Solved: A 60-kg uniform board 2.4 m long is supported by a pivot 80 cm from the left end

Give an example of an object on which the net force is zero, but that isnt in static equilibrium.

Give an example of an object on which the net torque about the center of gravity is zero, but that isnt in static equilibrium.

The best way to lift a heavy weight is to squat with your back vertical, rather than to lean over. Why?

Pregnant women often assume a posture with their shoulders held far back from their normal position. Why?

When you carry a bucket of water with one hand, you instinctively extend your opposite arm. Why?

Is a ladder more likely to slip when you stand near the top or the bottom? Explain.

How does a heavy keel help keep a boat from tipping over?

Does choosing a pivot point in an equilibrium problem mean that something is necessarily going to rotate about that point?

If you take the pivot point at the application point of one force in a static-equilibrium problem, that force doesnt enter the torque equation. Does that make the force irrelevant to the problem? Explain.

A short dog and a tall person are standing on a slope. If the slope angle increases, which will fall over first? Why?

A stiltwalker is standing motionless on one stilt. What can you say about the location of the stiltwalkers center of mass?

A body is subject to three forces: F S 1 = 1ni + 2nj N, applied at the point x = 2 m, y = 0 m; F S 2 = -2ni - 5nj N, applied at x = -1 m, y = 1 m; and F S 3 = 1ni + 3nj N, applied at x = -2 m, y = 5 m. Show that (a) the net force and (b) the net torque about the origin are both zero.

To demonstrate that the choice of pivot point doesnt matter, show that the torques in Exercise 12 sum to zero when evaluated about the points (3 m, 2 m) and (-7 m, 1 m).

In Fig. 12.11 the forces shown all have the same magnitude F. For each case shown, is it possible to place a third force so as to meet both conditions for static equilibrium? If so, specify the force and a suitable application point. If not, why not? - F2 S F2 S F1 S F1 S (a) (b) 2 1 -1 -2 x (m) x (m) -2 -1 1 2 -2 -1 1 2 1 y (m) y (m) 2 -1 -2 Figure 12.11 Exercise 14

Figure 12.12a shows a thin, uniform square plate of mass m and side L. The plate is in a vertical plane. Find the magnitude of the gravitational torque on the plate about each of the three points shown. (a) (b) A C B A B C Figure 12.12 Exercises 15 and 16

Repeat the preceding problem for the equilateral triangle in Fig. 12.12b, which has side L.

A 23-m-long log of irregular cross section lies horizontally, supported by a wall at one end and a cable attached 4.0 m from the other end, as shown in Fig. 12.13. The log weighs 7.5 kN and the tension in the cable is 6.2 kN. Find the logs center of gravity. 4.0 m 23 m Figure 12.13 Exercise 17

A 60-kg uniform board 2.4 m long is supported by a pivot 80 cm from the left end and by a scale at the right end (Fig. 12.14). How far from the left end should a 40-kg child sit for the scale to read zero? Figure 12.14 Exercises 18 and 19

Where should the child in Fig. 12.14 sit if the scale is to read (a) 100 N and (b) 300 N?

A 4.2-m-long beam is supported by a cable at its center. A 65-kg steelworker stands at one end of the beam. Where should a 190-kg bucket of concrete be suspended for the beam to be in static equilibrium?

Figure 12.15 shows how a scale with a capacity of only 250 N can be used to weigh a heavier person. The 3.4-kg board is 3.0 m long and has uniform density. Its free to pivot about the end farthest from the scale. Assume that the beam remains essentially horizontal. Whats the weight of a person standing 1.2 m from the pivot end if the scale reads 210 N? 0 250 N Board 1.2 m 3.0 m Figure 12.15 Exercise 21

A portion of a roller-coaster track is described by the equation h = 0.94x - 0.010x 2 , where h and x are the height and horizontal position in meters. (a) Find a point where the roller-coaster car could be in static equilibrium on this track. (b) Is this equilibrium stable or unstable?

The potential energy associated with a particle at position x is given by U = 2x 3 - 2x 2 - 7x + 10, with x in meters and U in joules. Find the positions of any stable and unstable equilibria.

