University Physics with Modern Physics

Roger A. Freedman, Hugh D. Young

Chapter 11

Equilibrium and Elasticity - all with Video Answers

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Chapter Questions

Problem 1

A $0.120 \mathrm{~kg},$ 50.0-cm-long uniform bar has a small $0.055 \mathrm{~kg}$ mass glued to its left end and a small $0.110 \mathrm{~kg}$ mass glued to the other end. The two small masses can each be treated as point masses. You want to balance this system horizontally on a fulcrum placed just under its center of gravity. How far from the left end should the fulcrum be placed?

Ben Nicholson

Ben Nicholson

Numerade Educator

Problem 2

The center of gravity of a $5.00 \mathrm{~kg}$ irregular object is shown in Fig. E11.2. You need to move the center of gravity $2.20 \mathrm{~cm}$ to the left by gluing on a $1.50 \mathrm{~kg}$ mass, which will then be considered as part of the object. Where should the center of gravity of this additional mass be located?

Averell Hause

Carnegie Mellon University

Problem 3

A uniform rod is $2.00 \mathrm{~m}$ long and has mass $1.80 \mathrm{~kg} .$ A $2.40 \mathrm{~kg}$ clamp is attached to the rod. How far should the center of gravity of the clamp be from the left-hand end of the rod in order for the center of gravity of the composite object to be $1.20 \mathrm{~m}$ from the left-hand end of the rod?

Ben Nicholson

Numerade Educator

Problem 4

Consider the free-body diagram shown in Fig. 11.9b. (a) What is the horizontal distance of the center of gravity of the person-ladder system from the point where the ladder touches the ground? (b) What is the torque about the rotation axis shown in the figure (point $B$ ) computed by taking the total weight of the person plus ladder acting at the center of gravity? (c) How does the result in part (b) compare to the sum of the individual torques computed in Example 11.3 for the weight of the person and the weight of the ladder?

Hubert Agamasu

Numerade Educator

Problem 5

A uniform steel rod has mass $0.300 \mathrm{~kg}$ and length $40.0 \mathrm{~cm}$ and is horizontal. A uniform sphere with radius $8.00 \mathrm{~cm}$ and mass $0.900 \mathrm{~kg}$ is welded to one end of the bar, and a uniform sphere with radius $6.00 \mathrm{~cm}$ and mass $0.380 \mathrm{~kg}$ is welded to the other end of the bar. The centers of the rod and of each sphere all lie along a horizontal line. How far is the center of grayity of the combined obiect from the center of the rod?

Vishal Gupta

Numerade Educator

Problem 6

A uniform $300 \mathrm{~N}$ trapdoor in a floor is hinged at one side. Find the net upward force needed to begin to open it and the total force exerted on the door by the hinges (a) if the upward force is applied at the center and
(b) if the upward force is applied at the center of the edge opposite the hinges.

Averell Hause

Carnegie Mellon University

Problem 7

A ladder carried by a fire truck is $20.0 \mathrm{~m}$ long. The ladder weighs $3400 \mathrm{~N}$ and its center of gravity is at its center. The ladder is pivoted at one end (A) about a pin (Fig. E11.7); ignore the friction torque at the pin. The ladder is raised into position by a force applied by a hydraulic piston at $C$. Point $C$ is $8.0 \mathrm{~m}$ from $A,$ and the force $\overrightarrow{\boldsymbol{F}}$ exerted by the piston makes an angle of $40^{\circ}$ with the ladder. What magnitude must $\boldsymbol{F}$ have to just lift the ladder off the support bracket at $B ?$ Start with a free-body diagram of the ladder.

Narayan Hari

Numerade Educator

Problem 8

Two people are carrying a uniform wooden board that is $3.00 \mathrm{~m}$ long and weighs $160 \mathrm{~N}$. If one person applies an upward force equal to $60 \mathrm{~N}$ at one end, at what point does the other person lift? Begin with a free-body diagram of the board.

Prashant Bana

Numerade Educator

Problem 9

Two people carry a heavy electric motor by placing it on a light board $2.00 \mathrm{~m}$ long. One person lifts at one end with a force of $400 \mathrm{~N}$, and the other lifts the opposite end with a force of $600 \mathrm{~N}$. (a) What is the weight of the motor, and where along the board is its center of gravity located? (b) Suppose the board is not light but weighs $200 \mathrm{~N},$ with its center of gravity at its center, and the two people exert the same forces as before. What is the weight of the motor in this case, and where is its center of gravity located?

Ben Nicholson

Numerade Educator

Problem 10

A $60.0 \mathrm{~cm}$, uniform, $50.0 \mathrm{~N}$ shelf is supported horizontally by two vertical wires attached to the sloping ceiling (Fig. E11.10). A very small $25.0 \mathrm{~N}$ tool is placed on the shelf midway between the points where the wires are attached to it. Find the tension in each wire. Begin by making a free-body diagram of the shelf.

Averell Hause

Carnegie Mellon University

Problem 11

A $350 \mathrm{~N}$, uniform, $1.50 \mathrm{~m}$ bar is suspended horizontally by two vertical cables at each end. Cable $A$ can support a maximum tension of $500.0 \mathrm{~N}$ without breaking, and cable $B$ can support up to $400.0 \mathrm{~N}$. You want to place a small weight on this bar.
(a) What is the heaviest weight you can put on without breaking either cable, and (b) where should you put this weight?

Ben Nicholson

Numerade Educator

Problem 12

A uniform ladder $5.0 \mathrm{~m}$ long rests against a frictionless, vertical wall with its lower end $3.0 \mathrm{~m}$ from the wall. The ladder weighs $160 \mathrm{~N}$. The coefficient of static friction between the foot of the ladder
and the ground is $0.40 .$ A man weighing $740 \mathrm{~N}$ climbs slowly up the ladder. Start by drawing a free-body diagram of the ladder. (a) What is the maximum friction force that the ground can exert on the ladder at its lower end? (b) What is the actual friction force when the man has climbed $1.0 \mathrm{~m}$ along the ladder? (c) How far along the ladder can the man climb before the ladder starts to slip?

Narayan Hari

Numerade Educator

Problem 13

A diving board $3.00 \mathrm{~m}$ long is supported at a point $1.00 \mathrm{~m}$ from the end, and a diver weighing $500 \mathrm{~N}$ stands at the free end (Fig. $\mathrm{E} 11.13$ ). The diving board is of uniform cross section and weighs $280 \mathrm{~N}$. Find
(a) the force at the support point and (b) the force at the left-hand end.

