Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway, John W. Jewett, Jr.

Chapter 12

Static Equilibrium and Elasticity - all with Video Answers

Educators

TG

Chapter Questions

Problem 1

A uniform beam of mass $m_{b}$ and length $\ell$ supports blocks with masses $m_{1}$ and $m_{2}$ at two positions as shown in Figure Pl2.1. The beam rests on two knife edges. For what value of $x$ will the beam be balanced at $P$ such that the normal force at $O$ is zero?

Mayukh Banik

Mayukh Banik

Numerade Educator

Problem 2

Write the necessary conditions for equilibrium of the object shown in Figure Pl2.2. Calculate torques about an axis through point $O$ .
GRAPH CANNOT COPY

Mayukh Banik

Numerade Educator

Problem 3

A carpenter's square has the shape of an L as shown in Figure P12.3. Locate its center of gravity.
GRAPH CANNOT COPY

Mayukh Banik

Numerade Educator

Problem 4

A circular pizza of radius $R$ has a circular piece of radius $R / 2$ removed from one side as shown in Figure P12.4. The center of gravity has moved from $C$ to $C^{\prime}$ along the $x$ axis. Show that the distance from $C$ to $C^{\prime}$ is $R / 6 .$ Assume the thickness and density of the pizza are uniform throughout.
GRAPH CANNOT COPY

Bret Rosen

Numerade Educator

Problem 5

A Consider the following distribution of objects: a $5.00-\mathrm{kg}$ object with its center of gravity at $(0,0) \mathrm{m},$ a $3.00-\mathrm{kg}$ object at $(0,4.00) \mathrm{m},$ and a $4.00-\mathrm{kg}$ object at $(3.00,0) \mathrm{m}$ . Where should a fourth object of mass 8.00 $\mathrm{kg}$ be placed so that the center of gravity of the four-object arrangement will be at $(0,0) ?$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 6

Pat builds a track for his model car out of solid wood as shown in Figure $\mathrm{P} 12.6 .$ The track is 5.00 $\mathrm{cm}$ wide, 1.00 $\mathrm{m}$ high, and 3.00 $\mathrm{m}$ long. The runway is cut so that it forms a parabola with the equation $y=(x-3)^{2} / 9 .$ Locate the horizontal coordinate of the center of gravity of this track.
GRAPH CANNOT COPY

Mayukh Banik

Numerade Educator

Problem 7

Figure $\mathrm{P} 12.7$ shows three uniform objects: a rod, a right triangle, and a square. Their masses and their coordinates in meters are given. Determine the center of gravity for the three-object system.
GRAPH CANNOT COPY

Mayukh Banik

Numerade Educator

Problem 8

A mobile is constructed of light rods, light strings, and beach souvenirs as shown in Figure $\mathrm{P} 12.8$ . Determine the masses of the objects (a) $m_{1},(\mathrm{b}) m_{2},$ and $(\mathrm{c}) m_{3}$ .
GRAPH CANNOT COPY

Mayukh Banik

Numerade Educator

Problem 9

Find the mass $m$ of the counterweight needed to balance the $1500-\mathrm{kg}$ truck on the incline shown in Figure $\mathrm{P} 12.9 .$ Assume all pulleys are frictionless and massless.
GRAPH CANNOT COPY

Mayukh Banik

Numerade Educator

Problem 10

Figure $\mathrm{P} 12.10$ shows a claw hammer being used to pull a nail out of a horizontal board. A force of 150 $\mathrm{N}$ is exerted horizontally as shown. Find (a) the force exerted by the hammer claws on the nail and (b) the force exerted by the surface on the point of contact with the hammer head. Assume the force the hammer exerts on the nail is parallel to the nail.

Mayukh Banik

Numerade Educator

Problem 11

A 15.0 -m uniform ladder weighing 500 $\mathrm{N}$ rests against a frictionless wall. The ladder makes a $60.0^{\circ}$ angle with the horizontal. ( a) Find the horizontal and vertical forces the ground exerts on the base of the ladder when an $800-\mathrm{N}$ firefighter is 4.00 $\mathrm{m}$ from the bottom. (b) If the ladder is just on the verge of slipping when the firefighter is 9.00 $\mathrm{m}$ up, what is the coefficient of static friction between ladder and ground?

