Chapter Questions
Determine the torques about the axis of rotation $P$ produced by each of the four forces shown in Figure P8.1. All forces have magnitudes of $120 \mathrm{N}$ and are exerted a distance of $2.0 \mathrm{m}$ from $P$ on some unshown object O.
Three $200-\mathrm{N}$ forces are exerted on the beam shown in Figure $\mathrm{P} 8.2$.(a) Determine the torques about the axis of rotation on the left produced by forces $\vec{F}_{1}$ on $\mathrm{B}$ and $\vec{F}_{2}$ on $\mathrm{B}$. (b) At what distance from the axis of rotation must $\vec{F}_{3 \text { on } \mathrm{B}}$ be exerted to cause a torque that balances those produced by $\vec{F}_{1}$ on $\mathrm{B}$ and $\vec{F}_{2 \text { on } \mathrm{B}} ?$
A 2.0 -m-long, 15 -kg ladder is resting against a house wall, making a $30^{\circ}$ angle with the vertical wall. The coefficient of static friction between theladder feet and the ground is $0.40,$ and between the top of the ladder and the wall the coefficient is 0. Make a list of the physical quantities you can determine or estimate using this information and calculate them.
Figure $P 8.4$ shows two different situations where three forces of equal magnitude are exerted on a square board hanging on a wall, supported by a nail. For each case, determine the sign of the total torque that the three forces exert on the board.
Three friends tie three ropes in a knot and pull on the ropes in different directions. Adrienne (rope 1 ) exerts a $20-\mathrm{N}$ force in the positive $x$ -direction, and Jim (rope 2) exerts a 40-N force at an angle $53^{\circ}$ above the negative $x$ -axis. Luis (rope 3 ) exerts a force that balances the first two so that the knot does not move. (a) Construct a force diagram for the knot.(b) Use equilibrium conditions to write equations that can be used to determine $F_{\mathrm{L}}$ on $\mathrm{K}_{x}$ and $F_{\mathrm{L}}$ on $\mathrm{K} \mathrm{y}$(c) Use equilibrium conditions to write equations that can be used to determine the magnitude and direction of $\vec{F}_{\mathrm{L} \text { on } \mathrm{K}}$.
Adrienne from Problem 8.5 now exerts a $100-\mathrm{N}$ force $\vec{F}_{\mathrm{A}}$ on $\mathrm{K}$ that points $30^{\circ}$ below the positive $x$ -axis, and Jim exerts a $150-\mathrm{N}$ force in the negative y-direction. How hard and in what direction does Luis now have to pull the knot so that it remains in equilibrium?
Kate joins Jim, Luis, and Adrienne in the rope-pulling exercise described in the previous two problems. This time, they tie four ropes to a ring. The three friends each pull on one rope, exerting the following forces: $\vec{T}_{1}$ on $\mathrm{R}$ (50 N in the positive $y$ -direction), $\vec{T}_{2 \text { on } \mathrm{R}}\left(20 \mathrm{N}, 25^{\circ}\right.$ above the negative $x$ -axis), and $\vec{T}_{3}$ on $\mathrm{R}\left(70 \mathrm{N}, 70^{\circ}\right.$ below the negative $x$ -axis). Kate pulls rope 4 , exerting a force $\vec{T}_{4 \mathrm{on} \mathrm{R}}$ so that the ring remains in equilibrium. (a) Constructa force diagram for the ring. (b) Use the first condition of equilibrium to write two equations that can be used to determine $T_{4}$ on $\mathrm{R} x$ and $T_{4}$ on $\mathrm{R} y$ "(c) Solve these equations and determine the magnitude and direction of $\vec{T}_{4 \mathrm{on} \mathrm{R}}$.
You hang a light in front of your house using an elaborate system to keep the 1.2-kg light in static equilibrium (see Figure $P 8.8$ ). What are the magnitudes of the forces that the ropes must exert on the knot connect ing the three ropes if $\theta_{2}=37^{\circ}$ and $\theta_{3}=0^{\circ} ?$ Rope 3 can be tied to the hook on the wall.
