A solid cube of edge-length balanced atop a cylinder of

True or false: (a) is sufficient for static equilibrium to exist. (b) is necessary for static equilibrium to exist. (c) In static equilibrium, the net torque about any point is zero. (d) An object in equilibrium cannot be moving.

True or false: (a) The center of gravity is always at the geometric center of a body. (b) The center of gravity must be located inside an object. (c) The center of gravity of a baton is located between the two ends. (d) The torque produced by the force of gravity about the center of gravity is always zero.

The horizontal bar in Figure 12-27 will remain horizontal if (a) and (b) (c) (d) (e) R1L1 _ R2L2L . 1M1 _ L2M2, M2R1 _ M1R2 M1R1 _ M2R2R , 1 _ R2L , 1 _ L2

Sit in a chair with your back straight. Now try to stand up without leaning forward. Explain why you cannot do it.

ENGINEERING APPLICATION You have a job digging holes for posts to support signs for a Louisiana restaurant (called Moscas). Explain why the higher above the ground a sign is mounted, the farther the posts should extend into the ground.

A father (mass ) and his son (mass ) begin walking out toward opposite ends of a balanced see-saw. As they walk, the see-saw stays exactly horizontal. What can be said about the relationship between the fathers speed and the sons speed ?

Travel mugs that people might set on the dashboards of their cars are often made with broad bases and relatively narrow mouths. Why would travel mugs be designed with this shape, rather than have the roughly cylindrical shape that mugs normally have?

ENGINEERING APPLICATION The sailors in the photo are using a technique called hiking out. What purpose does positioning themselves in this way serve? If the wind were stronger, what would they need to do to keep their craft stable?

An aluminum wire and a steel wire of the same length and diameter are joined end-to-end to form a wire of length One end of wire is then fastened to the ceiling and an object of mass is attached to the other end. Neglecting the mass of the wires, which of the following statements is true? (a) The aluminum portion will stretch by the same amount as the steel portion. (b) The tensions in the aluminum portion and the steel portion are equal. (c) The tension in the aluminum portion is greater than that in the steel portion. (d) None of the above SSM

A large crate weighing 4500 N rests on four 12-cm-high blocks on a horizontal surface (Figure 12-28). The crate is 2.0 m long, 1.2 m high, and 1.2 m deep. You are asked to lift one end of the crate using a long steel pry bar. The fulcrum on the pry bar is 10 cm from the end that lifts the crate. Estimate the length of the bar you will need to lift the end of the crate.

ENGINEERING APPLICATION Consider an atomic model for Youngs modulus. Assume that a large number of atoms are arranged in a cubic array, with each atom at a corner of a cube and each atom a distance a from its six nearest neighbors. Imagine that each atom is attached to its six nearest neighbors by little springs each with spring constant (a) Show that this material, if stretched, will have a Youngs modulus (b) Using Table 12-1 and assuming that estimate a typical value for the atomic spring constant in a metal.

By considering the torques about the centers of the ball joints in your shoulders, estimate the force your deltoid muscles (those muscles on top of the shoulder) must exert on your upper arm to keep your arm held out and extended at shoulder level. Then, estimate the force they must exert when you hold a 10-lb weight out to the side at arms length.

Your crutch is pressed against the sidewalk with a force along its own direction, as in Figure 12-29. This force is balanced by the normal force and a frictional force (a) Show that when the force of friction is at its maximum value, the coefficient of friction is related to the angle by (b) Explain how this result applies to the forces on your foot when you are not using a crutch. (c) Explain why it is advantageous to take short steps when walking on slippery surfaces?

A thin uniform rod of mass is suspended horizontally by two vertical wires. One wire is at the left end of the rod, and the other wire is of the length of the rod from the left end. (a) Determine the tension in each wire. (b) An object is now hung by a string attached to the right end of the rod. When this happens, it is noticed that the rod remains horizontal but the tension in the wire on the left vanishes. Determine the mass of the object.

An automobile has 58 percent of its weight on the front wheels. The front and back wheels on each side are separated by 2.0 m. Where is the center of gravity located?