Penny Riley

Numerade Educator

Problem 14

A uniform aluminum beam $9.00 \mathrm{~m}$ long, weighing $300 \mathrm{~N}$, rests symmetrically on two supports $5.00 \mathrm{~m}$ apart (Fig. E11.14). A boy weighing $600 \mathrm{~N}$ starts at point $A$ and walks toward the right. (a) In the same diagram construct two graphs showing the upward forces $F_{A}$ and $F_{B}$ exerted on the beam at points $A$ and $B$, as functions of the coordinate $x$ of the boy. Let $1 \mathrm{~cm}=100 \mathrm{~N}$ vertically and $1 \mathrm{~cm}=1.00 \mathrm{~m}$ horizontally. (b) From your diagram, how far beyond point $B$ can the boy walk before the beam tips? (c) How far from the right end of the beam should support $B$ be placed so that the boy can walk just to the end of the beam without causing it to tip?

Averell Hause

Carnegie Mellon University

Problem 15

Find the tension $T$ in each cable and the magnitude and direction of the force exerted on the strut by the pivot in each of the arrangements in Fig. $\mathbf{E} 11.15 .$ In each case let $w$ be the weight of the suspended crate full of priceless art objects. The strut is uniform and also has weight $w$. Start each case with a free-body diagram of the strut.

Narayan Hari

Numerade Educator

Problem 16

The horizontal beam in Fig. E11.16 weighs $190 \mathrm{~N},$ and its center of gravity is at its center. Find (a) the tension in the cable and
(b) the horizontal and vertical components of the force exerted on the beam
at the wall.

Averell Hause

Carnegie Mellon University

Problem 17

The boom shown in Fig. $\mathbf{E} \mathbf{1 1 . 1 7}$ weighs $2600 \mathrm{~N}$ and is attached to a frictionless pivot at its lower end. It is not uniform; the distance of its center of gravity from the pivot is $35 \%$ of its length. Find (a) the tension in the guy wire and (b) the horizontal and vertical components of the force exerted on the boom at its lower end. Start with a free-body diagram of the boom.

Hubert Agamasu

Numerade Educator

Problem 18

You are pushing an $80.0 \mathrm{~N}$ wheelbarrow as shown in Fig. E11.18. You lift upward on the handle of the wheelbarrow so that
the only point of contact between the wheelbarrow and the ground is at the front wheel. Assume the distances are as shown in the figure, where $0.50 \mathrm{~m}$ is the horizontal distance from the center of the wheel to the center of gravity of the wheelbarrow. The center of gravity of the dirt in the wheelbarrow is assumed to also be a horizontal distance of $0.50 \mathrm{~m}$ from the center of the wheel. Estimate the maximum total upward force that you can apply to the wheelbarrow handles. (a) If you apply this estimated force, what is the maximum weight of dirt that you can carry in the wheelbarrow? Express your answer in pounds. (b) If the dirt has the weight you calculated in part (a), what upward force does the ground apply to the wheel of the wheelbarrow?

Jerrah Biggerstaff

Jerrah Biggerstaff

Numerade Educator

Problem 19

A $9.00-\mathrm{m}$ -long uniform beam is hinged to a vertical wall and held horizontally by a $5.00-\mathrm{m}-$ long cable attached to the wall 4.00 $\mathrm{m}$ above the hinge (Fig. $\mathrm{E} 11.19)$. The metal of this cable has a test strength of $1.00 \mathrm{kN}$, which means that it will break if the tension in it exceeds that amount. (a) Draw a free-body diagram of the beam.
(b) What is the heaviest beam that the cable can support in this configuration?
(c) Find the horizontal and vertical components of the force the hinge exerts on the beam. Is the vertical component upward or downward?

Ben Nicholson

Numerade Educator

Problem 20

A $15,000 \mathrm{~N}$ crane pivots around a friction-free axle at
its base and is supported by a cable making a $25^{\circ}$ angle with the crane (Fig. E11.20). The crane is $16 \mathrm{~m}$ long and is not uniform, its center of gravity being $7.0 \mathrm{~m}$ from the axle as measured along the crane. The cable is attached
$3.0 \mathrm{~m}$ from the upper end of the crane. When the crane is raised to $55^{\circ}$ above the horizontal holding an $11,000 \mathrm{~N}$ pallet of bricks by a $2.2 \mathrm{~m},$ very light cord, find (a) the tension in the cable and (b) the horizontal and vertical components of the force that the axle exerts on the crane. Start with a free-body diagram of the crane.

Jilin Wang

Boston University

Problem 21

A 3.00-m-long, $190 \mathrm{~N}$, uniform rod at the $\mathrm{zoo}$ is held in a horizontal position by two ropes at its ends (Fig. E11.21). The left rope makes an angle of $150^{\circ}$ with the rod, and the right rope makes an angle $\theta$ with the horizontal. A $90 \mathrm{~N}$ howler monkey (Alouatta seniculus) hangs motionless $0.50 \mathrm{~m}$ from the right end of the rod as he carefully studies you. Calculate the tensions in the two ropes and the angle $\theta$. First make a free-body diagram of the rod.

John Barr

Numerade Educator

Problem 22

A nonuniform beam $4.50 \mathrm{~m}$ long and weighing $1.40 \mathrm{kN}$ makes an angle of $25.0^{\circ}$ below the horizontal. It is held in position by a frictionless pivot at its upper right end and by a cable $3.00 \mathrm{~m}$ farther down the beam and perpendicular to it (Fig. $\mathbf{E} 11.22$ ). The center of gravity of the beam is $2.00 \mathrm{~m}$ down the beam from the pivot. Lighting equipment exerts a $5.00 \mathrm{kN}$ downward force on the lower left end of the beam. Find the tension $T$ in the cable and the horizontal and vertical components of the force exerted on the beam by the pivot. Start by sketching a free-body diagram of the beam.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 23

Two forces equal in magnitude and opposite in direction, acting on an object at two different points, form what is called a couple. Two antiparallel forces with equal magnitudes $F_{1}=F_{2}=8.00 \mathrm{~N}$ are applied to a rod as shown in Fig. $\mathbf{E 1 1 . 2 3}$
(a) What should the distance $l$ between the forces be if they are to provide a net torque of $6.40 \mathrm{~N} \cdot \mathrm{m}$ about the left end of the rod? (b) Is the sense of this torque clockwise or counterclockwise?
(c) Repeat parts
(a) and (b) for a pivot at the point on the rod where $\overrightarrow{\boldsymbol{F}}_{2}$ is applied.

Vishal Gupta

Numerade Educator

Problem 24

You are doing exercises on a Nautilus machine in a gym to strengthen your deltoid (shoulder) muscles. Your arms are raised vertically and can pivot around the shoulder joint, and you grasp the cable of the machine in your hand $64.0 \mathrm{~cm}$ from your shoulder joint. The deltoid muscle is attached to the humerus $15.0 \mathrm{~cm}$ from the shoulder joint and makes a $12.0^{\circ}$ angle with that bone (Fig. $\mathrm{E} 11.24$ ) . If you have set the tension in the cable of the machine to $36.0 \mathrm{~N}$ on each arm, what is the tension in each deltoid muscle if you simply hold your outstretched arms in place? (Hint: Start by making a clear free-body diagram of your arm.