Find the values of the forces theropes exert on the knot if you replace the light in Problem 8.8 with a heavier 12 -kg object and the ropes make angles of $\theta_{2}=63^{\circ}$ and $\theta_{3}=45^{\circ}$ (see Figure P8.8).
Redraw Figure $\mathrm{P} 8.8$ with $\theta_{2}=50^{\circ}$ and $\theta_{3}=0^{\circ} .$ Rope 2 is found to exert a $100-\mathrm{N}$ force on the knot. Determine $m$ and the magnitudes of the forces thatthe other two ropes exert on the knot.
Determine the masses $m_{1}$ and $m_{2}$ of the two objects shown in Figure $P 8.11$ if the force exerted by the horizontal cable on the knot is $64 \mathrm{N}$.
You work in a machine shop and need to move a huge $640-\mathrm{kg}$ engine up and to the left in order to slide a cart under it. You use the system shown in Figure $P 8.12$. How hard and in what direction do you need to pull on rope 2 if the angle between rope 1 and the horizontal is $\theta_{1}=60^{\circ} ?$
You exert a $630-\mathrm{N}$ force on rope 2 in the previous problem (Figure $\mathrm{P} 8.12$ ). Write the two equations $(x$ and $y$ ) for the first condition of equilibrium using the pulley as the object of interest for a force diagram. Calculate $\theta_{1}$ and $\theta_{2}$. You may need to use the identity $(\sin \theta)^{2}+(\cos \theta)^{2}=1$.
A pulley system shown in Figure $P 8.14$ will allow you to lift heavy objects in the machine shop by exerting a relatively small force.(a) Construct a force diagram for each pulley.(b) Use the equations of equilibrium and the force diagrams to determine $T_{1}, T_{2}, T_{3},$ and $T_{4}$.
A tightrope walker wonders if her rope is safe. Her mass is $60 \mathrm{kg}$ and the length of the rope is about $20 \mathrm{m} .$ The rope will break if its tension exceeds $6700 \mathrm{N}$. What is the smallest angle at which the rope can bend up from the horizontal on either side of her to avoidbreaking?
An apparatus to lift hospital patients sitting at the sides of their beds is shown in Figure P8.16. At what angle above the horizontal does the rope going under the pulley bend while supporting the 78 -kg person hanging from the pulley?
A father $(80 \mathrm{kg}),$ mother $(56 \mathrm{kg}),$ daughter $(16 \mathrm{kg}),$ and son $(24 \mathrm{kg})$ try to occupy seats on the seesaw shown in Figure $P 8.17$ so that the seesaw is in equilibrium. Can they succeed? Explain.
You stand at the end of a uniform diving board a distance $d$ from support 2 (similar to that shown in Figure P8.18). Your mass is $m$ and the distance between the two supports is $a$. What can you determine from this information? Make a list of physical quantities and show how you will determine them.
You place a $3.0-\mathrm{m}$ -long board symmetrically across a $0.5-\mathrm{m}$ -wide chair to seat three physics students at a party at your house. If $70-\mathrm{kg}$ Dan sits on the left end of the board and 50 -kg Tahreen sits on the right end of the board, where should 54 -kg Komila sit to keep the board stable? What assumptions did you make?
You've got a flat tire. To lift your car, you make a homemade lever (see Figure $P 8.20$ ). A very light $1.6-\mathrm{m}$ -long handle part is pushed down on the right side of the fulcrum and a 0.10 -m-long part on the left side supports the back of the car. How hard must you push down on the handle so that the lever exerts an $8000-\mathrm{N}$ force to lift the back of the car?