Mayukh Banik

Numerade Educator

Problem 12

A uniform ladder of length $L$ and mass $m_{1}$ rests against a frictionless wall. The ladder makes an angle $\theta$ with the horizontal. (a) Find the horizontal and vertical forces the ground exerts on the base of the ladder when a fire-fighter of mass $m_{2}$ is a distance $x$ from the bottom. (b) If the ladder is just on the verge of slipping when the fire-fighter is a distance $d$ from the bottom, what is the coefficient of static friction between ladder and ground?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 13

A $1500-\mathrm{kg}$ automobile has a wheel base (the distance between the axles) of $3.00 \mathrm{m} .$ The automobile's center of mass is on the centerline at a point 1.20 $\mathrm{m}$ behind the front axle. Find the force exerted by the ground on each wheel.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 14

A $20.0-\mathrm{kg}$ floodlight in a park is supported at the end of a horizontal beam of negligible mass that is hinged to a pole as shown in Figure P12.14. A cable at an angle of $30.0^{\circ}$ with the beam helps support the light. Consider the equilibrium of the beam, drawing a free-body diagram of that object. Compute torques about an axis at the hinge at its left-hand end. Find (a) the tension in the cable, (b) the horizontal component of the force exerted by the pole on the beam, and $(c)$ the vertical component of this force. Now solve the same problem from the same free-body diagram by computing torques around the junction between the cable and the beam at the right-hand end of the beam. Find (d) the vertical component of the force exerted by the pole on the beam, (e) the tension in the cable, and (f) the horizontal component of the force exerted by the pole on the beam. (g) Compare the solution to parts (a) through (c) with the solution to parts (d) through (f). Is either solution more accurate? Simpler? Taking together the whole set of equations read from the free-body diagram in both solutions, how many equations do you have? How many unknown quantities can be determined?
GRAPH CANNOT COPY

Mayukh Banik

Numerade Educator

Problem 15

A flexible chain weighing 40.0 $\mathrm{N}$ hangs between two hooks located at the same height (Fig. Pl2.15). At each hook, the tangent to the chain makes an angle $\theta=42.0^{\circ}$ with the horizontal. Find (a) the magnitude of the force each hook exerts on the chain and (b) the tension in the chain at its midpoint. Suggestion: for part $(\mathrm{b}),$ make a free-body diagram for half of the chain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 16

Sir Lost-a-Lot dons his armor and sets out from the castle on his trusty steed in his quest to improve communication between damsels and dragons (Fig. Pl2.16). Unfortunately, his squire lowered the drawbridge too far and finally stopped it $20.0^{\circ}$ below the horizontal. Lost-a-Lot and his horse stop when their combined center of mass is 1.00 $\mathrm{m}$ from the end of the bridge. The uniform bridge is 8.00 $\mathrm{m}$ long and has mass $2000 \mathrm{kg} .$ The lift cable is attached to the bridge 5.00 $\mathrm{m}$ from the hinge at the castle end and to a point on the castle wall 12.0 $\mathrm{m}$ above the bridge. Lost-a-Lot's mass combined with his armor and steed is 1000 $\mathrm{kg}$ . Determine (a) the tension in the cable and the (b) horizontal and (c) vertical force components acting on the bridge at the hinge.

Mayukh Banik

Numerade Educator

Problem 17

Review problem. In the situation described in Problem 16 and illustrated in Figure $\mathrm{P} 12.16$ , the lift cable suddenly breaks! The hinge between the castle wall and the bridge is frictionless, and the bridge swings freely until it is vertical. (a) Find the angular acceleration of the bridge once it starts to move. (b) Find the angular speed of the bridge when it strikes the vertical castle wall below the hinge. (c) Find the force exerted by the hinge on the bridge immediately after the cable breaks. (d) Find the force exerted by the hinge on the bridge immediately before it strikes the castle wall.

Mayukh Banik

Numerade Educator

Problem 18

Stephen is pushing his sister Joyce in a wheelbarrow when it is stopped by a brick 8.00 $\mathrm{cm}$ high (Fig. P12.18). The wheelbarrow handles make an angle of $15.0^{\circ}$ below the horizontal. A downward force of 400 $\mathrm{N}$ is exerted on the wheel, which has a radius of $20.0 \mathrm{cm} .$ (a) What force must Stephen apply along the handles to just start the wheel over the brick? (b) What is the force (magnitude and direction) that the brick exerts on the wheel just as the wheel begins to lift over the brick? In both parts (a) and (b), assume the brick remains fixed and does not slide along the ground.