Averell Hause

Carnegie Mellon University

Problem 25

A uniform rod has one end attached to a vertical wall by a frictionless hinge. A horizontal wire runs from the other end of the rod to a point on the wall above the hinge and holds the rod at an angle $\theta$ above the horizontal. You vary the angle $\theta$ by changing the length of the wire, and for each $\theta$ you measure the tension $T$ in the wire. You plot your data in the form of a $T$ -versus-cot $\theta$ graph. The data lie close to a straight line that has slope $30.0 \mathrm{~N}$. What is the mass of the rod?

Vishal Gupta

Numerade Educator

Problem 26

A relaxed biceps muscle requires a force of $25.0 \mathrm{~N}$ for an elongation of $3.0 \mathrm{~cm} ;$ the same muscle under maximum tension requires a force of $500 \mathrm{~N}$ for the same elongation. Find Young's modulus for the muscle tissue under each of these conditions if the muscle is assumed to be a uniform cylinder with length $0.200 \mathrm{~m}$ and cross-sectional area $50.0 \mathrm{~cm}^{2}$.

Averell Hause

Carnegie Mellon University

Problem 27

A circular steel wire $2.00 \mathrm{~m}$ long must stretch no more than $0.25 \mathrm{~cm}$ when a tensile force of $700 \mathrm{~N}$ is applied to each end of the wire. What minimum diameter is required for the wire?

Ben Nicholson

Numerade Educator

Problem 28

Two cylindrical rods, one steel and the other copper, are joined end to end. Each rod is $0.750 \mathrm{~m}$ long and $1.50 \mathrm{~cm}$ in diameter. The combination is subjected to a tensile force with magnitude $4000 \mathrm{~N}$. For each rod, what are (a) the strain and (b) the elongation?

Narayan Hari

Numerade Educator

Problem 29

A metal rod that is $4.00 \mathrm{~m}$ long and $0.50 \mathrm{~cm}^{2}$ in crosssectional area is found to stretch $0.20 \mathrm{~cm}$ under a tension of $5000 \mathrm{~N}$. What is Young's modulus for this metal?

Ben Nicholson

Numerade Educator

Problem 30

A nylon rope used by mountaineers elongates $1.10 \mathrm{~m}$ under the weight of a $65.0 \mathrm{~kg}$ climber. If the rope is $45.0 \mathrm{~m}$ in length and $7.0 \mathrm{~mm}$ in diameter, what is Young's modulus for nylon?

Averell Hause

Carnegie Mellon University

Problem 31

A lead sphere has volume $6.0 \mathrm{~cm}^{3}$ when it is resting on a lab table, where the pressure applied to the sphere is atmospheric pressure. The sphere is then placed in the fluid of a hydraulic press. What increase in the pressure above atmospheric pressure produces a $0.50 \%$ decrease in the volume of the sphere?

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 32

A vertical, solid steel post $25 \mathrm{~cm}$ in diameter and $2.50 \mathrm{~m}$ long is required to support a load of $8000 \mathrm{~kg}$. You can ignore the weight of the post. What are (a) the stress in the post; (b) the strain in the post; and (c) the change in the post's length when the load is applied?

Averell Hause

Carnegie Mellon University

Problem 33

The bulk modulus for bone is $15 \mathrm{GPa}$. (a) If a diver-in-training is put into a pressurized suit, by how much would the pressure have to be raised (in atmospheres) above atmospheric pressure to compress her bones by $0.10 \%$ of their original volume? (b) Given that the pressure in the ocean increases by $1.0 \times 10^{4} \mathrm{~Pa}$ for every meter of depth below the surface, how deep would this diver have to go for her bones to compress by $0.10 \%$ ? Does it seem that bone compression is a problem she needs to be concerned with when diving?

Dominador Tan

Numerade Educator

Problem 34

A solid gold bar is pulled up from the hold of the sunken RMS Titanic. (a) What happens to its volume as it goes from the pressure at the ship to the lower pressure at the ocean's surface? (b) The pressure difference is proportional to the depth. How many times greater would the volume change have been had the ship been twice as deep? (c) The bulk modulus of lead is one-fourth that of gold. Find the ratio of the volume change of a solid lead bar to that of a gold bar of equal volume for the same pressure change.

Averell Hause

Carnegie Mellon University

Problem 35

A specimen of oil having an initial volume of $600 \mathrm{~cm}^{3}$ is subjected to a pressure increase of $3.6 \times 10^{6} \mathrm{~Pa}$, and the volume is found to decrease by $0.45 \mathrm{~cm}^{3} .$ What is the bulk modulus of the material? The compressibility?

Ajay Singhal

Numerade Educator

Problem 36

In the Challenger Deep of the Marianas Trench, the depth of seawater is $10.9 \mathrm{~km}$ and the pressure is $1.16 \times 10^{8} \mathrm{~Pa}$ (about $\left.1.15 \times 10^{3} \mathrm{~atm}\right) .$ (a) If a cubic meter of water is taken from the surface to this depth, what is the change in its volume? (Normal atmospheric pressure is about $1.0 \times 10^{5} \mathrm{~Pa}$. Assume that $k$ for seawater is the same as the freshwater value given in Table $11.2 .$ ) (b) What is the density of seawater at this depth? (At the surface, seawater has a density of $\left.1.03 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3} .\right)$

Averell Hause

Carnegie Mellon University

Problem 37

A square steel plate is $10.0 \mathrm{~cm}$ on a side and $0.500 \mathrm{~cm}$ thick. (a) Find the shear strain that results if two forces, each of magnitude $9.0 \times 10^{5} \mathrm{~N}$ and in opposite directions, act tangent to the surfaces of a pair of opposite sides of the object, as in Fig. $\mathrm{E} 11.18 .$ (b) Find the displacement $x$ in centimeters.

Dominador Tan

Numerade Educator

Problem 38

In lab tests on a $9.25 \mathrm{~cm}$ cube of a certain material, a force of $1375 \mathrm{~N}$ directed at $8.50^{\circ}$ to the cube (Fig. E11.38) causes the cube to deform through an angle of $1.24^{\circ} .$ What is the shear modulus of the material?

Ryan Hood

Numerade Educator

Problem 39

A steel wire with radius $r_{\text {steel }}$ has a fractional increase in length of $\left(\Delta l / l_{0}\right)_{\text {steel }}$ when the tension in the wire is increased from zero to $T_{\text {steel }}$. An aluminum wire has radius $r_{\text {al }}$ that is twice the radius of the steel wire: $r_{\mathrm{al}}=2 r_{\text {steel }} .$ In terms of $T_{\text {steel }},$ what tension in the aluminum wire produces the same fractional change in length as in the steel wire?