You are building a toy mobile, copying the design shown in Figure $P 8.21 .$ Object $A$ has a $1.0-\mathrm{kg}$ mass. What should be the mass of object B? The numbers in Figure $\mathrm{P} 8.21$ indicate the relative lengths of the rods on each side of their supporting cords.
You are building a toy mobile similar to that shown in Figure $\mathrm{P} 8.21$ but with different dimensions and replacing the objects with cups. The bottom rod is $20 \mathrm{cm}$ long, the middle rod is $15 \mathrm{cm}$ long, and the top rod is $8 \mathrm{cm}$ long. You put one penny in the bottomleft cup, three pennies in the bottom right cup, eleven pennies in the middle right cup, and five pennies in the top left cup. (a) Draw a force diagram for each rod. (b) Determine the cord attachment points and lengths on each side for each rod. (c) What assumptions did you make in order to solve the problem?
Compare the two different designs of nutcracker shown in Figure $P 8.23$ and decide which one is more efficient in cracking a nut. Estimate the forces exerted by each cracker on the nut when a $30-\mathrm{N}$ force is exerted on each handle. Indicate any assumptions that you made. You will need a ruler to solve this problem.
Ray decides to paint the outside of his uncle's house. He uses a $4.0-\mathrm{m}$ -long board supported by vertical cables at each end to paint the second floor. The board has a mass of $21 \mathrm{kg}$. Ray $(70 \mathrm{kg})$ stands $1.0 \mathrm{m}$ from the left cable. What are the forces that each cable exerts on the board?
A $2.0-\mathrm{m}$ -long uniform beam of mass $8.0 \mathrm{kg}$ supports a $12.0-\mathrm{kg}$ bag of vegetables at one end and a $6.0-\mathrm{kg}$ bag of fruit at the other end. At what distance from the vegetables should the beam rest on your shoulder to balance? What assumptions did you make?
A uniform beam of length $l$ and mass $m$ supports a bag of mass $m_{1}$ at the left end, another bag of mass $m_{2}$ at the right end, and a third bag $m_{3}$ at a distance $l_{3}$ from the left end $\left(l_{3} < 0.5 l\right)$. At what distance from the left end should you support the beam so that it balances?
A person whose height is $1.88 \mathrm{m}$ is lying on a light board placed on two scales so that scale 1 is under the person's head and scale 2 is under the person's feet. Scale 1 reads $48.3 \mathrm{kg}$ and scale 2 reads $39.3 \mathrm{kg} .$ Where is the center of mass of the person?
A seesaw has a mass of $30 \mathrm{kg}$, a length of $3.0 \mathrm{m},$ and a fulcrum beneath its midpoint. It is balanced when a $60-\mathrm{kg}$ person sits on one end and a 75-kg person sits on the other end. Locate the center of mass of the seesaw. Where is the center of mass of a uniform seesaw that is $3.0 \mathrm{m}$ long and has a mass of $30 \mathrm{kg}$ if two people of masses $60 \mathrm{kg}$ and $75 \mathrm{kg}$ sit on its ends?
You decide to cut an L-shaped object out of cardboard so that theobject is in static equilibrium if hung as shown in Figure $P 8.29 .$ Find the missing dimension of the object. Cut the object out of some cardboard and check if your result is correct.
You have a $1-\mathrm{m}-$ wide, $2-\mathrm{m}$ -long $14-\mathrm{kg}$ wooden board. If you place a $5.0-\mathrm{kg}$ pot of soup in one corner of the board, where is the center of mass of the board-pot system?
An $80-\mathrm{kg}$ clown sits on a $20-\mathrm{kg}$ bike on a tightrope attached between two trees. The center of mass of the clown is $1.6 \mathrm{m}$ above the rope, and the center of mass of the bike is $0.7 \mathrm{m}$ above the rope. A load of what mass should be fixed onto the bike and hang $1.5 \mathrm{m}$ below the rope so that the center of mass of the clown-bike-load system is $0.5 \mathrm{m}$ below the rope? What is the force that the rope exerts on each tree if the angle between the rope and the horizontal is $10^{\circ} ?$
Figure P8.32 shows a disk of radius $R$ with a circular hole of radius $r$ cut a distance $a$ from the center of the disk. Where is the disk'scenter of mass? (Hint: You can think of cutting the hole as adding material of negative mass to the original object.)