Mayukh Banik

Numerade Educator

Problem 19

One end of a uniform 4.00 -m-long rod of weight $F_{g}$ is supported by a cable. The other end rests against the wall, where it is held by friction, as shown in Figure Pl2.19. The coefficient of static friction between the wall and the rod is $\mu_{s}=0.500$ . Determine the minimum distance $x$ from point $A$ at which an additional object, also with the same weight $F_{g},$ can be hung without causing the rod to slip at point $A .$

Keshav Singh

Numerade Educator

Problem 20

In the What If? section of Example 12.2 , let $x$ represent the distance in meters between the person and the hinge at the left end of the beam. (a) Show that the cable tension in newtons is given by $T=93.9 x+125 .$ Argue that $T$ increases as $x$ increases. (b) Show that the direction angle $\theta$ of the hinge force is described by
$$
\tan \theta=\left(\frac{32}{3 x+4}-1\right) \tan 53.0^{\circ}
$$
How does $\theta$ change as $x$ increases? (c) Show that the magnitude of the hinge force is given by
$$
R=\sqrt{8.82 \times 10^{3} x^{2}-9.65 \times 10^{4} x+4.96 \times 10^{5}}
$$
How does $R$ change as $x$ increases?

Mayukh Banik

Numerade Educator

Problem 21

A vaulter holds a 29.4 $\mathrm{N}$ pole in equilibrium by exerting an upward force $\overrightarrow{\mathrm{U}}$ with her leading hand and a downward force $\overrightarrow{\mathbf{D}}$ with her trailing hand as shown in Figure $\mathrm{P} 12.21 .$ Point $C$ is the center of gravity of the pole. What are the magnitudes of $\overrightarrow{\mathbf{U}}$ and $\overrightarrow{\mathbf{D}}$ ?
GRAPH CANNOT COPY

Mayukh Banik

Numerade Educator

Problem 22

Evaluate Young's modulus for the material whose stress-strain curve is shown in Figure 12.13 .

Mayukh Banik

Numerade Educator

Problem 23

A 200 -kg load is hung on a wire of length $4.00 \mathrm{m},$ cross-sectional area $0.200 \times 10^{-4} \mathrm{m}^{2}$ , and Young's modulus $8.00 \times 10^{10} \mathrm{N} / \mathrm{m}^{2} .$ What is its increase in length?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 24

Assume Young's modulus for bone is $1.50 \times 10^{10} \mathrm{N} / \mathrm{m}^{2}$ . The bone breaks if stress greater than $1.50 \times 10^{8} \mathrm{N} / \mathrm{m}^{2}$ is imposed on it. (a) What is the maximum force that can be exerted on the femur bone in the leg if it has a minimum effective diameter of 2.50 $\mathrm{cm}$ ? (b) If this much force is applied compressively, by how much does the 25.0 -cm-long bone shorten?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 25

A child slides across a floor in a pair of rubber-soled shoes. The friction force acting on each foot is 20.0 $\mathrm{N}$ . The footprint area of each shoe sole is $14.0 \mathrm{cm}^{2},$ and the thickness of each sole is $5.00 \mathrm{mm} .$ Find the horizontal distance by which the upper and lower surfaces of each sole are offset. The shear modulus of the rubber is 3.00 $\mathrm{MN} / \mathrm{m}^{2}$ .

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 26

A steel wire of diameter 1 $\mathrm{mm}$ can support a tension of 0.2 $\mathrm{kN}$ . A cable to support a tension of 20 $\mathrm{kN}$ should have diameter of what order of magnitude?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 27

Assume that if the shear stress in steel exceeds about $4.00 \times 10^{8} \mathrm{N} / \mathrm{m}^{2}$ , the steel ruptures. Determine the shearing force necessary to (a) shear a steel bolt 1.00 $\mathrm{cm}$ in diameter and $(\mathrm{b})$ punch a 1.00 -cm-diameter hole in a steel plate 0.500 $\mathrm{cm}$ thick.