Dominador Tan

Numerade Educator

Problem 40

You apply a force of magnitude $F_{\perp}$ to one end of a wire and another force $F_{\perp}$ in the opposite direction to the other end of the wire. The cross-sectional area of the wire is $8.00 \mathrm{~mm}^{2}$. You measure the fractional change in the length of the wire, $\Delta l / l_{0},$ for several values of $F_{\perp}$ When you plot your data with $\Delta l / l_{0}$ on the vertical axis and $F_{\perp}$ (in units of $\mathrm{N}$ ) on the horizontal axis, the data lie close to a line that has slope $8.0 \times 10^{-7} \mathrm{~N}^{-1} .$ What is the value of Young's modulus for this wire?

Vishal Gupta

Numerade Educator

Problem 41

An increase in applied pressure $\Delta p_{1}$ produces a fractional volume change of $\left(\Delta V / V_{0}\right)_{1}$ for a sample of glycerin. In terms of $\Delta p_{1}$, what pressure increase above atmospheric pressure is required to produce the same volume change $\left(\Delta V / V_{0}\right)_{1}$ for a sample of ethyl alcohol?

Dominador Tan

Numerade Educator

Problem 42

A brass wire is to withstand a tensile force of $350 \mathrm{~N}$ without breaking. What minimum diameter must the wire have?

Averell Hause

Carnegie Mellon University

Problem 43

In a materials testing laboratory, a metal wire made from a new alloy is found to break when a tensile force of $90.8 \mathrm{~N}$ is applied perpendicular to each end. If the diameter of the wire is $1.84 \mathrm{~mm},$ what is the breaking stress of the alloy?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 44

A steel cable with cross-sectional area $3.00 \mathrm{~cm}^{2}$ has an elastic limit of $2.40 \times 10^{8} \mathrm{~Pa}$. Find the maximum upward acceleration that can be given a $1200 \mathrm{~kg}$ elevator supported by the cable if the stress is not to exceed one-third of the elastic limit.

Ben Nicholson

Numerade Educator

Problem 45

You are using a hammer to pull a nail from the floor, as shown in Fig. P11.45. Force $\vec{F}_{2}$ is the force you apply to the handle, and force $\vec{F}_{1}$ is the force the hammer applies to the nail. Estimate the maximum magnitude of the force $\vec{F}_{2}$ that you could apply to the hammer. If you apply this force, what is the magnitude of the force $\vec{F}_{1}$ that the hammer applies to the nail?

Manish Jain

Numerade Educator

Problem 46

A door $1.00 \mathrm{~m}$ wide and $2.00 \mathrm{~m}$ high weighs $330 \mathrm{~N}$ and is supported by two hinges, one $0.50 \mathrm{~m}$ from the top and the other $0.50 \mathrm{~m}$ from the bottom. Each hinge supports half the total weight of the door. Assuming that the door's center of gravity is at its center, find the horizontal components of force exerted on the door by each hinge.

Averell Hause

Carnegie Mellon University

Problem 47

A box of negligible mass rests at the left end of a $2.00 \mathrm{~m}$, $25.0 \mathrm{~kg}$ plank (Fig. $\mathrm{P} 11.47)$. The width of the box is $75.0 \mathrm{~cm}$, and sand is to be distributed uniformly throughout it. The center of gravity of the nonuniform plank is $50.0 \mathrm{~cm}$ from the right end. What mass of sand should be put into the box so that the plank balances horizontally on a fulcrum placed just below its midpoint?

Manish Jain

Numerade Educator

Problem 48

Sir Lancelot rides slowly out of the castle at Camelot and onto the 12.0 -m-long drawbridge that passes over the moat (Fig. $\mathbf{P 1 1 . 4 8}$ ). Unbeknownst to him, his enemies have partially severed the vertical cable holding up the front end of the bridge so that it will break under a tension of $5.80 \times 10^{3} \mathrm{~N}$. The bridge has mass $200 \mathrm{~kg}$ and its center of gravity is at its center. Lancelot, his lance, his armor, and his horse together have a combined mass of $600 \mathrm{~kg}$. Will the cable break before Lancelot reaches the end of the drawbridge? If so, how far from the castle end of the bridge will the center of gravity of the horse plus rider be when the cable breaks?

Dominador Tan

Numerade Educator

Problem 49

Mountaineers often use a rope to lower themselves down the face of a cliff (this is called rappelling). They do this with their body nearly horizontal and their feet pushing against the cliff (Fig. $\mathbf{P 1 1 . 4 9}$ ). Suppose that an $82.0 \mathrm{~kg}$ climber, who is $1.90 \mathrm{~m}$ tall and has a center of gravity $1.1 \mathrm{~m}$ from his feet, rappels down a vertical cliff with his body raised $35.0^{\circ}$ above the horizontal. He holds the rope $1.40 \mathrm{~m}$ from his feet, and it makes a $25.0^{\circ}$ angle with the cliff face.
(a) What tension does his rope need to support?
(b) Find the horizontal and vertical components of the force that the cliff face exerts on the
climber's feet. (c) What minimum coefficient of static friction is needed to prevent the climber's feet from slipping on the cliff face if he has one foot at a time against the cliff?

Manish Jain

Numerade Educator

Problem 50

A uniform, $8.0 \mathrm{~m}, 1150 \mathrm{~kg}$
beam is hinged to a wall and supported by a thin cable attached 2.0 $\mathrm{m}$ from the free end of the beam
(Fig. $\mathbf{P 1 1 . 5 0}$ ). The beam is supported at an angle of $30.0^{\circ}$ above the horizontal. (a) Draw a free-body diagram of the beam.
(b) Find the ten-
sion in the cable. (c) How hard does the beam push inward on the wall?

Dominador Tan

Numerade Educator

Problem 51

A uniform, $255 \mathrm{~N}$ rod that is $2.00 \mathrm{~m}$ long carries a $225 \mathrm{~N}$ weight at its right end and an unknown weight $W$ toward the left end (Fig. $\mathbf{P 1 1 . 5 1}$ ). When $W$ is placed $50.0 \mathrm{~cm}$ from the left end of the rod, the system just balances horizontally when the fulcrum is located 75.0 $\mathrm{cm}$ from the right end. (a) Find $W .$ (b) If $W$ is now moved $25.0 \mathrm{~cm}$ to the right, how far and in what direction must the fulcrum be moved to restore balance?

Ben Nicholson

Numerade Educator

Problem 52

A claw hammer is used to pull a nail out of a board (see Fig. $\mathrm{P} 11.45$ ). The nail is at an angle of $60^{\circ}$ to the board, and a force $\overrightarrow{\boldsymbol{F}}_{1}$ of magnitude $400 \mathrm{~N}$ applied to the nail is required to pull it from the board. The hammer head contacts the board at point $A,$ which is $0.080 \mathrm{~m}$ from where the nail enters the board. A horizontal force $\vec{F}_{2}$ is applied to the hammer handle at a distance of $0.300 \mathrm{~m}$ above the board. What magnitude of force $\overrightarrow{\boldsymbol{F}}_{2}$ is required to apply the required $400 \mathrm{~N}$ force $\left(F_{1}\right)$ to the nail? (Ignore the weight of the hammer.)