Youre a highway safety engineer, and youre asked to specify bolt sizes so the traffic signal in Fig. 12.16 wont fall over. The figure indicates the masses and positions of the structures various parts. The structure is mounted with two bolts, located symmetrically about the vertical members centerline, as shown. What tension force must the left-hand bolt be capable of withstanding? 8.22 m 3.60 m 76.0 cm 64.7 kg 321 kg Bolts 175 kg CM Figure 12.16 Problem 24

Figure 12.17a shows an outstretched arm with mass 4.2 kg. The arm is 56 cm long, and its center of gravity is 21 cm from the shoulder. The hand at the end of the arm holds a 6.0-kg mass. (a) Find the torque about the shoulder due to the weight of the arm and the 6.0-kg mass. (b) If the arm is held in equilibrium by the deltoid muscle, whose force on the arm acts below the horizontal at a point 18 cm from the shoulder joint (Fig. 12.17b), whats the force exerted by the muscle?

A uniform sphere of radius R is supported by a rope attached to a vertical wall, as shown in Fig. 12.18. The rope joins the sphere at a point where a continuation of the rope would intersect a horizontal line through the spheres center a distance 1 2 R beyond the center, as shown. Whats the smallest possible value for the coefficient of friction between wall and sphere?

You work for a garden equipment company, and youre designing a new garden cart. Specifications to be listed include the horizontal force that must be applied to push the fully loaded cart (mass 55 kg, 60-cm-diameter wheels) up an abrupt 8.0-cm step, as shown in Fig. 12.19. What do you specify for the force? F S 60 cm 8.0 cm Figure 12.19 Problem 27

Figure 12.20 shows the foot and lower leg of a person standing on the ball of one foot. Three forces act to maintain this equilibrium: the tension force T S in the Achilles tendon, the contact force F S c at the ankle joint, and the normal force n ! that supports the persons 697-N weight. The application points for these forces are shown in Fig. 12.20. The persons center of gravity is directly above the contact point with the ground, and you can treat the mass of the foot itself as being negligible. Find the magnitudes of (a) the tension in the Achilles tendon and (b) the contact force at the ankle joint. n F u c S T S 11.6 cm 4.55 cm 25 Ankle joint Achilles tendon Figure 12.20 Problem 28

A uniform 5.0-kg ladder is leaning against a frictionless vertical wall, with which it makes a 15 angle. The coefficient of friction between ladder and ground is 0.26. Can a 65-kg person climb to the top of the ladder without it slipping? If not, how high can that person climb? If so, how massive a person would make the ladder slip?

The boom in the crane of Fig. 12.21 is free to pivot about point P and is supported by the cable attached halfway along its 18-m length. The cable passes over a pulley and is anchored at the back of the crane. The boom has mass 1700 kg distributed uniformly along its length, and the mass hanging from the boom is 2200 kg. The boom makes a 50 angle with the horizontal. Find the tension in the cable. m 50 P 18 m Figure 12.21 Problem 30

A uniform board of length L and weight W is suspended between two vertical walls by ropes of length L/2 each. When a weight w is placed on the left end of the board, it assumes the configuration shown in Fig. 12.22. Find the weight w in terms of the board weight W. L 2 35 9.2 L 60 w L 2 Figure 12.22 Problem 31

Figure 12.23 shows a 1250-kg car that has slipped over an embankment. People are trying to hold the car in place by pulling on a horizontal rope. The cars bottom is pivoted on the edge of the embankment, and its center of mass lies farther back, as shown. If the car makes a 34 angle with the horizontal, what force must the people apply to hold it in place? 34 1.8 m 2.4 m CM Figure 12.23 Problem 32

Repeat Example 12.2, now assuming that the coefficient of friction at the ground is m1 and at the wall is m2. Show that the minimum angle at which the ladder wont slip is now given by f = tan-1 3(1 - m1m2)/2m14.