Averell Hause

Carnegie Mellon University

Problem 28

Review problem. A 30.0 -kg hammer, moving with speed $20.0 \mathrm{m} / \mathrm{s},$ strikes a steel spike 2.30 $\mathrm{cm}$ in diameter. The hammer rebounds with speed 10.0 $\mathrm{m} / \mathrm{s}$ after 0.110 s. What is the average strain in the spike during the impact?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 29

When water freezes, it expands by about 9.00$\%$ . What pressure increase would occur inside your automobile engine block if the water in it froze? (The bulk modulus of ice is $2.00 \times 10^{9} \mathrm{N} / \mathrm{m}^{2} . )$

TG

Troy Gabriele

Numerade Educator

Problem 30

Review problem. A 2.00 -m-long cylindrical steel wire with a cross-sectional diameter of 4.00 $\mathrm{mm}$ is placed over a light, frictionless pulley, with one end of the wire connected to a $5.00-\mathrm{kg}$ object and the other end connected to a $3.00-\mathrm{kg}$ object. By how much does the wire stretch while the objects are in motion?

Mayukh Banik

Numerade Educator

Problem 31

A walkway suspended across a hotel lobby is supported at numerous points along its edges by a vertical cable above each point and a vertical column underneath. The steel cable is 1.27 $\mathrm{cm}$ in diameter and is 5.75 $\mathrm{m}$ long before loading. The aluminum column is a hollow cylinder with an inside diameter of $16.14 \mathrm{cm},$ an outside diameter of $16.24 \mathrm{cm},$ and unloaded length of $3.25 \mathrm{m} .$ When the walk-way exerts a load force of 8500 $\mathrm{N}$ on one of the support points, how much does the point move down?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 32

The deepest point in any ocean is in the Mariana Trench, which is about 11 $\mathrm{km}$ deep, in the Pacific. The pressure at this depth is huge, about $1.13 \times 10^{8} \mathrm{N} / \mathrm{m}^{2}$ . (a) Calculate the change in volume of 1.00 $\mathrm{m}^{3}$ of seawater carried from the surface to this deepest point. (b) The density of seawater at the surface is $1.03 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}$ . Find its density at the bottom. (c) Explain whether or when it is a good approximation to think of water as incompressible.

Anand Jangid

Numerade Educator

Problem 33

A bridge of length 50.0 $\mathrm{m}$ and mass $8.00 \times 10^{4} \mathrm{kg}$ is supported on a smooth pier at each end as shown in Figure $\mathrm{P} 12.33$ . A truck of mass $3.00 \times 10^{4} \mathrm{kg}$ is located 15.0 $\mathrm{m}$ from one end. What are the forces on the bridge at the points of support?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 34

A new General Electric kitchen stove has a mass of 68.0 $\mathrm{kg}$ and the dimensions shown in Figure Pl2.34. The stove comes with a warning that it can tip forward if a person stands or sits on the oven door when it is open. What can you conclude about the weight of such a person? Could it be a child? List the assumptions you make in solving this problem. The stove is supplied with a wall bracket to prevent the accident.

Mayukh Banik

Numerade Educator

Problem 35

A uniform pole is propped between the floor and the ceiling of a room. The height of the room is 7.80 $\mathrm{ft}$ , and the coefficient of static friction between the pole and the ceiling is 0.576 . The coefficient of static friction between the pole and the floor is greater than that. What is the length of the longest pole that can be propped between the floor and the ceiling?

Mayukh Banik

Numerade Educator

Problem 36

Refer to Figure 12.16 $\mathrm{c}$ . A lintel of prestressed reinforced concrete is 1.50 $\mathrm{m}$ long. The cross-sectional area of the concrete is $50.0 \mathrm{cm}^{2} .$ The concrete encloses one steel reinforcing rod with cross-sectional area $1.50 \mathrm{cm}^{2} .$ The rod joins two strong end plates. Young's modulus for the concrete is $30.0 \times 10^{9} \mathrm{N} / \mathrm{m}^{2}$ . After the concrete cures and the original tension $T_{1}$ in the rod is released, the concrete is to be under compressive stress $8.00 \times 10^{6} \mathrm{N} / \mathrm{m}^{2} .$ (a) By what distance will the rod compress the concrete when the original tension in the rod is released? (b) What is the new tension $T_{2}$ in the rod? (c) The rod will then be how much longer than its unstressed length? (d) When the concrete was poured, the rod should have been stretched by what extension distance from its unstressed length? (e) Find the required original tension $T_{1}$ in the rod.

