College Physics: A Strategic Approach

Randall D. Knight, Brian Jones, Stuart Field

Chapter 8

Equilibrium and Elasticity - all with Video Answers

Educators

+ 2 more educators

Chapter Questions

Problem 1

A 64 kg student stands on a very light, rigid board that rests on a bathroom scale at each end, as shown in Figure P8.1. What is the reading on each of the scales?

Narayan Hari

Narayan Hari

Numerade Educator

Problem 2

Suppose the student in Figure $\mathrm{P} 8.1$ is $54 \mathrm{kg}$, and the board being stood on has a $10 \mathrm{kg}$ mass. What is the reading on each of the scales?

Narayan Hari

Numerade Educator

Problem 3

Il How close to the right edge of the 56 kg picnic table shown in Figure $\mathrm{P} 8.3$ can a $70 \mathrm{kg}$ man stand without the table tipping over? Hint: When the table is just about to tip, what is the force of the ground on the table's left leg?

Narayan Hari

Numerade Educator

Problem 4

In Figure $\mathrm{P} 8.4,$ a $70 \mathrm{kg}$ man walks out on a $10 \mathrm{kg}$ beam that rests on, but is not attached to, two supports. When the beam just starts to tip, what is the force exerted on the beam by the right support?

Narayan Hari

Numerade Educator

Problem 5

You're carrying a $3.6-\mathrm{m}$ -long, $25 \mathrm{kg}$ pole to a construction site when you decide to stop for a rest. You place one end of the pole on a fence post and hold the other end of the pole $35 \mathrm{cm}$ from its tip. How much force must you exert to keep the pole motionless in a horizontal position?

Narayan Hari

Numerade Educator

Problem 6

A typical horse weighs $5000 \mathrm{N}$. The distance between the front and rear hooves and the distance from the rear hooves to the center of mass for a typical horse are shown in Figure P8.6. What fraction of the horse's weight is borne by the front hooves?

Narayan Hari

Numerade Educator

Problem 7

A vendor hangs an $8.0 \mathrm{kg}$ sign in front of his shop with a cable held away from the building by a lightweight pole. The pole is free to pivot about the end where it touches the wall, as shown in Figure P8.7. What is the tension in the cable?

Narayan Hari

Numerade Educator

Problem 8

in Figure P8.8. The hippo carries $60 \%$ of its weight on its front feet. How far from its tail is the hippo's center of gravity?

Narayan Hari

Numerade Educator

Problem 9

The two objects in Figure $\mathrm{P} 8.9$ are balanced on the pivot. What is distance $d ?$

Narayan Hari

Numerade Educator

Problem 10

A bicycle mechanic is checking a road bike's chain. He applies a $45 \mathrm{N}$ force to a pedal at the angle shown in Figure P8.10 while keeping the wheel from rotating. The pedal is $17 \mathrm{cm}$ from the center of the crank; the gear has a diameter of $16 \mathrm{cm} .$ What is the tension in the chain?

Narayan Hari

Numerade Educator

Problem 11

A $60 \mathrm{kg}$ diver stands at the end of a $30 \mathrm{kg}$ springboard, as shown in Figure $\mathrm{P} 8.11 .$ The board is attached to a hinge at the left end but simply rests on the right support. What is the magnitude of the vertical force exerted by the hinge on the board?

Narayan Hari

Numerade Educator

Problem 12

A bike chain can support a tension of no more than $9800 \mathrm{N}$. The pedal connects to a crank $17 \mathrm{cm}$ from the axle, and the gear pulling the chain has a $9.1 \mathrm{cm}$ radius. When riding at a constant speed, with the crank and pedal horizontal, as in Figure P8.12, what is the maximum force that can be applied to the pedal before the chain breaks?

Narayan Hari

Numerade Educator

Problem 13

A uniform beam of length $1.0 \mathrm{m}$ and mass $10 \mathrm{kg}$ is attached to a wall by a cable, as shown in Figure $\mathrm{P} 8.13 .$ The beam is free to pivot at the point where it attaches to the wall. What is the tension in the cable?