You are headwaiter at a new restaurant, and your boss asks you to hang a sign for her. Youre to hang the sign, whose mass is 66 kg, in the configuration shown in Fig. 12.24. A uniform horizontal rod of mass 8.2 kg and length 2.3 m holds the sign. At one end the rod is attached to the wall by a pivot; at the other end its supported by a cable that can withstand a maximum tension of 800 N. Youre to determine the minimum height h above the pivot for anchoring the cable to the wall. 2.3 m h Figure 12.24 Problem 34

Climbers attempting to cross a stream place a 340-kg log against a vertical, frictionless ice cliff on the opposite side (Fig. 12.25). The log slopes up at 27 and its center of gravity is one-third of the way along its 6.3-m length. If the coefficient of friction between the left end of the log and the ground is 0.92, whats the maximum mass for a climber and pack to cross without the log slipping? 27 CG Figure 12.25 Problem 35

A crane in a marble quarry is mounted on the quarrys rock walls and is supporting a 2500-kg marble slab as shown in Fig. 12.26. The center of mass of the 830-kg boom is located one-third of the way from the pivot end of its 15-m length, as shown. Find the tension in the horizontal cable that supports the boom. 50 CM Figure 12.26 Problem 36

A rectangular block measures w * w * L, where L is the longer dimension. Its on a horizontal surface, resting on its long side. Use geometrical arguments to find an expression for the angle through you would have to tilt it in order to put it in an unstable equilibrium, resting on a short edge.

The potential energy as a function of position for a particle is given by U1x2 = U0a x 3 x 3 0 + a x 2 x 2 0 + 4 x x0 b where x0 and a are constants. For what values of a will there be two static equilibria? Comment on the stability of these equilibria.

A rectangular block of mass m measures w * w * L, where L is the longer dimension. Its on a horizontal surface, resting on its long side, as in the left-hand block in Fig. 12.8. (a) Taking the zero of potential energy when the block is lying on its long side, find an expression for its potential energy as a function of the angle u that the long dimension of the block makes with the horizontal, starting with u = 0 in the left-hand configuration of Fig. 12.8 and continuing through the upright position shown at the right (u = 90). (b) Use calculus to find the angle u where your function has a maximum, and check that it agrees with the answer to Problem 37. (c) Use calculus to show that this is a point of unstable equilibrium.

A 160-kg highway sign of uniform density is 2.3 m wide and 1.4 m high. At one side its secured to a pole with a single bolt, mounted a distance d from the top of the sign. The only other place where the sign contacts the pole is at its bottom corner. If the bolt can sustain a horizontal tension of 2.1 kN, whats the maximum permissible value for the distance d?

A 5.0-m-long ladder has mass 9.5 kg and is leaning against a frictionless wall, making a 66 angle with the horizontal. If the coefficient of friction between ladder and ground is 0.42, whats the mass of the heaviest person who can safely ascend to the top of the ladder? (The center of mass of the ladder is at its center.)

To what vertical height on the ladder in Problem 41 could a 95-kg person reach before the ladder starts to slip?

A uniform, solid cube of mass m and side s is in stable equilibrium when sitting on a level tabletop. How much energy is required to bring it to an unstable equilibrium where its resting on its corner?

An isosceles triangular block of mass m and height h is in stable equilibrium, resting on its base on a horizontal surface. How much energy does it take to bring it to unstable equilibrium, resting on its apex?

Youre investigating ladder safety for the Consumer Product Safety Commission. Your test case is a uniform ladder of mass m leaning against a frictionless vertical wall with which it makes an angle u. The coefficient of static friction at the floor is m. Your job is to find an expression for the maximum mass of a person who can climb to the top of the ladder without its slipping. With that result, youre to show that anyone can climb to the top if m tan u but that no one can if m 6 1 2 tan u.

A 2.0-m-long rod has density l in kilograms per meter of length described by l = a + bx, where a = 1.0 kg/m, b = 1.0 kg/m2 , and x is the distance from the left end of the rod. The rod rests horizontally with each end supported by a scale. What do the two scales read?

What horizontal force applied at its highest point is necessary to keep a wheel of mass M from rolling down a slope inclined at angle u to the horizontal?

A rectangular block twice as high as it is wide is resting on a board. The coefficient of static friction between board and incline is 0.63. If the boards inclination angle u (shown in Fig. 12.27) is gradually increased, will the block first tip over or first begin sliding? 1 2 1 3 u h h h Figure 12.27 Problems 48, 49, and 50

What condition on the coefficient of friction in Problem 48 will cause the block to slide before it tips?

A uniform solid cone of height h and base diameter 1 3 h sits on the board of Fig. 12.27. The coefficient of static friction between the cone and incline is 0.63. As the slope of the board is increased, will the cone first tip over or first begin sliding? (Hint: Start with an integration to find the center of mass.)

