# Daily Archives: August 12, 2019

1271 posts ## Puzzle Problem – Bridge Crossing at Night

Thanks to all of you who support me on Patreon. You da real mvps! \$1 per month helps!! 🙂 https://www.patreon.com/patrickjmt !! Puzzle Problem – Bridge Crossing at Night ## Lesson 32 – The set is a basis for Let is a linear transformation such that and

Leave a tip for good service: https://paypal.me/jjthetutor Let b_1=(1,1) and b_2=(1, 0). The set is a basis for. Let is a linear transformation such that. Then the matrix of relative to the basis is and the matrix of relative to the standard basis for Student Solution Manuals: https://amzn.to/2WZrFnD More help via http://jjthetutor.com Download my eBooks…

## 11.91 … CP An angler hangs a 4.50-kg fish from a vertical steel wire 1.50 m long and 5.00 * 10-3 cm2 in cross-sectional area. The upper end of the wire is securely fastened to a support. (a) Calculate the amount the wire is stretched by the hanging fish. The angler now applies a varying force F S at the lower end of the wire pulling it very slowly downward by 0.500 mm from its equilibrium position. For this downward motion calculate (b) the work done by gravity; (c) the work done by the force F S (d) the work done by the force the wire exerts on the fish; and (e) the change in the elastic potential energy (the potential energy associated with the tensile stress in the wire). Compare the answers in parts (d) and (e).

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## 11.90 … Knocking Over a Post. One end of a post weighing 400 N and with height h rests on a rough horizontal surface with ms = 0.30. The upper end is held by a rope fastened to the surface and making an angle of 36.9_ with the post (Fig. P11.90). A horizontal force F S is exerted on the post as shown. (a) If the force F S is applied at the midpoint of the post what is the largest value it can have without causing the post to slip? (b) How large can the force be without causing the post to slip if its point of application is 6 10 of the way from the ground to the top of the post? (c) Show that if the point of application of the force is too high the post cannot be made to slip no matter how great the force. Find the critical height for the point of application.

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## 11.89 … Two ladders 4.00 m and 3.00 m long are hinged at point A and tied together by a horizontal rope 0.90 m above the floor (Fig. P11.89). The ladders weigh 480 N and 360 N respectively and the center of gravity of each is at its center. Assume that the floor is freshly waxed and frictionless. (a) Find the upward force at the bottom of each ladder. (b) Find the tension in the rope. (c) Find the magnitude of the force one ladder exerts on the other at point A. (d) If an 800-N painter stands at point A find the tension in the horizontal rope.

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## 11.88 … DATA You are a construction engineer working on the interior design of a retail store in a mall. A 2.00-m-long uniform bar of mass 8.50 kg is to be attached at one end to a wall by means of a hinge that allows the bar to rotate freely with very little friction. The bar will be held in a horizontal position by a light cable from a point on the bar (a distance x from the hinge) to a point on the wall above the hinge. The cable makes an angle u with the bar. The architect has proposed four possible ways to connect the cable and asked you to assess them: (a) There is concern about the strength of the cable that will be required. Which set of x and u values in the table produces the smallest tension in the cable? The greatest? (b) There is concern about the breaking strength of the sheetrock wall where the hinge will be attached. Which set of x and u values produces the smallest horizontal component of the force the bar exerts on the hinge? The largest? (c) There is also concern about the required strength of the hinge and the strength of its attachment to the wall. Which set of x and u values produces the smallest magnitude of the vertical component of the force the bar exerts on the hinge? The largest? (Hint: Does the direction of the vertical component of the force the hinge exerts on the bar depend on where along the bar the cable is attached?) (d) Is one of the alternatives given in the table preferable? Should any of the alternatives be avoided? Discuss.