Dominador Tan

Numerade Educator

Problem 53

You open a restaurant and hope to entice customers by hanging out a sign (Fig. $\mathbf{P} 1 \mathbf{1 . 5 3}$ ). The uniform horizontal beam supporting the sign is $1.50 \mathrm{~m}$ long, has a mass of $16.0 \mathrm{~kg},$ and is hinged to the wall. The sign itself is uniform with a mass of $28.0 \mathrm{~kg}$ and overall length of $1.20 \mathrm{~m}$. The two wires supporting the sign are each $32.0 \mathrm{~cm}$ long, are $90.0 \mathrm{~cm}$ apart, and are equally spaced from the middle of the sign. The cable supporting the beam is $2.00 \mathrm{~m}$ long.
(a) What minimum tension must your cable be able to support without having your sign come crashing down? (b) What minimum vertical force must the hinge be able to support without pulling out of the wall?

Dominador Tan

Numerade Educator

Problem 54

End $A$ of the bar $A B$ in Fig. $\mathbf{P 1 1 . 5 4}$ rests on a friction-
less horizontal surface, and end $B$ is hinged. A horizontal force $\vec{F}$ of magnitude $220 \mathrm{~N}$ is exerted on end
A. Ignore the weight of the bar. What are the horizontal and vertical com-
ponents of the force exerted by the bar on the hinge at $B ?$

Dominador Tan

Numerade Educator

Problem 55

A therapist tells a $74 \mathrm{~kg}$ patient with a broken leg that he must have his leg in a cast suspended horizontally. For minimum discomfort, the leg should be supported by a vertical strap attached at the center of mass of the leg-cast system (Fig. $\mathbf{P} 11.55$ ). To comply with these instructions, the patient consults a table of typical mass distributions and finds that both upper legs (thighs) together typically account for $21.5 \%$ of body weight and the center of mass of each thigh is $18.0 \mathrm{~cm}$ from the hip joint. The patient also reads that the two lower legs (including the feet) are $14.0 \%$ of body weight, with a center of mass $69.0 \mathrm{~cm}$ from the hip joint. The cast has a mass of $5.50 \mathrm{~kg}$, and its center of mass is $78.0 \mathrm{~cm}$ from the hip joint. How far from the hip joint should the supporting strap be attached to the cast?

Dominador Tan

Numerade Educator

Problem 56

A loaded cement mixer drives onto an old drawbridge, where it stalls with its center of gravity threequarters of the way across the span. The truck driver radios for help, sets the handbrake, and waits. Meanwhile, a boat approaches, so the drawbridge is raised by means of a cable attached to the end opposite the hinge (Fig. $\mathbf{P} 1 \mathbf{1 . 5 6}$ ). The drawbridge is $40.0 \mathrm{~m}$ long and has a mass of $18,000 \mathrm{~kg} ;$ its center of gravity is at its midpoint. The cement mixer, with driver, has mass $30,000 \mathrm{~kg}$. When the drawbridge has been raised to an angle of $30^{\circ}$ above the horizontal, the cable makes an angle of $70^{\circ}$ with the surface of the bridge. (a) What is the tension $T$ in the cable when the drawbridge is held in this position?
(b) What are the horizontal and vertical components of the force the hinge exerts on the span?

Anand Jangid

Numerade Educator

Problem 57

The left-hand end of a uniform rod of length $L$ and mass $m$ is attached to a vertical wall by a frictionless hinge. The rod is held at an angle $\theta$ above the horizontal by a horizontal wire that runs between the wall and the right-hand end of the rod. (a) If the tension in the wire is $T$, what is the magnitude of the angle $\theta$ that the rod makes with the horizontal? (b) The wire breaks and the rod rotates about the hinge. What is the angular speed of the rod as the rod passes through a horizontal position?

Dominador Tan

Numerade Educator

Problem 58

A $72.0 \mathrm{~kg}$ weightlifter doing arm raises holds a $7.50 \mathrm{~kg}$ weight. Her arm pivots around the elbow joint, starting $40.0^{\circ}$ below the horizontal (Fig. $\mathbf{P 1 1 . 5 8}$ ). Biometric measurements have shown that, together, the forearms and the hands account for $6.00 \%$ of a person's weight. since the upper arm is held vertically, the biceps muscle always acts vertically and is attached to the bones of the forearm $5.50 \mathrm{~cm}$ from the elbow joint. The center of mass of this person's forearm-hand combination is $16.0 \mathrm{~cm}$ from the elbow joint, along the bones of the forearm, and she holds the weight $38.0 \mathrm{~cm}$ from her elbow joint. (a) Draw a freebody diagram of the forearm. (b) What force does the biceps muscle exert on the forearm? (c) Find the magnitude and direction of the force that the elbow joint exerts on the forearm. (d) As the weightlifter raises her arm toward a horizontal position, will the force in the biceps muscle increase, decrease, or stay the same? Why?

Dominador Tan

Numerade Educator

Problem 59

The left-hand end of a light rod of length $L$ is attached to a vertical wall by a frictionless hinge. An object of mass $m$ is suspended from the rod at a point a distance $\alpha L$ from the hinge, where $0<\alpha \leq 1.00 .$ The rod is held in a horizontal position by a light wire that runs from the right-hand end of the rod to the wall. The wire makes an angle $\theta$ with the rod. (a) What is the angle $\beta$ that the net force exerted by the hinge on the rod makes with the horizontal? (b) What is the value of $\alpha$ for which $\beta=\theta ?$ (c) What is $\beta$ when $\alpha=1.00 ?$

Dominador Tan

Numerade Educator

Problem 60

The left-hand end of a slender uniform rod of mass $m$ is placed against a vertical wall. The rod is held in a horizontal position by friction at the wall and by a light wire that runs from the right-hand end of the rod to a point on the wall above the rod. The wire makes an angle $\theta$ with the rod. (a) What must the magnitude of the friction force be in order for the rod to remain at rest? (b) If the coefficient of static friction between the rod and the wall is $\mu_{\mathrm{s}},$ what is the maximum angle between the wire and the rod at which the rod doesn't slip at the wall?

Dominador Tan

Numerade Educator

Problem 61

A uniform, 7.5-m-long beam weighing $6490 \mathrm{~N}$ is hinged to a wall and supported by a thin cable attached $1.5 \mathrm{~m}$ from the free end of the beam. The cable runs between the beam and the wall and makes a $40^{\circ}$ angle with the beam. What is the tension in the cable when the beam is at an angle of $30^{\circ}$ above the horizontal?