Figure 12-30 shows a lever of negligible mass with a vertical force \(F_\mathrm{app}\) being applied to lift a load F. The mechanical advantage of the lever is defined as \(M=F/F_{\text{app min}}\), where \(F_\text{app min}\) is the smallest force necessary to lift the load F. Show that for this simple lever system, M = x/X where x is the moment arm (distance to the pivot) for the applied force, and X is the moment arm for the load.

ENGINEERING APPLICATION Figure 12-31 shows a 25-foot sailboat. The mast is a uniform 120-kg pole that is supported on the deck and held fore and aft by wires as shown. The tension in the forestay (wire leading to the bow) is 1000 N. Determine the tension in the backstay (wire leading aft) and the normal force that the deck exerts on the mast. (Assume that the frictional force the deck exerts on the mast to be negligible.) SSM

A uniform 10.0-m beam of mass 300 kg extends over a ledge as in Figure 12-32. The beam is not attached, but simply rests on the surface. A 60.0-kg student intends to position the beam so that he can walk to the end of it. What is the maximum distance the beam can extend past end of the ledge and still allow him to perform this feat?

BIOLOGICAL APPLICATION A gravity board is a convenient and quick way to determine the location of the center of gravity of a person. It consists of a horizontal board supported by a fulcrum at one end and a scale at the other end. To demonstrate this in class, your physics professor calls on you to lie horizontally on the board with the top of your head directly above the fulcrum point as shown in Figure 12-33. The fulcrum is 2.00 m from the scale. In preparation for this experiment, you had accurately weighed yourself and determined your mass to be 70.0 kg. When you are at rest on the gravity board, the scale advances 250 N beyond its reading when the board is there by itself. Use this data to determine the location of your center of gravity relative to your feet. SSM

A stationary 3.0-m board of mass 5.0 kg is hinged at one end. A force is applied vertically at the other end, and the board makes a angle with the horizontal. A 60-kg block rests on the board 80 cm from the hinge as shown in Figure 12-34. (a) Find the magnitude of the force . (b) Find the force exerted by the hinge. (c) Find the magnitude of the force as well as the force exerted by the hinge, if is exerted, instead, at right angles to the board.

A cylinder of mass is supported by a frictionless trough formed by a plane inclined at to the horizontal on the left and one inclined at on the right, as shown in Figure 12-35. Find the force exerted by each plane on the cylinder.

A uniform 18-kg door that is 2.0 m high by 0.80 m wide is hung from two hinges that are 20 cm from the top and 20 cm from the bottom. If each hinge supports half the weight of the door, find the magnitude and direction of the horizontal components of the forces exerted by the two hinges on the door.

Find the force exerted on the strut by the hinge at A for the arrangement in Figure 12-36 if (a) the strut is weightless, and (b) the strut weighs 20 N.

Julie has been hired to help paint the trim of a building, but she is not convinced of the safety of the apparatus. A 5.0-m plank is suspended horizontally from the top of the building by ropes attached at each end. Julie knows from previous experience that the ropes being used will break if the tension exceeds 1.0 kN. Her 80-kg boss dismisses Julies worries and begins painting while standing 1.0 m from the end of the plank. If Julies mass is 60 kg and the plank has a mass of 20 kg, over what range of positions can Julie stand to join her boss without causing the ropes to break?

Acylinder of mass and radius rolls against a step of height as shown in Figure 12-37. When a horizontal force of magnitude is applied to the top of the cylinder, the cylinder remains at rest. (a) Find an expression for the normal force exerted by the floor on the cylinder. (b) Find an expression for the horizontal force exerted by the edge of the step on the cylinder. (c) Find an expression for the vertical component of the force exerted by the edge of the step on the cylinder. SSM

For the cylinder in Problem 25, find an expression for the minimum magnitude of the horizontal force that will roll the cylinder over the step if the cylinder does not slide on the edge.

CONTEXT-RICH Figure 12-38 shows a hand holding an epee, a weapon used in the sport of fencing, which you are taking as a physical education elective. The center of mass of your epee is 24 cm from the pommel (the end of the epee at the grip). You have weighed it so you know that the epees mass is 0.700 kg and its full length is 110 cm. (a) At the beginning of a match you hold it straight out in static equilibrium. Find the total force exerted by your hand on the epee. (b) Find the torque exerted by your hand on the epee. (c) Your hand, being an extended object, actually exerts its force along the length of the epee grip. Model the total force exerted by your hand as two oppositely directed forces whose lines of action are separated by the width of your hand (taken to be 10.0 cm). Find the magnitudes and directions of these two forces.