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## 11.87 .. DATA You need to measure the mass M of a 4.00-mlong bar. The bar has a square cross section but has some holes drilled along its length so you suspect that its center of gravity isn’t in the middle of the bar. The bar is too long for you to weigh on your scale. So first you balance the bar on a knife-edge pivot and determine that the bar’s center of gravity is 1.88 m from its left-hand end. You then place the bar on the pivot so that the point of support is 1.50 m from the left-hand end of the bar. Next you suspend a 2.00-kg mass 1m12 from the bar at a point 0.200 m from the left-hand end. Finally you suspend a mass m2 = 1.00 kg from the bar at a distance x from the left-hand end and adjust x so that the bar is balanced. You repeat this step for other values of m2 and record each corresponding value of x. The table gives your results. (a) Draw a free-body diagram for the bar when m1 and m2 are suspended from it. (b) Apply the static equilibrium equation gtz = 0 with the axis at the location of the knife-edge pivot. Solve the equation for x as a function of m2. (c) Plot x versus 1>m2. Use the slope of the best-fit straight line and the equation you derived in part (b) to calculate that bar’s mass M. Use g = 9.80 m>s2. (d) What is the y-intercept of the straight line that fits the data? Explain why it has this value.

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## 11.86 .. DATA You are to use a long thin wire to build a pendulum in a science museum. The wire has an unstretched length of 22.0 m and a circular cross section of diameter 0.860 mm; it is made of an alloy that has a large breaking stress. One end of the wire will be attached to the ceiling and a 9.50-kg metal sphere will be attached to the other end. As the pendulum swings back and forth the wire’s maximum angular displacement from the vertical will be 36.0_. You must determine the maximum amount the wire will stretch during this motion. So before you attach the metal sphere you suspend a test mass (mass m) from the wire’s lower end. You then measure the increase in length _l of the wire for several different test masses. Figure P11.86 a graph of _l versus m shows the results and the straight line that gives the best fit to the data. The equation for this line is _l = 10.422 mm>kg2m. (a) Assume that g = 9.80 m>s2 and use Fig. P11.86 to calculate Young’s modulus Y for this wire. (b) You remove the test masses attach the 9.50-kg sphere and release the sphere from rest with the wire displaced by 36.0_. Calculate the amount the wire will stretch as it swings through the vertical. Ignore air resistance.

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## 11.85 … CP BIO Stress on the Shin Bone. The compressive strength of our bones is important in everyday life. Young’s modulus for bone is about 1.4 * 1010 Pa. Bone can take only about a 1.0% change in its length before fracturing. (a) What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 3.0 cm2 ? (This is approximately the crosssectional area of a tibia or shin bone at its narrowest point.) (b) Estimate the maximum height from which a 70-kg man could jump and not fracture his tibia. Take the time between when he first touches the floor and when he has stopped to be 0.030 s and assume that the stress on his two legs is distributed equally.

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## 11.84 … CP An amusement park ride consists of airplane-shaped cars attached to steel rods (Fig. P11.84). Each rod has a length of 15.0 m and a cross-sectional area of 8.00 cm2. (a) How much is each rod stretched when it is vertical and the ride is at rest? (Assume that each car plus two people seated in it has a total weight of 1900 N.) (b) When operating the ride has a maximum angular speed of 12.0 rev>min. How much is the rod stretched then?

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## 11.83 … A 1.05-m-long rod of negligible weight is supported at its ends by wires A and B of equal length (Fig. P11.83). The cross-sectional area of A is 2.00 mm2 and that of B is 4.00 mm2. Young’s modulus for wire A is 1.80 * 1011 Pa; that for B is 1.20 * 1011 Pa. At what point along the rod should a weight w be suspended to produce (a) equal stresses in A and B and (b) equal strains in A and B?

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## 11.82 .. Hooke’s Law for a Wire. A wire of length l0 and cross-sectional area A supports a hanging weight W. (a) Show that if the wire obeys Eq. (11.7) it behaves like a spring of force constant AY>l0 where Y is Young’s modulus for the wire material. (b) What would the force constant be for a 75.0-cm length of 16-gauge 1diameter = 1.291 mm2 copper wire? See Table 11.1. (c) What would W have to be to stretch the wire in part (b) by 1.25 mm?

