of mass remains over your feet. 3) Two equal forces are applied to a door at the doorknob. The first force is applied perpendicular to the door; the second force is applied at 30° to the plane of the door. Which force exerts the greater torque? A) the first applied perpendicular to the door B) the second applied at an angle C) both exert equal non-zero torques D) both exert zero torques E) Additional information is needed. 4) A heavy boy and a lightweight girl are balanced on a massless seesaw. If they both move forward so that they are one-half their original distance from the pivot point, what will happen to the seesaw? A) It is impossible to say without knowing the masses. B) It is impossible to say without knowing the distances. C) The side the boy is sitting on will tilt downward. D) Nothing, the seesaw will still be balanced. E) The side the girl is sitting on will tilt downward. 5) A figure skater is spinning slowly with arms outstretched. She brings her arms in close to her body and her angular speed increases dramatically. The speed increase is a demonstration of A) conservation of energy: her moment of inertia is decreased, and so her angular speed must increase to conserve energy. B) conservation of angular momentum: her moment of inertia is decreased, and so her angular speed must increase to conserve angular momentum. C) Newton’s second law for rotational motion: she exerts a torque and so her angular speed increases. D) This has nothing to do with mechanics, it is simply a result of her natural ability to perform. 6) A girl weighing 450. N sits on one end of a seesaw that is 3.0 m long and is pivoted 1.3 m from the girl. If the seesaw is just balanced when a boy sits at the opposite end, what is his weight? Neglect the weight of the seesaw. 7) An 82.0 kg painter stands on a long horizontal board 1.55 m from one end. The 15.5 kg board is 5.50 m long. The board is supported at each end. (a) What is the total force provided by both supports? (b) With what force does the support, closest to the painter, push upward? FIGURE 11-4 8) The mobile shown in Figure 11-4 is perfectly balanced. What must be the masses of m1, m2, and m3?

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WEEKLY ASSIGNMENT #5 (WOW THAT TOOK A WHILE) NAME: 1. Find the linear approximation of the function f(x; y; z) = p x2 + y2 + z2 at some point to approximate a value of the number p (3:02)2 + (1:97)2 + (5:99)2. 1 2. Consider your favorite function, the Cobb-Douglas production function. P(L;K) = 1:5L:65K:35 modeling the production of the state of Idaho. Over time we discover that capitol is gradually increasing at an approximate rate of 0:02 units per year. If we decide as a group that we are perfectly happy with our production level and would rather have additional vacation time then how much can we decrease labor by each year and keep the same level of production. In how long(rounded up to the nearest year) will we have an additional week of vacation? 2 3. Use the chain rule to find dz dt or dw=dt (a) z = x?y x+2y x = et; y = e?t. (b) w = sin x cos x x = p t; y = 1=t. 4. Use the chain rule to find @z=@t or @z=@s (a) z = (x ? y)5 x = s2t; y = st2 (b) z = er cos  r = st;  = p x2 + y2. 3 5. The temperature at a point (x; y; z) is given by the function T(x; y; z) = 200e?x2?3y2?9z2 where T is measure in C and x; y; z in meters. (a) Find the rate of change of temperature at the point (2;?1; 2) in the direction toward the point (3;?3; 3). (b) In which direction does the temperature increase fastest, and what is that fastest rate? 4 6. Suppose (1; 1) is a critical point of a function f with continuous second derivatives. In each case, what can you say about f. (a) fxx(1; 1) = 4; fxy(1; 1) = 1; fyy(1; 1) = 2 (b) fxx(1; 1) = 4; fxy(1; 1) = 3; fyy(1; 1) = 2 (c) fxx(1; 1) = ?1; fxy(1; 1) = 6; fyy(1; 1) = 1 (d) fxx(1; 1) = ?1; fxy(1; 1) = 2; fyy(1; 1) = ?8 (e) fxx(1; 1) = 4; fxy(1; 1) = 6; fyy(1; 1) = 9 5 Bonus Show that f(x; y) = x2 + 4y2 ? 4xy + 2 has an infinite number of critical points, and for all of them D = 0 at each one. Then show that f has a local (and absolute) minimum at each critical point. 6