A large gate weighing 200 N is supported by hinges at the top and bottom, and is further supported by a wire, as shown in Figure 12-39. (a) What must the tension in the wire be for the force on the upper hinge to have no horizontal component? (b) What is the horizontal force on the lower hinge? (c) What are the vertical forces on the hinges?

CONTEXT-RICH On a camping trip, you moor your boat at the end of a dock in a river that is rapidly flowing to the right. The boat is anchored to the dock by a chain 5.0 m long, as shown in Figure 12-40. A 100-N weight is suspended from the center in the chain. This will allow the tension in the chain to change as the force of the current which pulls the boat away from the dock and to the right varies. The drag force by the water on the boat depends on the speed of the water. You decide to apply the principles of statics you learned in physics class. (Ignore the weight of the chain.) The drag force on the boat is 50 N. (a) What is the tension in the chain? (b) How far is the boat from the dock? (c) The maximum tension the chain can sustain is 500 N. What is the minimum drag force on the boat that would snap the chain?

Romeo takes a uniform 10-m ladder and leans it against the smooth (frictionless) wall of the Capulet residence. The ladders mass is 22 kg and the bottom rests on the ground 2.8 m from the wall. When Romeo, whose mass is 70 kg, gets 90 percent of the way to the top, the ladder begins to slip. What is the coefficient of static friction between the ground and the ladder?

Two 80-N forces are applied to opposite corners of a rectangular plate, as shown in Figure 12-41. (a) Find the torque produced by this couple using the Equation 12-6. (b) Show that the result is the same as if you determine the torque about the lower left-hand corner. SSM

A uniform cube of side and mass rests on a horizontal surface. A horizontal force is applied to the top of the cube, as in Figure 12-42. This force is not sufficient to move or tip the cube. (a) Show that the force of static friction exerted by the surface and the applied force constitute a couple, and find the torque exerted by the couple. (b) The torque exerted by the couple is balanced by the torque exerted by the couple consisting of the normal force on the cube and the gravitational force on the cube. Use this fact to find the effective point of application of the normal force when (c) Find the greatest magnitude of for which the cube will not tip. (Assuming the cube does not slip.)

A ladder of negligible mass and of length leans against a slick wall making an angle of with the horizontal floor. The coefficient of friction between the ladder and the floor is A man climbs the ladder. What height can he reach before the ladder slips?

A uniform ladder of length and mass leans against a frictionless vertical wall, making an angle of with the horizontal. The coefficient of static friction between the ladder and the ground is 0.45. If your mass is four times that of the ladder, how high can you climb before the ladder begins to slip?

A ladder of mass and length leans against a frictionless vertical wall, so that it makes an angle with the horizontal. The center of mass of the ladder is a height above the floor. Aforce directly away from the wall pulls on the ladder at its midpoint. Find the minimum coefficient of static friction for which the top end of the ladder will separate from the wall before the lower end begins to slip.

A 900-N man sits on top of a stepladder of negligible mass that rests on a frictionless floor as in Figure 12-43. There is a cross brace halfway up the ladder. The angle at the apex is (a) What is the force exerted by the floor on each leg of the ladder? (b) What is the tension in the cross brace? (c) If the cross brace is moved down toward the bottom of the ladder (maintaining the same angle ), will its tension be the same, greater, or less than when it was at its higher position? Explain your answer.

Auniform ladder rests against a frictionless vertical wall. The coefficient of static friction between the ladder and the floor is 0.30. What is the smallest angle between the ladder and the horizontal such that the ladder will not slip?

A uniform log with a mass of 100 kg, a length of 4.0 m, and a radius of 12 cm is held in an inclined position, as shown in Figure 12-44. The coefficient of static friction between the log and the horizontal surface is 0.60. The log is on the verge of slipping to the right. Find the tension in the support wire and the angle the wire makes with the vertical wall.