Mayukh Banik

Numerade Educator

Problem 37

A hungry bear weighing 700 $\mathrm{N}$ walks out on a beam in an attempt to retrieve a basket of food hanging at the end of the beam (Fig. Pl2.37). The beam is uniform, weighs $200 \mathrm{N},$ and is 6.00 $\mathrm{m}$ long; the basket weighs 80.0 $\mathrm{N}$ . (a) Draw a free-body diagram for the beam. (b) When the bear is at $x=1.00 \mathrm{m},$ find the tension in the wire and the components of the force exerted by the wall on the left end of the beam. (c) What If? If the wire can withstand a maximum tension of $900 \mathrm{N},$ what is the maximum distance the bear can walk before the wire breaks?

Mayukh Banik

Numerade Educator

Problem 38

The following equations are obtained from a free-body diagram of a rectangular farm gate, supported by two hinges on the left-hand side. A bucket of grain is hanging from the latch.
$$
\begin{array}{c}{-A+C=0} \\ {+B-392 \mathrm{N}-50.0 \mathrm{N}=0} \\ {A(0)+B(0)+C(1.80 \mathrm{m})-392 \mathrm{N}(1.50 \mathrm{m})} \\ {-50.0 \mathrm{N}(3.00 \mathrm{m})=0}\end{array}
$$
(a) Draw the free-body diagram and complete the statement of the problem, specifying the unknowns. (b) Determine the values of the unknowns and state the physical meaning of each.

Mayukh Banik

Numerade Educator

Problem 39

A uniform sign of weight $F_{g}$ and width 2$L$ hangs from a light, horizontal beam hinged at the wall and supported by a cable (Fig. Pl2.39). Determine (a) the tension in the cable and $(b)$ the components of the reaction force exerted by the wall on the beam, in terms of $F_{g}, d, L,$ and $\theta .$

Rehmat Kazmi

Numerade Educator

Problem 40

A $1200-\mathrm{N}$ uniform boom is supported by a cable as shown in Figure $\mathrm{P} 12.40 .$ The boom is pivoted at the bottom, and a $2000-\mathrm{N}$ object hangs from its top. Find the tension in the cable and the components of the reaction force exerted by the floor on the boom.

Mayukh Banik

Numerade Educator

Problem 41

A crane of mass 3000 $\mathrm{kg}$ supports a load of 10000 $\mathrm{kg}$ as shown in Figure $\mathrm{P} 12.41$ . The crane is pivoted with a frictionless pin at $A$ and rests against a smooth support at $B$ . Find the reaction forces at $A$ and $B .$

Mayukh Banik

Numerade Educator

Problem 42

Assume a person bends forward to lift a load "with his back" as shown in Figure P12.42a. The person's spine pivots mainly at the fifth lumbar vertebra, with the principal supporting force provided by the erector spinalis muscle in the back. To estimate the magnitude of the forces involved, consider the model shown in Figure Pl2.42b for a person bending forward to lift a 200 -N object. The person's spine and upper body are represented as a uniform horizontal rod of weight 350 $\mathrm{N}$ , pivoted at the base of the spine. The erector spinalis muscle, attached at a point two thirds of the way up the spine, maintains the position of the back. The angle between the spine and this muscle is $12.0^{\circ} .$ Find $(a)$ the tension in the back muscle and (b) the compressional force in the spine. (c) Is this method a good way to lift a load? Explain your answer, using the results of parts (a) and (b). It can be instructive to compare a human to other animals. Can you suggest a better method to lift a load?

Mayukh Banik

Numerade Educator

Problem 43

A $10000-\mathrm{N}$ shark is supported by a cable attached to a 4.00 $\mathrm{m}$ rod that can pivot at the base. Calculate the tension in the tie rope between the rod and the wall, assuming the tie rope is holding the system in the position shown in Figure $\mathrm{P} 12.43$ . Find the horizontal and vertical forces exerted on the base of the rod. Ignore the weight of the rod.

Mayukh Banik

Numerade Educator

Problem 44

A uniform rod of weight $F_{g}$ and length $L$ is supported at its ends by a frictionless trough as shown in Figure Pl2.44 (a) Show that the center of gravity of the rod must be vertically over point $O$ when the rod is in equilibrium. (b) Determine the equilibrium value of the angle $\theta$ .

Mayukh Banik

Numerade Educator

Problem 45

A force is exerted on a uniform rectangular cabinet weighing 400 $\mathrm{N}$ as shown in Figure $\mathrm{P} 12.45$ . (a) The cabinet slides with constant speed when $F=200 \mathrm{N}$ and $h=0.400 \mathrm{m} .$ Find the coefficient of kinetic friction and the position of the resultant normal force. (b) Taking $F=300 \mathrm{N}$ , find the value of $h$ for which the cabinet just begins to tip.