Vishal Gupta

Numerade Educator

Problem 14

The towers holding small wind turbines are often raised and lowered for easy servicing of the turbine. Figure P8.14 shows a $1000 \mathrm{kg}$ wind turbine mounted on the end of a 24-m-long, 700 kg tower that connects to a support column at a pivot. A piston connected 3.0 $\mathrm{m}$ from the pivot applies the force needed to raise or lower the tower. At the instant shown, the wind turbine is being raised at a very slow, constant speed. What magnitude force is the piston applying?

Narayan Hari

Numerade Educator

Problem 15

A standard four-drawer filing cabinet is 52 inches high and 15 inches wide. If it is evenly loaded, the center of gravity is at the center of the cabinet. A worker is tilting a filing cabinet to the side to clean under it. To what angle can he tilt the cabinet before it tips over?

Narayan Hari

Numerade Educator

Problem 16

A double-decker London bus might be in danger of rolling over in a highway accident, but at the low speeds of its urban environment, it's plenty stable. The track width is $2.05 \mathrm{m}$. With no passengers, the height of the center of gravity is $1.45 \mathrm{m},$ rising to $1.73 \mathrm{m}$ when the bus is loaded to capacity. What are the critical angles for both the unloaded and loaded bus?

Narayan Hari

Numerade Educator

Problem 17

The stability of a vehicle is often rated by the static stability factor, which is one-half the track width divided by the height of the center of gravity above the road. A typical SUV has a static stability factor of $1.2 .$ What is the critical angle?

Narayan Hari

Numerade Educator

Problem 18

A magazine rack has a center of gravity 16 $\mathrm{cm}$ above the floor, as shown in Figure P8.18. Through what maximum angle, in degrees, can the rack be tilted without falling over?

Narayan Hari

Numerade Educator

Problem 19

A car manufacturer claims that you can drive its new vehicle across a hill with a $47^{\circ}$ slope before the vehicle starts to tip. If the vehicle is $2.0 \mathrm{m}$ wide, how high is its center of gravity?

Narayan Hari

Numerade Educator

Problem 20

A thin $2.00 \mathrm{kg}$ box rests on a $6.00 \mathrm{kg}$ board that hangs over the end of a table, as shown in Figure $\mathrm{P} 8.20 .$ How far can the center of the box be from the end of the table before the board begins to tilt?

Anand Jangid

Numerade Educator

Problem 21

An orthodontic spring, connected between the upper and lower jaws, is adjusted to provide no force with the mouth open. When the patient closes her mouth, however, the spring compresses by $6.0 \mathrm{mm} .$ What force is exerted if the spring constant is $160 \mathrm{N} / \mathrm{m} ?$

Narayan Hari

Numerade Educator

Problem 22

The four wheels of a car are connected to the car's body by spring assemblies that let the wheels move up and down over bumps and dips in the road. When a $68 \mathrm{kg}$ (about $150 \mathrm{lb}$ ) person sits on the left front fender of a small car, this corner of the car dips by about $1.2 \mathrm{cm}$ (about $1 / 2$ in). If we treat the spring assembly as a single spring, what is the approximate spring constant?

Narayan Hari

Numerade Educator

Problem 23

Experiments using "optical tweezers" measure the elasticity of individual DNA molecules. For small enough changes in length, the elasticity has the same form as that of a spring. A DNA molecule is anchored at one end, then a force of $1.5 \mathrm{nN}$ $\left(1.5 \times 10^{-9} \mathrm{N}\right)$ pulls on the other end, causing the molecule to stretch by $5.0 \mathrm{nm}\left(5.0 \times 10^{-9} \mathrm{m}\right) .$ What is the spring constant of that DNA molecule?

Vishal Gupta

Numerade Educator

Problem 24

A passenger railroad car has a total of 8 wheels. Springs on each wheel compress - slightly -when the car is loaded. Ratings for the car give the stiffness per wheel (the spring constant, treating the entire spring assembly as a single spring as $2.8 \times 10^{7} \mathrm{N} / \mathrm{m} .$ When 30 passengers, each with average mass $80 \mathrm{kg},$ board the car, how much does the car move down on its spring suspension? Assume that each wheel supports $1 / 8$ the weight of the car.