Atall, uniform, rectangular block sits on an inclined plane, as shown in Figure 12-45. Acord attached to the top of the block prevents it from falling down the incline. What is the maximum angle for which the block will not slide on the incline? Assume the block has a height-to-width ratio, of 4.0 and the coefficient of static friction between it and the incline is ms SSM

A 50-kg ball is suspended from a steel wire of length 5.0 m and radius 2.0 mm. By how much does the wire stretch?

Copper has a tensile strength of about (a) What is the maximum load that can be hung from a copper wire of diameter 0.42 mm? (b) If half this maximum load is hung from the copper wire, by what percentage of its length will it stretch?

A 4.0-kg mass is supported by a steel wire of diameter 0.60 mm and length 1.2 m. How much will the wire stretch under this load?

As a runners foot pushes off on the ground, the shearing force acting on an 8.0-mm-thick sole is shown in Figure 12-46. If the force of 25 N is distributed over an area of find the angle of shear given that the shear modulus of the sole is SSM

A steel wire of length 1.50 m and diameter 1.00 mm is joined to an aluminum wire of identical dimensions to make a composite wire of length 3.00 m. Find the resulting change in length of this composite wire if an object with a mass of 5.00 kg is hung vertically from one of its ends. (Neglect any effects the masses of two wires have on the changes in their lengths.)

Equal but opposite forces of magnitude are applied to both ends of a thin wire of length and cross-sectional area Show that if the wire is modeled as a spring, the force constant is given by and the potential energy stored in the wire is where is Youngs modulus and is the amount the wire has stretched.

The steel E string of a violin is under a tension of 53.0 N. The diameter of the string is 0.200 mm and the length under tension is 35.0 cm. Find (a) the unstretched length of this string, and (b) the work needed to stretch the string.

ENGINEERING APPLICATION, SPREADSHEET During a materials science experiment on the Youngs modulus of rubber, the teaching assistant supplies you and your team with a rubber strip that is rectangular in cross section. She tells you to first measure the cross section dimensions, and you find their values are The lab write-up calls for the rubber strip to be suspended vertically and various (known) masses attached to its lower end. Your team obtains the following data for the length of the strip as a function of the load (mass) suspended from the end of the strip: Load, kg 0.0 0.10 0.20 0.30 0.40 0.50 Length, cm 5.0 5.6 6.2 6.9 7.8 8.8 (a) Use a spreadsheet or graphing calculator to find Youngs modulus for the rubber strip over this range of loads. Hint: It is probably best to plot versus Why? (b) Find the energy stored in the strip when the load is 0.15 kg. (See Problem 45.) (c) Find the energy stored in the strip when the load is 0.30 kg. Is it twice as much as your answer to Part (b)? Explain.

A large mirror is hung from a nail, as shown in Figure 12-47. The supporting steel wire has a diameter of 0.20 mm and an unstretched length of 1.7 m. The distance between the points of support at the top of the mirrors frame is 1.5 m. The mass of the mirror is 2.4 kg. How much will the distance between the nail and the mirror increase due to the stretching of the wire as the mirror is hung?

Two masses, and are supported by wires that have equal lengths when unstretched. The wire supporting is an aluminum wire 0.70 mm in diameter, and the one supporting is a steel wire 0.50 mm in diameter. What is the ratio if the two wires stretch by the same amount?

A 0.50-kg ball is attached to one end of an aluminum wire that has a diameter of 1.6 mm and an unstretched length of 0.70 m. The other end of the wire is fixed to the top of a post. The ball rotates about the post in a horizontal plane at a rotational speed such that the angle between the wire and the horizontal is Find the tension in the wire and the increase in its length due to the tension in the wire.

An elevator cable is to be made of a new type of composite developed by Acme Laboratories. In the lab, a sample of the cable that is 2.00 m long and has a cross-sectional area of fails under a load of 1000 N. The actual cable used to support the elevator will be 20.0 m long and have a cross-sectional area of It will need to support a load of 20,000 N safely. Will it?

If a materials density remains constant when it is stretched in one direction, then (because its total volume remains constant), its length must decrease in one or both of the other directions. Take a rectangular block of length width and depth and pull on it so that its new length If and show that

You are given a wire with a circular cross section of radius and a length If the wire is made from a material whose density remains constant when it is stretched in one direction, then show that assuming that (See Problem 52.)