Prove the statement in Section 12.1 that the choice of pivot point doesnt matter when applying conditions for static equilibrium. Figure 12.28 shows an object on which the net force is assumed to be zero. The net torque about the point O is also zero. Show that the net torque about any other point P is also zero. To do so, write the net torque about P as t ! P = a r ! Pi * F S i , where the vectors r ! P go from P to the force-application points, and the index i labels the different forces. In Fig. 12.28, note that r ! Pi = r ! Oi + R S , where R S is a vector from P to O. Use this result in your expression for t ! P and apply the distributive law to get two separate sums. Use the assumptions that F S net = 0 ! and t ! O = 0 ! to argue that both terms are zero. This completes the proof. rP2 u rO2 u R S F3 S F2 S F1 S O P Figure 12.28 Problem 51

Three identical books of length L are stacked over the edge of a table as shown in Fig. 12.29. The top book overhangs the middle one by L/2, so it just barely avoids falling. The middle book overhangs the bottom one by L/4. How much of the bottom book can overhang the edge of the table without the books falling? 1 2 1 4 L L L ? Figure 12.29 Problem 52

A uniform pole of mass M is at rest on an incline of angle u secured by a horizontal rope as shown in Fig. 12.30. Find the minimum frictional coefficient that will keep the pole from slipping. u M Figure 12.30 Problems 53 and 54

For what angle does the situation in Problem 53 require the greatest coefficient of friction?

Figure 12.31 shows a popular system for mounting bookshelves. An aluminum bracket is mounted on a vertical aluminum support by small tabs inserted into vertical slots. Contact between the bracket and support occurs only at the upper tab and at the bottom of the bracket, 4.5 cm below the upper tab. If each bracket in the shelf system supports 32 kg of books, with the center of gravity 12 cm out from the vertical support, what is the horizontal component of the force exerted on the upper bracket tab? Fg S Shelf Bracket 4.5 cm 12 cm PHYSICS Figure 12.31 Problem 55

The nuchal ligament is a thick, cordlike structure that supports the head and neck in animals like horses. Figure 12.32 shows the nuchal ligament and its attachment points on a horses skeleton, along with an approximation to the spine as a rigid rod. Centers of mass of head and neck are also shown. If the masses of head and neck are 29 kg and 68 kg, respectively, whats the tension in the nuchal ligament? (Note: Your answer will be an overestimate because muscles also provide support.)

A 4.2-kg plant hangs from the bracket shown in Fig. 12.33. The brackets mass is 0.85 kg, and its center of mass lies 9.0 cm from the wall. A single screw holds the bracket to the wall, as shown. Find the horizontal tension in the screw. (Hint: Imagine that the bracket is slightly loose and pivoting about its bottom end. Assume the wall is frictionless.) CM 28 cm 9 cm 7.2 cm Figure 12.33 Problem 57

The wheel in Fig. 12.34 has mass M and is weighted with an additional mass m as shown. The coefficient of friction is sufficient to keep the wheel from sliding; however, it might still roll. Show that it wont roll only if m 7 M sin u 1 - sin u . u m M Figure 12.34 Problem 58

An interstellar spacecraft from an advanced civilization is hovering above Earth, as shown in Fig. 12.35. The ship consists of two pods of mass m separated by a rigid shaft of negligible mass and one Earth radius (RE) long. Find (a) the magnitude and direction of the net gravitational force on the ship and (b) the net torque about the center of mass. (c) Show that the ships center of gravity is displaced approximately 0.083RE from its center of mass. m m RE RE 2RE Earth Figure 12.35 Problem 59

Youre called to testify in a product liability lawsuit. An infant sitting in the portable seat shown in Fig. 12.36 was injured when it fell to the floor. The manufacturer claims the child was too heavy for the seat; the parents claim the seat was defective. Tests show that the seat can safely hold a child if the force F S of the table on the chair doesnt exceed 229 N. The seats mass is 2.0 kg, the injured childs is 10 kg, and the center of mass of child and seat was 16 cm from the table edge. In whose favor should the jury rule? F S Tabletop 22 cm 16 cm Figure 12.36 Problem 60