Narayan Hari

Numerade Educator

Problem 25

One end of a 10-cm-long spring is attached to the ceiling. When a $2.0 \mathrm{kg}$ mass is hung from the other end, the spring stretches to a length of $15 \mathrm{cm}$.
a. What is the spring constant?
b. How long is the spring when a $3.0 \mathrm{kg}$ mass is suspended from it?

Narayan Hari

Numerade Educator

Problem 26

A scale used to weigh fish consists of a spring hung from a support. The spring's equilibrium length is $10.0 \mathrm{cm} .$ When a $4.0 \mathrm{kg}$ fish is suspended from the end of the spring, it stretches to a length of $12.4 \mathrm{cm}$.
a. What is the spring constant $k$ for this spring?
b. If an $8.0 \mathrm{kg}$ fish is suspended from the spring, what will be the length of the spring?

Narayan Hari

Numerade Educator

Problem 27

A spring has an unstretched length of $10 \mathrm{cm}$. It exerts a restoring force $F$ when stretched to a length of $11 \mathrm{cm} .$
a. For what total stretched length of the spring is its restoring force $3 F ?$
b. At what compressed length is the restoring force $2 F ?$

Nafis Fuad

Numerade Educator

Problem 28

A spring stretches $5.0 \mathrm{cm}$ when a $0.20 \mathrm{kg}$ block is hung from it. If a $0.70 \mathrm{kg}$ block replaces the $0.20 \mathrm{kg}$ block, how far does the spring stretch?

Narayan Hari

Numerade Educator

Problem 29

You need to make a spring scale to measure the mass of objects hung from it. You want each $1.0 \mathrm{cm}$ length along the scale to correspond to a mass difference of 0.10 kg. What should be the value of the spring constant?

Narayan Hari

Numerade Educator

Problem 30

Dynamic climbing ropes are designed to be quite pliant, allowing a falling climber to slow down over a long distance. The graph in Figure $\mathrm{P} 8.30$ shows force-versus-strain data for an 11-mm-diameter climbing rope. What is the Young's modulus for this rope?

Narayan Hari

Numerade Educator

Problem 31

A force stretches a wire by $1.0 \mathrm{mm}$.
a. A second wire of the same material has the same cross section and twice the length. How far will it be stretched by the same force?
b. A third wire of the same material has the same length and twice the diameter as the first. How far will it be stretched by the same force?

Narayan Hari

Numerade Educator

Problem 32

Static climbing ropes are designed to be relatively stiff so that they stretch less than dynamic ropes. To meet a certain specification, an 11 -mm-diameter rope must experience a maximum elongation of $5.0 \%$ when supporting a $150 \mathrm{kg}$ load. What is the minimum Young's modulus?

Narayan Hari

Numerade Educator

Problem 33

What hanging mass will stretch a 2.0-m-long, $0.50-\mathrm{mm}$ diameter steel wire by $1.0 \mathrm{mm} ?$

Averell Hause

Carnegie Mellon University

Problem 34

An 80-cm-long, 1.0-mm-diameter steel guitar string must be tightened to a tension of $2.0 \mathrm{kN}$ by turning the tuning screws. By how much is the string stretched?

Narayan Hari

Numerade Educator

Problem 35

A mineshaft has an ore elevator hung from a single braided cable of diameter $2.5 \mathrm{cm} .$ Young's modulus of the cable is $10 \times 10^{10} \mathrm{N} / \mathrm{m}^{2} .$ When the cable is fully extended, the end of the cable is $800 \mathrm{m}$ below the support. How much does the fully extended cable stretch when $1000 \mathrm{kg}$ of ore is loaded?

Narayan Hari

Numerade Educator

Problem 36

The normal force of the ground on the foot can reach three times a runner's body weight when the foot strikes the pavement. By what amount does the 52 -cm-long femur of an $80 \mathrm{kg}$ runner compress at this moment? The cross-section area of the bone of the femur can be taken as $5.2 \times 10^{-4} \mathrm{m}^{2}$.

Narayan Hari

Numerade Educator

Problem 37

A three-legged wooden bar stool made out of solid Douglas fir has legs that are $2.0 \mathrm{cm}$ in diameter. When a $75 \mathrm{kg}$ man sits on the stool, by what percent does the length of the legs decrease? Assume, for simplicity, that the stool's legs are vertical and that each bears the same load.