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## 11.81 … CP A 12.0-kg mass fastened to the end of an aluminum wire with an unstretched length of 0.70 m is whirled in a vertical circle with a constant angular speed of 120 rev>min. The cross-sectional area of the wire is 0.014 cm2. Calculate the elongation of the wire when the mass is (a) at the lowest point of the path and (b) at the highest point of its path.

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## 11.80 … Pyramid Builders. Ancient pyramid builders are balancing a uniform rectangular slab of stone tipped at an angle u above the horizontal using a rope (Fig. P11.80). The rope is held by five workers who share the force equally. (a) If u = 20.0_ what force does each worker exert on the rope? (b) As u increases does each worker have to exert more or less force than in part (a) assuming they do not change the angle of the rope? Why? (c) At what angle do the workers need to exert no force to balance the slab? What happens if u exceeds this value?

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## 11.79 .. A garage door is mounted on an overhead rail (Fig. P11.79). The wheels at A and B have rusted so that they do not roll but rather slide along the track. The coefficient of kinetic friction is 0.52. The distance between the wheels is 2.00 m and each is 0.50 m from the vertical sides of the door. The door is uniform and weighs 950 N. It is pushed to the left at constant speed by a horizontal force F S . (a) If the distance h is 1.60 m what is the vertical component of the force exerted on each wheel by the track? (b) Find the maximum value h can have without causing one wheel to leave the track.

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## 11.78 . A weight W is supported by attaching it to a vertical uniform metal pole by a thin cord passing over a pulley having negligible mass and friction. The cord is attached to the pole 40.0 cm below the top and pulls horizontally on it (Fig. P11.78). The pole is pivoted about a hinge at its base is 1.75 m tall and weighs 55.0 N. A thin wire connects the top of the pole to a vertical wall. The nail that holds this wire to the wall will pull out if an outward force greater than 22.0 N acts on it. (a) What is the greatest weight W that can be supported this way without pulling out the nail? (b) What is the magnitude of the force that the hinge exerts on the pole?

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## 11.77 . An engineer is designing a conveyor system for loading hay bales into a wagon (Fig. P11.77). Each bale is 0.25 m wide 0.50 m high and 0.80 m long (the dimension perpendicular to the plane of the figure) with mass 30.0 kg. The center of gravity of each bale is at its geometrical center. The coefficient of static friction between a bale and the conveyor belt is 0.60 and the belt moves with constant speed. (a) The angle b of the conveyor is slowly increased. At some critical angle a bale will tip (if it doesn’t slip first) and at some different critical angle it will slip (if it doesn’t tip first). Find the two critical angles and determine which happens at the smaller angle. (b) Would the outcome of part (a) be different if the coefficient of friction were 0.40?

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## 11.76 .. Two identical uniform beams weighing 260 N each are connected at one end by a frictionless hinge. A light horizontal crossbar attached at the midpoints of the beams maintains an angle of 53.0° between the beams. The beams are suspended from the ceiling by vertical wires such that they form a “V” (Fig. P11.76). (a) What force does the crossbar exert on each beam? (b) Is the crossbar under tension or compression? (c) What force (magnitude and direction) does the hinge at point A exert on each beam?

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## 11.75 … Two uniform 75.0-g marbles 2.00 cm in diameter are stacked as shown in Fig. P11.75 in a container that is 3.00 cm wide. (a) Find the force that the container exerts on the marbles at the points of contact A B and C. (b) What force does each marble exert on the other?

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## 11.74 . If you put a uniform block at the edge of a table the center of the block must be over the table for the block not to fall off. (a) If you stack two identical blocks at the table edge the center of the top block must be over the bottom block and the center of gravity of the two blocks together must be over the table. In terms of the length L of each block what is the maximum overhang possible (Fig. P11.74)? (b) Repeat part (a) for three identical blocks and for four identical blocks. (c) Is it possible to make a stack of blocks such that the uppermost block is not directly over the table at all? How many blocks would it take to do this? (Try.)

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## 11.73 . The Farmyard Gate. A gate 4.00 m wide and 2.00 m high weighs 700 N. Its center of gravity is at its center and it is hinged at A and B. To relieve the strain on the top hinge a wire CD is connected as shown in Fig. P11.73. The tension in CD is increased until the horizontal force at hinge A is zero. What is (a) the tension in the wire CD; (b) the magnitude of the horizontal component of the force at hinge B; (c) the combined vertical force exerted by hinges A and B?