Ben Nicholson

Numerade Educator

Problem 62

A uniform drawbridge must be held at a $37^{\circ}$ angle above the horizontal to allow ships to pass underneath. The drawbridge weighs $45,000 \mathrm{~N}$ and is $14.0 \mathrm{~m}$ long. A cable is connected $3.5 \mathrm{~m}$ from the hinge where the bridge pivots (measured along the bridge) and pulls horizontally on the bridge to hold it in place. (a) What is the tension in the cable? (b) Find the magnitude and direction of the force the hinge exerts on the bridge. (c) If the cable suddenly breaks, what is the magnitude of the angular acceleration of the drawbridge just after the cable breaks? (d) What is the angular speed of the drawbridge as it becomes horizontal?

Averell Hause

Carnegie Mellon University

Problem 63

As part of an exercise program, a $75 \mathrm{~kg}$ person does toe raises in which he raises his entire body weight on the ball of one foot (Fig. $\mathbf{P 1 1 . 6 3}$ ). The Achilles tendon pulls straight upward on the heel bone of his foot. This tendon is $25 \mathrm{~cm}$ long and has a crosssectional area of $78 \mathrm{~mm}^{2}$ and a Young's modulus of 1470 MPa.
(a) Draw a free-body diagram of the person's foot (everything below the ankle joint). Ignore the weight of the foot. (b) What force does the Achilles tendon exert on the heel during this exercise? Express your answer in newtons and in multiples of his weight. (c) By how many millimeters does the exercise stretch his Achilles tendon?

Dominador Tan

Numerade Educator

Problem 64

(a) In Fig. P11.64 a $6.00-\mathrm{m}$ -long, uniform beam is hanging from a point $1.00 \mathrm{~m}$ to the right of its center. The beam weighs $140 \mathrm{~N}$ and makes an angle of $30.0^{\circ}$ with the vertical. At the right-hand end of the beam a $100.0 \mathrm{~N}$ weight is hung; an unknown weight $w$ hangs at the left end. If the system is in equilibrium, what is $w ?$ You can ignore the thickness of the beam. (b) If the beam makes, instead, an angle of $45.0^{\circ}$ with the vertical, what is $w ?$

Averell Hause

Carnegie Mellon University

Problem 65

The left-hand end of a uniform rod of mass $2.00 \mathrm{~kg}$ andlength $1.20 \mathrm{~m}$ is attached to a vertical wall by a frictionless hinge. The rod is held in a horizontal position by an aluminum wire that runs between the right-hand end of the rod and a point on the wall that is above the hinge. The cross-sectional radius of the wire is $2.50 \mathrm{~mm}$ and the wire makes an angle of $30.0^{\circ}$ with the rod. (a) What is the length of the wire? (b) An object of mass $90.0 \mathrm{~kg}$ is suspended from the right-hand end of the rod. What is the increase in the length of the wire when this object is added? In your analysis do you need to be concerned that the lengthening of the wire means that the rod is no longer horizontal?

Dominador Tan

Numerade Educator

Problem 66

A holiday decoration consists of two shiny glass spheres with $\begin{array}{lll}\text { masses } & 0.0240 \mathrm{~kg} \text { and } 0.0360 \mathrm{~kg}\end{array}$
suspended from a uniform rod with mass $0.120 \mathrm{~kg}$ and length $1.00 \mathrm{~m}$ (Fig. $\mathrm{P} 11.66$ ). The rod is suspended from the ceiling by a vertical cord at each end, so that it is horizontal. Calculate the tension in each of the cords $A$ through $F$

Averell Hause

Carnegie Mellon University

Problem 67

The yoga exercise "Downward- Facing Dog" requires stretching your hands straight out above your head and bending down to lean against the floor. This exercise is performed by a $750 \mathrm{~N}$ person as shown in Fig. $\mathrm{P} 11.67$. When he bends his body at the hip to a $90^{\circ}$ angle between his legs and trunk, his legs, trunk, head, and arms have the dimensions indicated. Furthermore, his legs and feet weigh a total of $277 \mathrm{~N}$, and their center of mass is $41 \mathrm{~cm}$ from his hip, measured along his legs. The person's trunk, head, and arms weigh $473 \mathrm{~N}$, and their center of gravity is $65 \mathrm{~cm}$ from his hip, measured along the upper body.
(a) Find the normal force that the floor exerts on each foot and on each hand, assuming that the person does not favor either hand or either foot.
(b) Find the friction force on each foot and on each hand, assuming that it is the same on both feet and on both hands (but not necessarily the same on the feet as on the hands). [Hint: First treat his entire body as a system; then isolate his legs (or his upper body).]

Ben Nicholson

Numerade Educator

Problem 68

A brass wire is $1.40 \mathrm{~m}$ long and has a cross-sectional area of $6.00 \mathrm{~mm}^{2}$. A small steel ball with mass $0.0800 \mathrm{~kg}$ is attached to the end of the wire. You hold the other end of the wire and whirl the ball in a vertical circle of radius $1.40 \mathrm{~m}$. What speed must the ball have at the lowest point of its path if its fractional change in length of the brass wire at this point from its unstretched length is $2.0 \times 10^{-5}$ ? Treat the ball as a point mass.

Prashant Bana

Numerade Educator

Problem 69

A worker wants to turn over a uniform, $1250 \mathrm{~N}$, rectangular crate by pulling at $53.0^{\circ}$ on one of its vertical sides (Fig. $\mathbf{P} 1 \mathbf{1 . 6 9}$ ). The floor is rough enough to prevent the crate from slipping.
(a) What pull is needed to just start the crate to tip?
(b) How hard does the floor push upward on the crate? (c) Find the friction force on the crate.
(d) What is the minimum coefficient of static friction needed to prevent the crate from slipping on the floor?

Dominador Tan

Numerade Educator

Problem 70

One end of a uniform meter stick is placed against a vertical wall (Fig. $\mathbf{P 1 1 . 7 0}$ ). The other end is held by a lightweight cord that makes an angle $\theta$ with the stick. The coefficient of static friction between the end of the meter stick and the wall is $0.40 .$
(a) What is the maximum value the angle $\theta$ can have if the stick is to remain in equilibrium?
(b) Let the angle $\theta$ be $15^{\circ} .$ A block of the same weight as the meter stick is suspended from the stick, as shown, at a distance $x$ from the wall. What is the minimum value of $x$ for which the stick will remain in equilibrium?
(c) When $\theta=15^{\circ}$, how large must the coefficient of static friction be so that the block can be attached $10 \mathrm{~cm}$ from the left end of the stick without causing it to slip?

Abhinav Roy

Numerade Educator

Problem 71

Two friends are carrying a $200 \mathrm{~kg}$ crate up a flight of stairs. The crate is $1.25 \mathrm{~m}$ long and $0.500 \mathrm{~m}$ high, and its center of gravity is at its center. The stairs make a $45.0^{\circ}$ angle with respect to the floor. The crate also is carried at a $45.0^{\circ}$ angle, so that its bottom side is parallel to the slope of the stairs (Fig. $\mathbf{P 1 1 . 7 1}$ ). 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?