Narayan Hari

Numerade Educator

Problem 38

To penetrate armor, a projectile's point concentrates force in a small area, creating a stress large enough that the armor fails. A species of jellyfish launches a pointed needle that can penetrate the hard shell of a crustacean. The rapid deceleration on impact creates a $32 \mu \mathrm{N}$ force on the tip, which has a very small $15 \mathrm{nm}$ radius. What is the resulting stress? How does this compare to the ultimate stress of steel?

Narayan Hari

Numerade Educator

Problem 39

A 3.0-m-tall, 50-cm-diameter concrete column supports a $200,000 \mathrm{kg}$ load. By how much is the column compressed?

Narayan Hari

Numerade Educator

Problem 40

You've just put a new wood floor in your house. An object will dent the flooring if the stress $-$ the force divided by the area -exerted by the object is great enough. Who is more likely to dent your floor: a 50 kg woman in high-heeled shoes (assume a circular heel pad $0.50 \mathrm{cm}$ in diameter $)$ with all of her weight on one heel or a $5000 \mathrm{kg}$ African elephant (assume a circular contact area of $40 \mathrm{cm}$ in diameter for one foot) standing on all four feet?

Narayan Hari

Numerade Educator

Problem 41

A glass optical fiber in a communications system has a diameter of $9.0 \mu \mathrm{m}$
a. What maximum tension could this fiber support without breaking?
b. Assume that the fiber stretches in a linear fashion until the instant it breaks. By how much will a 10-m-long fiber have stretched when it is at the breaking point?

Narayan Hari

Numerade Educator

Problem 42

If you tethered a space station to the earth by a long cable, you could get to space in an elevator that rides up the cable—much simpler and cheaper than riding to space on a rocket. There’s one big problem, however: There is no way to create a cable that is long enough. The cable would need to reach 36,000 km upward, to the height where a satellite orbits at the same speed as the earth rotates; a cable this long made of ordinary materials couldn’t even support its own weight. Consider a steel cable suspended from a point high above the earth. The stress in the cable is highest at the top; it must support the weight of cable below it. What is the greatest length the cable could have without failing? For the purposes of this problem, you can ignore the variation in gravity near the surface of the earth. Hint: The mass of the cable is the volume of the cable multiplied by the density. The density of steel is $7900 \mathrm{kg} / \mathrm{m}^{3}$.

Narayan Hari

Numerade Educator

Problem 43

The Achilles tendon connects the muscles in your calf to the back of your foot. When you are sprinting, your Achilles tendon alternately stretches, as you bring your weight down onto your forward foot, and contracts to push you off the ground. A $70 \mathrm{kg}$ runner has an Achilles tendon that is $15 \mathrm{cm}$ long with a typical $1.1 \times 10^{-4} \mathrm{m}^{2}$ area.
a. By how much will the runner's Achilles tendon stretch if the maximum force on it is 8.0 times his weight, a typical value while running?
b. What fraction of the tendon's length does this correspond to?

Vishal Gupta

Numerade Educator

Problem 44

A woman is pushing a load in a wheelbarrow, as in Figure P8.44. The combined mass of the wheel-barrow and the load is $110 \mathrm{kg}$, with a center of gravity $0.25 \mathrm{m}$ behind the axle. The woman supports the wheelbarrow at the handles, $1.1 \mathrm{m}$ behind the axle.
a. What is the force required to support the wheelbarrow?
b. What fraction of the weight of the wheelbarrow and the load does this force represent?

Narayan Hari

Numerade Educator

Problem 45

Figure $\mathrm{P} 8.45$ shows the operation of a garlic press. The lower part of the press is held steady, and the upper handle is pushed down, thereby crushing a garlic clove through a screen. Approximate distances are shown in the figure. If the user exerts a $12 \mathrm{N}$ force on the upper handle, estimate the force on the clove.

Supratim Pal

Numerade Educator

Problem 46

Consider a rower in a scull as in Figure $\mathrm{P} 8.46 .$ The oars aren't accelerating, and they are rotating at a constant speed, so the net force and net torque on the oars are zero. An oar is $2.8 \mathrm{m}$ long, and the rower pulls with a $250 \mathrm{N}$ force on the handle, which is $0.92 \mathrm{m}$ from the pivot.
a. Assume that the oar touches the water at its very end. What is the drag force from the water on the oar? Assume that the oar is perpendicular to the boat, and that the force of the rower and the drag force are both perpendicular to the oar.
b. Given that both oars are the same, what is the total force propelling the boat forward?