Mayukh Banik

Numerade Educator

Problem 46

Consider the rectangular cabinet of Problem $45,$ but with a force $\overrightarrow{\mathbf{F}}$ applied horizontally at the upper edge. (a) What is the minimum force required to start to tip the cabinet? (b) What is the minimum coefficient of static friction required for the cabinet not to slide with the application of a force of this magnitude? (c) Find the magnitude and direction of the minimum force required to tip the cabinet if the point of application can be chosen anywhere on the cabinet.

Mayukh Banik

Numerade Educator

Problem 47

A uniform beam of mass $m$ is inclined at an angle $\theta$ to the horizontal. Its upper end produces a $90^{\circ}$ bend in a very rough rope tied to a wall, and its lower end rests on a rough floor (Fig. Pl2.47). (a) Let $\mu_{s}$ represent the coeffcient of static friction between beam and floor. Assume $\mu_{s}$ is less than the cotangent of $\theta .$ Determine an expression for the maximum mass $M$ that can be suspended from the top before the beam slips. (b) Determine the magnitude of the reaction force at the floor and the magnitude of the force exerted by the beam on the rope at $P$ in terms of $m, M,$ and $\mu_{s}$ .

Mayukh Banik

Numerade Educator

Problem 48

Consider a light truss, with weight negligible compared with the load it supports. Suppose it is formed
from struts lying in a plane and joined by smooth hinge pins at their ends. External forces act on the truss only at the joints. Figure $\mathrm{P} 12.48$ shows one example of the simplest truss, with three struts and three pins. State reasoning to prove that the force any strut exerts on a pin must be directed along the length of the strut, as a force of tension or compression.

Mayukh Banik

Numerade Educator

Problem 49

Figure $\mathrm{P} 12.48$ shows a truss that supports a downward force of 1000 $\mathrm{N}$ applied at the point $B$ . The truss has negligible weight. The piers at $A$ and $C$ are smooth. (a) Apply the conditions of equilibrium to prove that $n_{A}=366 \mathrm{N}$ and $n_{C}=634 \mathrm{N} .$ (b) Use the result proved in Problem 48 to identify the directions of the forces that the bars exert on the pins joining them. Find the force of tension or of compression in each of the three bars.

Mayukh Banik

Numerade Educator

Problem 50

One side of a plant shelf is supported by a bracket mounted on a vertical wall by a single screw as shown in Figure P12.50. Ignore the weight of the bracket. (a) Find the horizontal component of the force that the screw exerts on the bracket when an 80.0 $\mathrm{N}$ vertical force is applied as shown. (b) As your grandfather waters his geraniums, the $80.0-\mathrm{N}$ load force is increasing at the rate 0.150 $\mathrm{N} / \mathrm{s}$ . At what rate is the force exerted by the screw changing? Suggestions: Imagine that the bracket is slightly loose. You can do parts (a) and (b) most efficiently if you call the load force $W$ and solve symbolically for the screw force $F .$

Mayukh Banik

Numerade Educator

Problem 51

A steplader of negligible weight is constructed as shown in Figure $\mathrm{P} 12.51 .$ A painter of mass 70.0 $\mathrm{kg}$ stands on the ladder 3.00 $\mathrm{m}$ from the bottom. Assume the floor is frictionless. Find (a) the tension in the horizontal bar connecting the two halves of the ladder, (b) the normal forces at $A$ and $B$ , and (c) the components of the reaction force at the single hinge $C$ that the left half of the ladder exerts on the right half. Suggestion: Treat the ladder as a single object, but also each half of the ladder separately.

Mayukh Banik

Numerade Educator

Problem 52

Figure $\mathrm{P} 12.52$ shows a vertical force applied tangentially to a uniform cylinder of weight $F_{g} .$ The coefficient of static friction between the cylinder and all surfaces is 0.500 . In terms of $F_{g},$ find the maximum force $P$ that can be applied without causing the cylinder to rotate. As a first step, explain why both friction forces will be at their maximum values when the cylinder is on the verge of slipping.