For most materials listed in Table 12-1, the tensile strength is two to three orders of magnitude lower than Youngs modulus. Consequently, most of these materials will break before their strain exceeds 1 percent. Of man-made materials, nylon has about the greatest extensibilityit can take strains of about 0.2 before breaking. But spider silk beats anything man-made. Certain forms of spider silk can take strains on the order of 10 before breaking! (a) If such a thread has a circular cross-section of radius and unstretched length find its new radius when stretched to a length (Assume that the density of the thread remains constant as it stretches.) (b) If the Youngs modulus of the spider thread is calculate the tension needed to break the thread in terms of and

BIOLOGICAL APPLICATION A standard bowling ball weighs 16 pounds. You wish to hold a bowling ball in front of you, with your elbow bent at a right angle. Assume that your biceps attaches to your forearm at 2.5 cm out from the elbow joint, and that your biceps muscle pulls vertically upward, that is, it acts at right angles to the forearm. Also assume that the ball is held 38 cm out from the elbow joint. Let the mass of your forearm be 5.0 kg and assume its center of gravity is located 19 cm out from the elbow joint. How much force must your biceps muscle apply to the forearm in order to hold out the bowling ball at desired angle?

BIOLOGICAL APPLICATION, CONTEXT-RICH A biology laboratory at your university is studying the location of a persons center of gravity as a function of her or his body weight. They pay well and you decide to volunteer. The location of your center of gravity when standing erect is to be determined by having you lie on a uniform board (mass of 5.00 kg, length 2.00 m) supported by two scales, as shown in Figure 12-48. If your height is 188 cm and the left scale reads 445 N while the right scale reads 400 N, where is your center of gravity relative to your feet? Assume the scales are both exactly the same distance from the two ends of the board, are separated by 178 cm, and are set to each read zero before you get on the platform.

Figure 12-49 shows a mobile consisting of four objects hanging on three rods of negligible mass. Find the values of the unknown masses of the objects if the mobile is to balance. Hint: Find the mass m first.

ENGINEERING APPLICATION, CONTEXT-RICH Steel construction beams, with an industry designation of have a weight of 22 pounds per foot. A new business in town has hired you to place its sign on a 4.0 m long steel beam of this type. The design calls for the beam to extend outward horizontally from the front brick wall (Figure 12-50). It is to be held in place by a 5.0-mlong steel cable. The cable is attached to one end of the beam and to the wall above the point at which the beam is in contact with the wall. During an initial stage of construction, the beam is not to be bolted to the wall, but to be held in place solely by friction. (a) What is the minimum coefficient of friction between the beam and the wall for the beam to remain in static equilibrium? (b) What is the tension in the cable in this case?

Consider a rigid 2.5-m-long beam (Figure 12-51) that is supported by a fixed 1.25-m-high post through its center and pivots on a frictionless bearing at its center atop the vertical 1.25-m-high post. One end of the beam is connected to the floor by a spring that has a force constant When the beam is horizontal, the spring is vertical and unstressed. If an object is hung from the opposite end of the beam, the beam settles into an equilibrium position where it makes an angle of with the horizontal. What is the mass of the object? SSM

A rope and pulley system, called a block and tackle, is used to raise an object of mass (Figure 12-52) at constant speed. When the end of the rope moves downward through a distance L, the height of the lower pulley is increased by (a) What is the ratio (b) Assume that the mass of the block and tackle is negligible and that the pulley bearings are frictionless. Show that by applying the workenergy principle to the blocktackle object.

A plate of mass in the shape of an equilateral triangle is suspended from one of its corners, and a mass is suspended from another of its corners. If the base of the triangle makes an angle of with the horizontal, what is the ratio m>M?

A standard six-sided pencil is placed on a notebook (Figure 12-53). Find the minimum coefficient of static friction such that, if the upper cover is raised, the pencil rolls down the incline rather than sliding.

An 8.0-kg box that has a uniform density and is twice as tall as it is wide rests on the floor of a truck. What is the maximum coefficient of static friction between the box and floor so that the box will slide toward the rear of the truck rather than tip when the truck accelerates forward on a level road?