Narayan Hari

Numerade Educator

Problem 47

Hold your upper arm vertical and your lower arm horizontal with your hand palm-down on a table, as shown in Figure P8.47. If you now push down on the table, you'll feel that your triceps muscle has contracted and is trying to pivot your lower arm about the elbow joint. If a person with the arm dimensions shown pushes down hard with a $90 \mathrm{N}$ force (about $20 \mathrm{lb}),$ what force must the triceps muscle provide? You can ignore the mass of the arm and hand in your calculation.

Narayan Hari

Numerade Educator

Problem 48

If you stand on one foot while holding your other leg up behind you, your muscles apply a force to hold your leg in this raised position. We can model this situation as in Figure $\mathrm{P} 8.48$. The leg pivots at the knee joint, and the force that holds the leg up is provided by a tendon attached to the lower leg as shown. Assume that the lower leg and the foot have a combined mass of $4.0 \mathrm{kg},$ and that their combined center of gravity is at the center of the lower leg.
a. How much force must the tendon exert to keep the leg in this position?
b. As you hold your leg in this position, the upper leg exerts a force on the lower leg at the knee joint. What are the magnitude and direction of this force?

Narayan Hari

Numerade Educator

Problem 49

If you hold your arm outstretched with palm upward, as in Figure $\mathrm{P} 8.49,$ the force to keep your arm from falling comes from your deltoid muscle. The arm of a typical person has mass $4.0 \mathrm{kg}$ and the distances and angles shown in the figure.
a. What force must the deltoid muscle provide to keep the arm in this position?
b. By what factor does this force exceed the weight of the arm?

Narayan Hari

Numerade Educator

Problem 50

Dogs -like many animals - stand and walk on their toes. A photo of the rear foot of a dog is shown in Figure $\mathrm{P} 8.50 \mathrm{a} ;$ Figure $\mathrm{P} 8.50 \mathrm{b}$ shows the bones of the leg and foot along with relevant distances. The colored element corresponds to your foot, and the connection with the leg corresponds to your ankle. The Achilles tendon pulls on the end of the foot, along a line $4.0 \mathrm{cm}$ from the ankle. What is the tension in the tendon if a $20 \mathrm{kg}$ dog is supporting $1 / 4$ of its weight on one rear foot?

Rashmi Sinha

Numerade Educator

Problem 51

A 3.0-m-long rigid beam with a mass of $100 \mathrm{kg}$ is supported at each end, as shown in Figure $\mathrm{P} 8.51$. An $80 \mathrm{kg}$ student stands $2.0 \mathrm{m}$ from support 1 How much upward force does each support exert on the beam?

Narayan Hari

Numerade Educator

Problem 52

An $80 \mathrm{kg}$ construction worker sits down $2.0 \mathrm{m}$ from the end of a $1450 \mathrm{kg}$ steel beam to eat his lunch, as shown in Figure P8.52. The cable supporting the beam is rated at $15,000 \mathrm{N}$. Should the worker be worried?

Narayan Hari

Numerade Educator

Problem 53

A man is attempting to raise a 7.5 -m-long, $28 \mathrm{kg}$ flagpole that has a hinge at the base by pulling on a rope attached to the top of the pole, as shown in Figure P8.53. With what force does the man have to pull on the rope to hold the pole motionless in this position?

Massimo Antonelli

Massimo Antonelli

Numerade Educator

Problem 54

An 85 kg man stands in a very strong wind moving at $14 \mathrm{m} / \mathrm{s}$ at torso height. As you know, he will need to lean in to the wind, and we can model the situation to see why. Assume that the man has a mass of $85 \mathrm{kg},$ with a center of gravity $1.0 \mathrm{m}$ above the ground. The action of the wind on his torso, which we approximate as a cylinder $50 \mathrm{cm}$ wide and $90 \mathrm{cm}$ long centered $1.2 \mathrm{m}$ above the ground, produces a force that tries to tip him over backward. To keep from falling over, he must lean forward.
a. What is the magnitude of the torque provided by the wind force? Take the pivot point at his feet. Assume that he is standing vertically.
b. At what angle to the vertical must the man lean to provide a gravitational torque that is equal to this torque due to the wind force?