Youre designing a vacation cabin at a ski resort. The cabin has a cathedral ceiling as shown in Fig. 12.37, and you estimate that each roof rafter needs to support up to 170 kg of snow and building materials. The horizontal tie beam near the roof peak can withstand a 7.5-kN force. You can neglect any horizontal force from the vertical walls, and treat contact forces as concentrated at the roof peak and the outside edge of the rafter/wall junctions. Will the tie beam hold? Will it be in tension or compression? 0.8 m 3.2 m Tie beam Roof rafter 9.6 m Wall Figure 12.37 Problem 61

Youll need to study the Application on page 209 to do this problem. An SUV without ECS has SSF = 1.12 with its two passengers on board. (a) Can it successfully negotiate an 85-m-radius turn on a flat road, going at the speed limit of 100 km/h? (b) With its passengers, the SUVs total mass is 1940 kg, and the left-toright spacing between its tires is 1.71 m. If a 315-kg load of cargo is secured to the roof, with its center of gravity 2.1 m above the road, whats the maximum safe speed on the same road?

Engineers designing a new semiconductor device measure the potential energy that results when they move an electron to different positions within their device. The device is one-dimensional, so the positions all lie along a line. The table below gives the resulting data. Plot these data and from your plot, determine the approximate positions of any equilibria and whether such equilibria are stable or unstable.

Youve been hired by your states environmental agency to monitor carbon dioxide levels just above rivers, with the goal of understanding whether river water acts as a source or sink of CO2. Youve constructed the apparatus shown in Fig. 12.38, consisting of a boom mounted on a pivot, a vertical support, and a rope with pulley for raising and lowering the boom so its end can extend different distances over the river. In addition, theres a separate rope and pulley for dropping the sampling apparatus so its just above the river. Boom rope Sampling rope Pulley Sampling apparatus Boom Pulley Pivot Figure 12.38 Passage Problems 6467 When the boom rope is horizontal, it cant exert any vertical force. Therefore, a. its impossible to hold the boom with the boom rope horizontal. b. the boom rope tension becomes infinite. c. the pivot supplies the necessary vertical force. d. the boom rope exerts no torque.

Youve been hired by your states environmental agency to monitor carbon dioxide levels just above rivers, with the goal of understanding whether river water acts as a source or sink of CO2. Youve constructed the apparatus shown in Fig. 12.38, consisting of a boom mounted on a pivot, a vertical support, and a rope with pulley for raising and lowering the boom so its end can extend different distances over the river. In addition, theres a separate rope and pulley for dropping the sampling apparatus so its just above the river. Boom rope Sampling rope Pulley Sampling apparatus Boom Pulley Pivot Figure 12.38 Passage Problems 6467 The tension in the boom rope will be greatest when a. the boom is horizontal. b. the boom rope is horizontal. c. the boom is vertical. d. in some orientation other than (a), (b), or (c).

Youve been hired by your states environmental agency to monitor carbon dioxide levels just above rivers, with the goal of understanding whether river water acts as a source or sink of CO2. Youve constructed the apparatus shown in Fig. 12.38, consisting of a boom mounted on a pivot, a vertical support, and a rope with pulley for raising and lowering the boom so its end can extend different distances over the river. In addition, theres a separate rope and pulley for dropping the sampling apparatus so its just above the river. Boom rope Sampling rope Pulley Sampling apparatus Boom Pulley Pivot Figure 12.38 Passage Problems 6467 If you secure the boom at a fixed angle and lower the sampling apparatus at constant speed, the boom rope tension will a. increase. b. decrease. c. remain the same. d. increase only if the sampling apparatus is more massive than the boom.

Youve been hired by your states environmental agency to monitor carbon dioxide levels just above rivers, with the goal of understanding whether river water acts as a source or sink of CO2. Youve constructed the apparatus shown in Fig. 12.38, consisting of a boom mounted on a pivot, a vertical support, and a rope with pulley for raising and lowering the boom so its end can extend different distances over the river. In addition, theres a separate rope and pulley for dropping the sampling apparatus so its just above the river. Boom rope Sampling rope Pulley Sampling apparatus Boom Pulley Pivot Figure 12.38 Passage Problems 6467 If you pull the boom rope with constant speed, the angle the boom makes with the horizontal will a. increase at a constant rate. b. increase at an increasing rate. c. increase at a decreasing rate. d. decrease.