A person's broken leg is kept in place by the apparatus shown in Figure $P 8.33 .$ If the rope pulling on the leg exerts a $120-N$ force on it, how massive should be the block hanging from the rope that passes over the pulley? First, derive a general expression for $m$ in terms of relevant parameters and then determine the mass of the block.
Diving board The diving board shown in Figure P8.18 has a mass of $28 \mathrm{kg}$ and its center of mass is at the board's geometrical center. Determine the forces that support posts 1 and 2 (separated by $1.4 \mathrm{m}$ ) exert on the board when a $60-\mathrm{kg}$ person stands on the end of the board $2.8 \mathrm{m}$ from support post 2.
A uniform cubical box of mass $m$ and side $L$ sits on the floor with its bottom left edge pressing against a ridge. Derive the expression for the least force you need to exert horizontally at the top right edge of the box that will cause its bottom right edge to be slightly off the floor, as shown in Figure $P 8.35 .$ (Note: With the right edge slightly off the floor, the ground and ridge exert their forces on the bottom left edge of the box.)
If the force $F$ shown in Figure $P 8.35$ is $840 N$ and the bottom right edge of the box is slightly off the ground, what is the mass of the cubical box of side $1.2 \mathrm{m} ?$
You decide to hang a new 10 -kg flowerpot using the arrangement shown in Figure $P 8.37$. Can you use a slanted rope attached from the wall to the end of the beam if that rope breaks when the tension exceeds $170 \mathrm{N} ?$ The mass of the beam is not known, but it looks light.
You decide to hang another plant from a 1.5 -m-long 2.0 -kg horizontal beam that is attached by a hinge to the wall on the left. A cable attached to the right end goes $37^{\circ}$ above the beam to a connecting point above the hinge on the wall. You hang a $100-\mathrm{N}$ pot from the beam $1.4 \mathrm{m}$ away from the wall. What is the force that the cable exerts on the beam?
Now you decide to change the way you hang the pot described in Problems 8.37 and $8.38 .$ You orient the beam at a $37^{\circ}$ angle above the horizontal and orient the cable horizontally from the wall to the end of the beam. The beam still holds the $2.0-\mathrm{kg}$ pot and plant hanging $0.1 \mathrm{m}$ from its end. Now determine the force that the cable exerts on the beam and the force that the wall hinge exerts on the beam (its $x$ - and $y$ -components and the magnitude and direction of that force).
What mechanical work must you do to lift a log that is $3.0 \mathrm{m}$ long and has a mass of $100 \mathrm{kg}$ from the horizontal to a vertical position? (Hint: Use the work-energy principle.) You are lifting one end of the log while the other end is on the ground all the time.
A 70 -g meter stick has a 30 -g piece of modeling clay attached to the end. Where should you drill a hole in the meter stick so that you can hang the stick horizontally in equilibrium on a nail in the wall? Draw a picture to help explain your decision.
You are trying to tilt a very tall refrigerator $(2.0 \mathrm{m}$ high, $1.0 \mathrm{m}$ deep, $1.4 \mathrm{m}$ wide, and $100 \mathrm{kg}$ ) so that your friend can put a blanket underneath to slide it out of the kitchen. Determine the force that you need to exert on the front of the refrigerator at the start of its tipping. You push horizontally $1.4 \mathrm{m}$ above the floor.
Can you put a $0.2-\mathrm{kg}$ candle on a $0.6-\mathrm{kg}$ candleholder, as shown in Figure $\mathrm{P} 8.43,$ without tipping the candleholder over? Explain. The centers of mass of the candle and candleholder are marked on the figure.