A balance scale has unequal arms. The scale is balanced with a 1.50-kg block on the left pan and a 1.95 kg block on the right pan (Figure 12-54). If the 1.95-kg block is removed from the right pan and the 1.50-kg block is then moved to the right pan, what mass on the left pan will balance the scale?

A cube leans against a frictionless wall, making an angle of with the floor, as shown in Figure 12-55. Find the minimum coefficient of static friction between the cube and the floor that is needed to prevent the cube from slipping. SSM

Figure 12-56 shows a 5.00-kg 1.00-m-long rod hinged to a vertical wall and supported by a thin wire. The wire and rod each make angles of with the vertical. When a 10.0-kg block is suspended from the midpoint of the rod, the tension in the supporting wire is 52.0 N. If the wire will break when the tension exceeds 75 N, what is the maximum distance from the hinge at which the block can be suspended? T

Figure 12-57 shows a 20.0-kg ladder leaning against a frictionless wall and resting on a frictionless horizontal surface. To keep the ladder from slipping, the bottom of the ladder is tied to the wall by a thin wire. When no one is on the ladder, the tension in the wire is 29.4 N. (The wire will break if the tension exceeds 200 N.) (a) If an 80.0-kg person climbs halfway up the ladder, what force will be exerted by the ladder against the wall? (b) How far from the bottom end of the ladder can an 80.0-kg person climb? SSM

A 360-kg object is supported on a wire attached to a 15-m-long steel bar that is pivoted at a vertical wall and supported by a cable as shown in Figure 12-58. The mass of the bar is 85 kg. With the cable attached to the bar 5.0 m from the lower end, as shown, what are the tension in the cable and the force exerted by the wall on the steel bar?

Repeat Problem 63 if the truck accelerates up a hill that makes an angle of 9.0 with the horizontal.

A thin uniform rod 60 cm long is balanced 20 cm from one end when an object whose mass is is at the end nearest the pivot and an object of mass is at the opposite end (Figure 12-59a). Balance is again achieved if the object whose mass is is replaced by the object of mass and no object is placed at the other end of the rod (Figure 12-59b). Determine the mass of the rod. (2m _ 2.0 grams) m m (2m _ 2.0 grams)

SPREADSHEET There are a large number of identical uniform bricks, each of length If they are stacked one on top of another lengthwise (see Figure 12-60), the maximum offset that will allow the top brick to rest on the bottom brick is (a) Show that if this two-brick stack is placed on top of a third brick, the maximum offset of the second brick on the third brick is (b) Show that, in general, if you have a stack of bricks, the maximum offset of the brick (counting down from the top) on the th brick is (c) Write a spreadsheet program to calculate total offset (the sum of the individual offsets) for a stack of bricks, and calculate this for and and 100. (d) Does the sum of the individual offsets approach a finite limit as If so, what is that limit? SSM

A uniform sphere of radius and mass is held at rest on an inclined plane of angle by a horizontal string, as shown in Figure 12-61. Let and (a) What is the tension in the string? (b) What is the normal force exerted on the sphere by the inclined plane? (c) What is the frictional force acting on the sphere?

The legs of a tripod make equal angles of with each other at the apex, where they join together. A 100-kg block hangs from the apex. What are the compressional forces in the three legs?

Figure 12-62 shows a 20-cm-long uniform beam resting on a cylinder that has a radius of 4.0-cm. The mass of the beam is 5.0 kg, and that of the cylinder is 8.0 kg. The coefficient of static friction between beam and cylinder is zero, whereas the coefficients of static friction between the cylinder and the floor, and between the beam and the floor, are not zero. Are there any values for these coefficients of static friction such that the system is in static equilibrium? If so, what are these values? If not, explain why none exist.

Two solid smooth (frictionless) spheres of radius are placed inside a cylinder of radius as in Figure 12-63. The mass of each sphere is Find the force exerted by the bottom of the cylinder on the bottom sphere, the force exerted by the wall of the cylinder on each sphere, and the force exerted by one sphere on the other. Express all forces in terms of m, R, and r. SSM

A solid cube of edge-length balanced atop a cylinder of diameter is in unstable equilibrium if (Figure 12-64), and is in stable equilibrium if . The cube does not slip on the cylinder. Determine the minimum value of the ratio for which the cube is in stable equilibrium.