Kevin Hayakawa

University of California - Los Angeles

Problem 55

A 40 kg, 5.0-m-long beam is supported by, but not attached to, the two posts in Figure P8.55. A 20 kg boy starts walking along the beam. How close can he get to the right end of the beam without it tipping?

Vishal Gupta

Numerade Educator

Problem 56

Two identical, side-by-side springs with spring constant $240 \mathrm{N} / \mathrm{m}$ support a $2.00 \mathrm{kg}$ hanging box. Each spring supports the same weight. By how much is each spring stretched?

Narayan Hari

Numerade Educator

Problem 57

A 5.0 kg mass hanging from a spring scale is slowly lowered onto a vertical spring, as shown in Figure P8.57. The scale reads in newtons.
a. What does the spring scale read just before the mass touches the lower spring?
b. The scale reads $20 \mathrm{N}$ when the lower spring has been compressed by $2.0 \mathrm{cm}$ What is the value of the spring constant for the lower spring?
c. At what compression distance will the scale read zero?

Nafis Fuad

Numerade Educator

Problem 58

DNA molecules are typically folded tightly. Stretching a strand of DNA means straightening it, and the molecules resist this straightening. Investigators can attach beads to the ends of a strand of DNA and, using "optical tweezers," measure the force required to produce a certain extension. Data for the stretch of a 3500 base pair strand of DNA approximately follow the line in the graph in Figure $\mathrm{P} 8.58$. What is the spring constant for this strand of DNA?

Satpal Satpal

Numerade Educator

Problem 59

Figure P8.59 shows two springs attached to a block that can slide on a frictionless surface. In the block's equilibrium position, the left spring is compressed by $2.0 \mathrm{cm}$.
a. By how much is the right spring compressed?
b. What is the net force on the block if it is moved $15 \mathrm{cm}$ to the right of its equilibrium position?

Narayan Hari

Numerade Educator

Problem 60

A $25 \mathrm{kg}$ child bounces on a pogo stick. The pogo stick has a spring with spring constant $2.0 \times 10^{4} \mathrm{N} / \mathrm{m}$. When the child makes a nice big bounce, she finds that at the bottom of the bounce she is accelerating upward at $9.8 \mathrm{m} / \mathrm{s}^{2} .$ How much is the spring compressed?

Narayan Hari

Numerade Educator

Problem 61

Figure $\mathrm{P} 8.61$ shows a lightweight plank supported at its right end by a 7.0 -mm-diameter rope with a tensile strength of $6.0 \times 10^{7} \mathrm{N} / \mathrm{m}^{2}.$
a. What is the maximum force that the rope can support?
b. What is the greatest distance, measured from the pivot, that the center of gravity of an $800 \mathrm{kg}$ piece of heavy machinery can be placed without snapping the rope?

Narayan Hari

Numerade Educator

Problem 62

In the hammer throw, an athlete spins a heavy mass in a circle at the end of a cable before releasing it for distance. For male athletes, the "hammer" is a mass of $7.3 \mathrm{kg}$ at the end of a $1.2 \mathrm{m}$ cable, which is typically a $3.0-\mathrm{mm}-$ diameter steel cable. A world-class thrower can get the hammer up to a speed of $29 \mathrm{m} / \mathrm{s}$. If an athlete swings the mass in a horizontal circle centered on the handle he uses to hold the cable
a. What is the tension in the cable?
b. How much does the cable stretch?

Narayan Hari

Numerade Educator

Problem 63

There is a disk of cartilage between each pair of vertebrae in your spine. Suppose a disk is $0.50 \mathrm{cm}$ thick and $4.0 \mathrm{cm}$ in diameter. If this disk supports half the weight of a $65 \mathrm{kg}$ person, by what fraction of its thickness does the disk compress?

Narayan Hari

Numerade Educator

Problem 64

Orb spiders make silk with a typical diameter of $0.15 \mathrm{mm}$.
a. A typical large orb spider has a mass of 0.50 g. If this spider suspends itself from a single 12-cm long strand of silk, by how much will the silk stretch?
b. What is the maximum weight that a single thread of this silk could support?