You have an Atwood machine (see Figure 4.9) with two blocks each of mass $m$ attached to the ends of a string of length $l$. The string passes over a frictionless pulley down to the blocks hanging on each side. While pulling down on one block, you release it. Both blocks continue to move at constant speed, one up and the other down. Is the system still in equilibrium? Find the vertical component of the center of mass of the two-block system. Indicate all of your assumptions and the coordinate system used.
You stand sideways in a moving train. Estimate how far apart you should keep your feet so that when the train accelerates at $2.0 \mathrm{m} / \mathrm{s}^{2}$ you can still stand without holding anything. List all your assumptions.
Your hand holds a liter of milk (mass about $1 \mathrm{kg}$ ) while your arm is bent at the elbow in a $90^{\circ}$ angle. Estimate the torque caused by the milk on your arm about the elbow joint. Indicate all numbers used in your calculations. This is an estimate, and your answer may differ by 10 to $50 \%$ from the answers of others.
You hold a 4.0-kg computer. Estimate the torques exerted on your forearm about the elbow joint caused by the downward force exerted by the computer on the forearm and the upward $340-\mathrm{N}$ force exerted by the biceps muscle on the forearm. Ignore the mass of the arm. Indicate any assumptions you make.
A man is holding a $16-\mathrm{kg}$ child using both hands with his elbows bent in a $90^{\circ}$ angle. The biceps muscle provides the positive torque he needs to support the child. Determine the force that each of his biceps muscles must exert on the forearm in order to hold the child safely in this position. Ignore the triceps muscle and the mass of the arm.
A man pushes on a table exerting a 20-N downward force with his hand. Determine the force that his triceps muscle must exert on his forearm in order to balance the upward force that the table exerts on his hand. Ignore the biceps muscle and the mass of the If you did not ignore the mass of the arm, would the force you determined be smaller or larger? Explain.
Find the force that the biceps muscle shown in Example 8.5 exerts on the forearm when you lift a $16-\mathrm{kg}$ dumbbell with your hand. Also determine the force that the bone in the upper arm (the humerus) exerts on the bone in the forearm at the elbow joint. The mass of the forearm is about $5.0 \mathrm{kg}$ and its center of mass is $16 \mathrm{cm}$ from the elbow joint. Ignore the triceps muscle.
You are exercising your hamstring muscle (the large muscle in the back of the thigh). You use an elastic cord attached to a hook on the wall while keeping your leg in a bent position (Figure $P 8.51$ ). Determine the magnitude of the tension force $\vec{T}_{\mathrm{H} \text { on } \mathrm{L}}$ exerted by the hamstring muscles on the leg and the magnitude of compression force $\vec{F}_{\mathrm{F}}$ on $\mathrm{B}$ at the knee joint that the femur exerts on the calf bone. The cord exerts a $20-1 b$ force $\vec{F}_{\mathrm{C} \text { on } \mathrm{F}}$ on the foot.
You injure your back at work lifting a $420-N$ radiator. To understand how it happened, you model your back as a weightless beam (Figure $P 8.52$ ), analogous to the backbone of a person in a bent position when lifting an object. (a) Determine the tension force that the horizontal cable exerts on the beam (which is analogous to the force the back muscle exerts on the backbone) and the force that the wall exerts on the beam at the hinge (which is analogous to the force that a disk in the lower back exerts on the backbone). (b) Why do doctors recommend lifting objects with the legs bent?
A woman lifts a 3.6-kg dumbbell in each hand with her arm in a horizontal position at the side of her body and holds it there for $3 \mathrm{s}$ (see Figure $\mathrm{P} 8.53$ ). What force does the deltoid muscle in her shoulder exert on the humerus bone while holding the dumbbell? The deltoid attaches $13 \mathrm{cm}$ from the shoulder joint and makes a $13^{\circ}$ angle with the humerus. The dumbbell in her hand is $0.55 \mathrm{m}$ from the shoulder joint, and the center of mass of her $4.0-\mathrm{kg}$ arm is $0.24 \mathrm{m}$ from the joint.