Dominador Tan

Numerade Educator

Problem 72

In a city park a nonuniform wooden beam $4.00 \mathrm{~m}$ long is suspended horizontally by a light steel cable at each end. The cable at the left-hand end makes an angle of $30.0^{\circ}$ with the vertical and has tension $620 \mathrm{~N}$. The cable at the right-hand end of the beam makes an angle of $50.0^{\circ}$ 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.

Averell Hause

Carnegie Mellon University

Problem 73

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 $T_{\max }$ represent the maximum value of the tendon tension. Use the results of Example 11.4 to express $w_{\max }$ (the maximum weight that can be held) in terms of $T_{\max }$ $L, D,$ and $h .$ Your expression should $n o t$ include the angle $\theta .$ (b) The tendons of different primates are attached to the forearm at different values of $D .$ Calculate the derivative of $w_{\max }$ with respect to $D,$ and

Dominador Tan

Numerade Educator

Problem 74

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 $\overrightarrow{\boldsymbol{F}}$ (Fig. $\mathbf{P} 11.74) .$ What is the smallest magnitude of the force $\overrightarrow{\boldsymbol{F}}$ 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?

Averell Hause

Carnegie Mellon University

Problem 75

A gate $4.00 \mathrm{~m}$ wide and $2.00 \mathrm{~m}$ high weighs $700 \mathrm{~N}$. 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 $C D$ is connected as shown in Fig. $\mathbf{P} 1 \mathbf{1 . 7 5}$. The tension in $C D$ is increased until the horizontal force at hinge $A$ is zero. What are
(a) the tension in the wire $C D ;$ (b) the magnitude of the horizontal component of the force at hinge $B ;$ (c) the combined vertical force exerted by hinges $A$ and $B ?$

Dominador Tan

Numerade Educator

Problem 76

If you put a uniform block at the edge of a table, the centerof 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. $\mathbf{P 1 1 . 7 6 ) ?}$
(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.)

Dominador Tan

Numerade Educator

Problem 77

Two uniform, $75.0 \mathrm{~g}$ marbles $2.00 \mathrm{~cm}$ in diameter are stacked as shown in Fig. $\mathrm{P} 11.77$ in a container that is $3.00 \mathrm{~cm}$ 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?

Dominador Tan

Numerade Educator

Problem 78

Two identical, uniform beams weighing $260 \mathrm{~N}$ 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 $53.0^{\circ}$ between the beams. The beams are suspended from the ceiling by vertical wires such that they form a "V" (Fig. $\mathbf{P 1 1 . 7 8}$ ). (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?

Dominador Tan

Numerade Educator

Problem 79

An engineer is designing a conveyor system for loading hay bales into a wagon (Fig. $\mathbf{P} 1 \mathbf{1} .79)$. Each bale is $0.25 \mathrm{~m}$ wide, $0.50 \mathrm{~m}$ high, and $0.80 \mathrm{~m}$ long (the dimension perpendicular to the plane of the figure), with mass $30.0 \mathrm{~kg}$. 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 $\beta$ 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 $0.40 ?$

Dominador Tan

Numerade Educator

Problem 80

Ancient pyramid builders are balancing a uniform rectangular stone slab of weight $w$, tipped at an angle $\theta$ above the horizontal, using a rope (Fig. $\mathbf{P} 11.80$ ). The rope is held by five workers who share the force equally. (a) If $\theta=20.0^{\circ},$ what force does each worker exert on the rope? (b) As $\theta$ 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 $\theta$ exceeds this value?

Dominador Tan

Numerade Educator

Problem 81

A garage door is mounted on an overhead rail (Fig. $\mathbf{P 1 1 . 8 1}$ ). 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 $2.00 \mathrm{~m}$ and each is $0.50 \mathrm{~m}$ from the vertical
sides of the door. The door is uniform and weighs $950 \mathrm{~N}$. It is pushed to the left at constant speed by a horizontal force $\overrightarrow{\boldsymbol{F}},$ that is applied as shown in the figure. (a) If the distance $h$ is $1.60 \mathrm{~m},$ 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.

Dominador Tan

Numerade Educator

Problem 82

A $12.0 \mathrm{~kg}$ mass, fastened to the end of an aluminum rod with an unstretched length of $0.70 \mathrm{~m},$ is whirled in a vertical circle with a constant angular speed of $120 \mathrm{rev} / \mathrm{min}$. The cross-sectional area of the rod is $0.014 \mathrm{~cm}^{2}$. Calculate the elongation of the rod when the mass is (a) at the lowest point of the path and (b) at the highest point of its path.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 83

A 1.05 -m-long rod of negligible weight is supported at its ends by wires $A$ and $B$ of equal length (Fig. $\mathbf{P} \mathbf{1 1 . 8 3}$ ). The cross-sectional area of $A$ is $2.00 \mathrm{~mm}^{2}$ and that of $B$ is $4.00 \mathrm{~mm}^{2}$. Young's modulus for wire $A$ is $1.80 \times 10^{11} \mathrm{~Pa} ;$ that
for $B$ is $1.20 \times 10^{11} \mathrm{~Pa}$. 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 $B ?$

Vishal Gupta

Numerade Educator

Problem 84

An amusement park ride consists of airplane-shaped cars attached to steel rods (Fig. $\mathbf{P} \mathbf{1 1 . 8 4}$ ). Each rod has a length of $15.0 \mathrm{~m}$ and a cross-sectional area of $8.00 \mathrm{~cm}^{2}$. The rods are attached to a frictionless
hinge at the top, so that the cars can swing outward when the ride rotates.
(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 $1900 \mathrm{~N}$.) (b) When operating, the ride has a maximum angular speed of 12.0 rev $/$ min. How much is the rod stretched then?

Dominador Tan

Numerade Educator

Problem 85

The compressive strength of our bones is important in everyday life. Young's modulus for bone is about $1.4 \times 10^{10} \mathrm{~Pa}$. 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 $3.0 \mathrm{~cm}^{2} ?$ (This is approximately the cross-sectional area of a tibia, or shin bone, at its narrowest point.) (b) Estimate the maximum height from which a $70 \mathrm{~kg}$ 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 $0.030 \mathrm{~s}$, and assume that the stress on his two legs is distributed equally.