Mayukh Banik

Numerade Educator

Problem 53

A Review problem. A wire of length $L,$ Young's modulus $Y,$ and cross-sectional area $A$ is stretched elastically by an amount $\Delta L$ . By Hooke's law, the restoring force is $-k \Delta L .$ (a) Show that $k=Y A / L .$ (b) Show that the work done in stretching the wire by an amount $\Delta L$ is
$$
W=\frac{1}{2} Y A \frac{(\Delta L)^{2}}{L}
$$

Mayukh Banik

Numerade Educator

Problem 54

Two racquetballs each having a mass of 170 $\mathrm{g}$ are placed in a glass jar as shown in Figure P12.54. Their centers and the point $A$ lie on a straight line. Assume the walls are frictionless. (a) Determine $P_{1}, P_{2},$ and $P_{3} .$ (b) Determine the magnitude of the force exerted by the left ball on the right ball.

Mayukh Banik

Numerade Educator

Problem 55

In exercise physiology studies, it is sometimes important to determine the location of a person's center of mass. This determination can be done with the arrangement shown in Figure $\mathrm{P} 12.55$ . A light plank rests on two scales, which read $F_{g 1}=380 \mathrm{N}$ and $F_{g 2}=320 \mathrm{N}$ . A distance of 2.00 $\mathrm{m}$ separates the scales. How far from the woman's feet is her center of mass?

Mayukh Banik

Numerade Educator

Problem 56

A steel cable 3.00 $\mathrm{cm}^{2}$ in cross-sectional area has a mass of 2.40 $\mathrm{kg}$ per meter of length. If 500 $\mathrm{m}$ of the cable is hung over a vertical cliff, how much does the cable stretch under its own weight? Take $Y_{\text { steel }}=2.00 \times 10^{11} \mathrm{N} / \mathrm{m}^{2}$ .

Mayukh Banik

Numerade Educator

Problem 57

(a) Estimate the force with which a karate master strikes a board, assuming the hand's speed at the moment of impact is 10.0 $\mathrm{m} / \mathrm{s}$ , decreasing to 1.00 $\mathrm{m} / \mathrm{s}$ during a 0.00200 -s time interval of contact between the hand and the board. The mass of his hand and arm is $1.00 \mathrm{kg} .$ (b) Estimate the shear stress, assuming this force is exerted on a 1.00 -cm-
thick pine board that is 10.0 $\mathrm{cm}$ wide. (C) If the maximum shear stress a pine board can support before breaking is $3.60 \times 10^{6} \mathrm{N} / \mathrm{m}^{2},$ will the board break?

Mayukh Banik

Numerade Educator

Problem 58

Review problem. An aluminum wire is 0.850 $\mathrm{m}$ long and has a circular cross section of diameter 0.780 $\mathrm{mm}$ . Fixed at the top end, the wire supports a $1.20-\mathrm{kg}$ object that swings in a horizontal circle. Determine the angular velocity required to produce a strain of $1.00 \times 10^{-3}$ .

Mayukh Banik

Numerade Educator

Problem 59

Review problem. A trailer with loaded weight $\overrightarrow{\mathbf{F}}_{g}$ is being pulled by a vehicle with a force $\overrightarrow{\mathbf{P}}$ as shown in Figure P12.59. The trailer is loaded such that its center of mass is located as shown. Ignore the force of rolling friction and let $a$ represent the $x$ component of the acceleration of the trailer. (a) Find the vertical component of $\overrightarrow{\mathbf{P}}$ in terms of the given parameters. (b) Assume $a=2.00 \mathrm{m} / \mathrm{s}^{2}$ and $h=1.50 \mathrm{m} .$ What must be the value of $d$ so that $P_{y}=0$ (no vertical load on the vehicle)? (c) Find the values of $P_{x}$ and $P_{y}$ given that $F_{g}=1500 \mathrm{N}, d=0.800 \mathrm{m}, L=3.00 \mathrm{m},$ $h=1.50 \mathrm{m},$ and $a=-2.00 \mathrm{m} / \mathrm{s}^{2} .$

Mayukh Banik

Numerade Educator

Problem 60

Review problem. A car moves with speed $v$ on a horizontal circular track of radius $R$ . A head-on view of the car is shown in Figure $\mathrm{P} 12.60$ . The height of the car's center of mass above the ground is $h,$ and the separation between its inner and outer wheels is $d .$ The road is dry, and the car does not skid. Show that the maximum speed the car can have without overturning is given by
$$
v_{\max }=\sqrt{\frac{g R d}{2 h}}
$$
To reduce the risk of rollover, should one increase or decrease $h$ ? Should one increase or decrease the width $d$ of the wheel base?

Mayukh Banik

Numerade Educator