Narayan Hari

Numerade Educator

Problem 65

Larger animals have sturdier bones than smaller animals. A mouse's skeleton is only a few percent of its body weight, compared to $16 \%$ for an elephant. To see why this must be so, recall, from Example $8.10,$ that the stress on the femur for a man standing on one leg is $1.4 \%$ of the bone's tensile strength. Suppose we scale this man up by a factor of 10 in all dimensions, keeping the same body proportions. Use the data for Example 8.10 to compute the following.
a. Both the inside and outside diameter of the femur, the region of compact bone, will increase by a factor of $10 .$ What will be the new cross-section area?
b. The man's body will increase by a factor of 10 in each dimension. What will be his new mass?
c. If the scaled-up man now stands on one leg, what fraction of the tensile strength is the stress on the femur?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 66

The main muscles that hold your head upright attach to your spine in back of the point where your head pivots on your neck. Figure $\mathrm{P} 8.66$ shows typical numbers for the distance from the pivot to the muscle attachment point and the distance from the pivot to the center of gravity of the head. The muscles pull down to keep your head upright. If the muscle relaxes $-$ if, for instance, you doze in one of your classes besides Physics - your head tips forward. In the questions that follow, assume that your head has a mass of $4.8 \mathrm{kg}$, and that you maintain the relative angle between your head and your spine.
a. With the head held level, as in Figure $\mathrm{P} 8.66,$ what muscle force is needed to keep a $4.8 \mathrm{kg}$ head upright?
b. If you tip your body forward so that your spine is level with the ground, what muscle force is needed to keep your head in the same orientation relative to the spine?
c. If you tip your body backward, you will reach a point where no muscle force is needed to keep your head upright. For the distances given in Figure $\mathrm{P} 8.66,$ at what angle does this balance occur?

Narayan Hari

Numerade Educator

Problem 67

A woman weighing $580 \mathrm{N}$ does a pushup from her knees, as shown in Figure $\mathrm{P} 8.67 .$ What are the normal forces of the floor on (a) each of her hands and (b) each of her knees?

Narayan Hari

Numerade Educator

Problem 68

When you bend over, a series of large muscles, the erector spinae, pull on your spine to hold you up. Figure $\mathrm{P} 8.68$ shows a simplified model of the spine as a rod of length $L$ that pivots at its lower end. In this model, the center of gravity of the $320 \mathrm{N}$ weight of the upper torso is at the center of the spine. The $160 \mathrm{N}$ weight of the head and arms acts at the top of the spine. The erector spinae muscles are modeled as a single muscle that acts at an $12^{\circ}$ angle to the spine. Suppose the person in Figure $\mathrm{P} 8.68$ bends over to an angle of $30^{\circ}$ from the horizontal.
a. What is the tension in the erector muscle?
Hint: Align your $x$ -axis with the axis of the spine.
b. A force from the pelvic girdle acts on the base of the spine. What is the component of this force in the direction of the spine? (This large force is the cause of many back injuries).

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 69

What is the approximate Young's modulus for the ligament?
A. $4 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}$
B. $8 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}$
C. $3 \times 10^{6} \mathrm{N} / \mathrm{m}^{2}$
D. $6 \times 10^{6} \mathrm{N} / \mathrm{m}^{2}$

Narayan Hari

Numerade Educator

Problem 70

The segment of ligament tested has a resting length of $40 \mathrm{mm} .$ How long is the ligament at a strain of $0.60 ?$
A. $46 \mathrm{mm}$
B. $52 \mathrm{mm}$
C. 58 mm
D. $64 \mathrm{mm}$

Narayan Hari

Numerade Educator

Problem 71

Suppose the ligament has a circular cross section. For a certain ligament, an investigator measures the restoring force at a strain of $0.40 .$ If the ligament is replaced with one that has twice the diameter, by what factor does the restoring force increase?
A. 1.4
B. 2
C. 4
D. 8

Narayan Hari

Numerade Educator

Problem 72

The volume of the ligament stays the same as it stretches, so the cross-section area decreases as the length increases. Given this, how would a force $F$ versus change in length $\Delta L$ curve appear?

Mukesh Devi

Numerade Educator