Dominador Tan

Numerade Educator

Problem 86

You are to use a long, thin wire to build a pendulum in a science mu-
seum. The wire has
an unstretched length of $22.0 \mathrm{~m}$ and a cir-
cular cross section of
$\begin{array}{lll}\text { diameter } & 0.860 & \mathrm{~mm}\end{array}$
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 $9.50 \mathrm{~kg}$ 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 $36.0^{\circ}$. 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 $\Delta l$ of the wire for several different test masses. Figure $\mathbf{P} 1 \mathbf{1} .86,$ a graph of $\Delta l$ versus $m$ shows the results and the straight line that gives the best fit to the data. The equation for this line is $\Delta l=(0.422 \mathrm{~mm} / \mathrm{kg}) m$
(a) Assume that $g=9.80 \mathrm{~m} / \mathrm{s}^{2},$ and use Fig. $\mathrm{P} 11.86$ to calculate Young's modulus $Y$ for this wire. (b) You remove the test masses, attach the $9.50 \mathrm{~kg}$ sphere, and release the sphere from rest, with the wire displaced by $36.0^{\circ} .$ Calculate the amount the wire will stretch as it swings through the vertical. Ignore air resistance.

Dominador Tan

Numerade Educator

Problem 87

You need to measure the mass $M$ of a $4.00-\mathrm{m}$ -long 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 $1.88 \mathrm{~m}$ from its left-hand end. You then place the bar on the pivot so that the point of support is $1.50 \mathrm{~m}$ from the left-hand end of the bar. Next you suspend a $2.00 \mathrm{~kg}$ mass $\left(m_{1}\right)$ from the bar at a point $0.200 \mathrm{~m}$ from the left-hand end. Finally, you suspend a mass $m_{2}=1.00 \mathrm{~kg}$ 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 $m_{2}$ and record each corresponding value of $x$. The table gives your results.
$$
\begin{array}{l|llllll}
m_{2}(\mathrm{~kg}) & 1.00 & 1.50 & 2.00 & 2.50 & 3.00 & 4.00 \\
\hline x(\mathrm{~m}) & 3.50 & 2.83 & 2.50 & 2.32 & 2.16 & 2.00
\end{array}
$$
(a) Draw a free-body diagram for the bar when $m_{1}$ and $m_{2}$ are suspended from it. (b) Apply the static equilibrium equation $\Sigma \tau_{z}=0$ with the axis at the location of the knife-edge pivot. Solve the equation for $x$ as a function of $m_{2}$. (c) Plot $x$ versus $1 / m_{2}$. 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 $g=9.80 \mathrm{~m} / \mathrm{s}^{2} .$ (d) What is the $y$ -intercept of the straight line that fits the data? Explain why it has this value.

Dominador Tan

Numerade Educator

Problem 88

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 $8.50 \mathrm{~kg}$ 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 $\theta$ with the bar. The architect has proposed four possible ways to connect the cable and asked you to assess them:
$$
\begin{array}{lllll}
\text { Alternative } & \text { A } & \text { B } & \text { C } & \text { D } \\
\hline x(\mathrm{~m}) & 2.00 & 1.50 & 0.75 & 0.50 \\
\theta(\text { degrees }) & 30 & 60 & 37 & 75
\end{array}
$$
(a) There is concern about the strength of the cable that will be required. Which set of $x$ and $\theta$ 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 $\theta$ 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 $\theta$ 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.

Mariano Edutestprod

Mariano Edutestprod

Numerade Educator

Problem 89

Two ladders, $4.00 \mathrm{~m}$ and $3.00 \mathrm{~m}$ long, are hinged at point $A$ and tied together by a horizontal rope $0.90 \mathrm{~m}$ above the floor (Fig. $\mathbf{P 1 1 . 8 9}$ ). The ladders weigh $480 \mathrm{~N}$ and $360 \mathrm{~N}$, 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 $800 \mathrm{~N}$ painter stands at point $A,$ find the tension in the horizontal rope.

Mariano Edutestprod

Mariano Edutestprod

Numerade Educator

Problem 90

One end of a post weighing $400 \mathrm{~N}$ and with height $h$ rests on a rough horizontal surface with $\mu_{\mathrm{s}}=0.30$. The upper end is held by a rope fastened to the surface and mak-
ing an angle of $36.9^{\circ}$ with the post (Fig. $\mathrm{P} 11.90$ ). A horizontal force $\vec{F}$ is exerted on the post as shown. (a) If the force $\overrightarrow{\boldsymbol{F}}$ 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 $\frac{6}{10}$ 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.

Mariano Edutestprod

Mariano Edutestprod

Numerade Educator

Problem 91

An angler hangs a $4.50 \mathrm{~kg}$ fish from a vertical steel wire $1.50 \mathrm{~m}$ long and $5.00 \times 10^{-3} \mathrm{~cm}^{2}$ 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 $\vec{F}$ at the lower end of the wire, pulling it very slowly downward by $0.500 \mathrm{~mm}$ from its equilibrium position. For this downward motion, calculate (b) the work done by gravity; (c) the work done by the force $\overrightarrow{\boldsymbol{F}} ;$ (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).

Mariano Edutestprod

Mariano Edutestprod

Numerade Educator

Problem 92

What is tension $T_{2}$ in the rope behind him? (a) $590 \mathrm{~N} ;$ (b) $650 \mathrm{~N} ;$ (c) $860 \mathrm{~N} ;$ (d) $1100 \mathrm{~N}$.

Averell Hause

Carnegie Mellon University

Problem 93

If he leans slightly farther back (increasing the angle between his body and the vertical) but remains stationary in this new position, which of the following statements is true? Assume that the rope remains horizontal. (a) The difference between $T_{1}$ and $T_{2}$ will increase, balancing the increased torque about his feet that his weight produces when he leans farther back; (b) the difference between $T_{1}$ and $T_{2}$ will decrease, balancing the increased torque about his feet that his weight produces when he leans farther back; (c) neither $T_{1}$ nor $T_{2}$ will change, because no other forces are changing; (d) both $T_{1}$ and $T_{2}$ will change, but the difference between them will remain the same.

Ben Nicholson

Numerade Educator

Problem 94

His body is again leaning back at $30.0^{\circ}$ to the vertical, but now the height at which the rope is held above-but still parallel to - the ground is varied. The tension in the rope in front of the competitor $\left(T_{1}\right)$ is measured as a function of the shortest distance between the rope and the ground (the holding height). Tension $T_{1}$ is found to decrease as the holding height increases. What could explain this observation? As the holding height increases, (a) the moment arm of the rope about his feet decreases due to the angle that his body makes with the vertical; (b) the moment arm of the weight about his feet decreases due to the angle that his body makes with the vertical; (c) a smaller tension in the rope is needed to produce a torque sufficient to balance the torque of the weight about his feet; (d) his center of mass moves down to compensate, so less tension in the rope is required to maintain equilibrium.

Averell Hause

Carnegie Mellon University

Problem 95

His body is leaning back at $30.0^{\circ}$ to the vertical, but the coefficient of static friction between his feet and the ground is suddenly reduced to $0.50 .$ What will happen? (a) His entire body will accelerate forward; (b) his feet will slip forward; (c) his feet will slip backward; (d) his feet will not slip.

Ben Nicholson

Numerade Educator