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## 11.72 .. You are trying to raise a bicycle wheel of mass m and radius R up over a curb of height h. To do this you apply a horizontal force F S (Fig. P11.72). What is the smallest magnitude of the force F S that will succeed in raising the wheel onto the curb when the force is applied (a) at the center of the wheel and (b) at the top of the wheel? (c) In which case is less force required?

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## 11.71 .. You are a summer intern for an architectural firm. An 8.00-m-long uniform steel rod is to be attached to a wall by a frictionless hinge at one end. The rod is to be held at 22.0_ below the horizontal by a light cable that is attached to the end of the rod opposite the hinge. The cable makes an angle of 30.0_ with the rod and is attached to the wall at a point above the hinge. The cable will break if its tension exceeds 650 N. (a) For what mass of the rod will the cable break? (b) If the rod has a mass that is 10.0 kg less than the value calculated in part (a) what are the magnitude and direction of the force that the hinge exerts on the rod?

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## 11.70 … In a city park a nonuniform wooden beam 4.00 m long is suspended horizontally by a light steel cable at each end. The cable at the left-hand end makes an angle of 30.0_ with the vertical and has tension 620 N. The cable at the right-hand end of the beam makes an angle of 50.0_ with the vertical. As an employee of the Parks and Recreation Department you are asked to find the weight of the beam and the location of its center of gravity.

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## 11.69 .. BIO CALC Refer to the discussion of holding a dumbbell in Example 11.4 (Section 11.3). The maximum weight that can be held in this way is limited by the maximum allowable tendon tension T (determined by the strength of the tendons) and by the distance D from the elbow to where the tendon attaches to the forearm. (a) Let Tmax represent the maximum value of the tendon tension. Use the results of Example 11.4 to express wmax (the maximum weight that can be held) in terms of Tmax L D and h. Your expression should not include the angle u. (b) The tendons of different primates are attached to the forearm at different values of D. Calculate the derivative of wmax with respect to D and determine whether the derivative is positive or negative. (c) A chimpanzee tendon is attached to the forearm at a point farther from the elbow than for humans. Use this to explain why chimpanzees have stronger arms than humans. (The disadvantage is that chimpanzees have less flexible arms than do humans.)

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## 11.68 .. BIO Forearm. In the human arm the forearm and hand pivot about the elbow joint. Consider a simplified model in which the biceps muscle is attached to the forearm 3.80 cm from the elbow joint. Assume that the person’s hand and forearm together weigh 15.0 N and that their center of gravity is 15.0 cm from the elbow (not quite halfway to the hand). The forearm is held horizontally at a right angle to the upper arm with the biceps muscle exerting its force perpendicular to the forearm. (a) Draw a free-body diagram for the forearm and find the force exerted by the biceps when the hand is empty. (b) Now the person holds an 80.0-N weight in his hand with the forearm still horizontal. Assume that the center of gravity of this weight is 33.0 cm from the elbow. Draw a free-body diagram for the forearm and find the force now exerted by the biceps. Explain why the biceps muscle needs to be very strong. (c) Under the conditions of part (b) find the magnitude and direction of the force that the elbow joint exerts on the forearm. (d) While holding the 80.0-N weight the person raises his forearm until it is at an angle of 53.0_ above the horizontal. If the biceps muscle continues to exert its force perpendicular to the forearm what is this force now? Has the force increased or decreased from its value in part (b)? Explain why this is so and test your answer by doing this with your own arm.

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## 11.67 .. Two friends are carrying a 200-kg crate up a flight of stairs. The crate is 1.25 m long and 0.500 m high and its center of gravity is at its center. The stairs make a 45.0_ angle with respect to the floor. The crate also is carried at a 45.0_ angle so that its bottom side is parallel to the slope of the stairs (Fig. P11.67). If the force each person applies is vertical what is the magnitude of each of these forces? Is it better to be the person above or below on the stairs?

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