Repeat the previous problem with a $7.2-\mathrm{kg}$ dumbbell. Determine both the force that the deltoid exerts on the humerusand the force that the lifter's shoulder joint exerts on her humerus.
The head of a football running back (see Figure $\mathrm{P} 8.55$ ) can be considered as a lever with the vertebra at the bottom of the skull as a fulcrum (the axis of rotation). The center of mass is about $0.025 \mathrm{m}$ in front of the axis of rotation. The torque caused by the force that Earth exerts on the 8.0 -kg head/helmet is balanced by the torque caused by the downward forces exerted by a complex muscle system in the neck. That muscle system includes the trapezius and levator scapulae muscles, among others (effectively 0.057 $\mathrm{m}$ from the axis of rotation). (a) Determine the magnitude of the force exerted by the neck muscle system pulling down to balance the torque caused by the force exerted by Earth on the head.(b) If an opposing player exerts a downward $180-\mathrm{N}(40-\mathrm{lb})$ force on the facemask, what muscle force would these neck muscles now need to exert to keep the head in equilibrium?
Eiichi has purchased an adjustable hand grip to use for strengthening hands, fingers, and wrists. The hand grip consists of two handles that can rotate around a common axis and a spring that connects the handles (see Figure $P 8.56$ ). The force needed to squeeze the hand grip can be adjusted from small (Figure P8.56a) to large (Figure P8.56b) by changing the position at which the lower end of the spring is hooked to the left handle. Eiichi, Yuko, and Lars have different explanations for why it is harder to squeeze the hand grip in case (b) compared to (a).Eiichi: It is harder to squeeze the hand grip because the distance between the axis of rotation and the point where the force is exerted on the left handle by the spring is larger in case (b) than in case (a). Yuko: It is harder to squeeze the hand grip because the extension of the spring in case (b) is larger than in case (a). Lars: It is harder to squeeze the hand grip because the angle $\theta$ in case(b) is smaller than in case (a).First, comment on the students' explanations and decide whose ideas are correct and whose aren't. Then construct a complete correct explanation. Indicate any assumptions that you made.
While browsing books on neurophysiology, you come across a book published in 1967 by Soviet neurophysiologist Nikolai Aleksandrovich Bernstein, The Co-ordination and Regulation of Movements, in which he describes a technique to determine the mass of a part of the human body if the position of the center of mass of that body part is known. His technique for determining the mass of the forearm is described as follows: The person lies on a support board, as shown in Figure $P 8.57 .$ Two readings of the scale are taken: first with the forearm held in position $1\left(m_{1}\right)$ and second with it in position $2\left(m_{2}\right) .$ Knowing the distance from the elbow to the center of mass of the forearm $\left(d_{\mathrm{c}}\right)$ and the distance between the knife edges supporting the board $(D),$ the mass of the forearm and hand can be calculated from the following expression:$$m_{\mathrm{fh}}=\frac{D\left(m_{2}-m_{1}\right)}{d_{\mathrm{c}}}$$(a) First, without deriving the expression, evaluate it to see if it is reasonable. Are the units correct? Is the sign of the expression positive? Are qualitative dependences reasonable? (b) Now derive the expression.(c) Why do you think it is important to place the support board on knife edges instead of rigid blocks?
You have two force sensors connected to a computer and a meter stick of known mass. The sensors are used to keep the stick horizontal; there are no other supports. You push on the stick with your finger in an arbitrary location. (a) Design an experimental setup that will allow you to determine the magnitude of your pushing force $F$ and the location of your finger $x$ based on the readings of the two force sensors.(b) Derive an expression that can be used as a computer algorithm to calculate $x$ and $F$ using the readings of the force sensors and the parameters of your setup. (c) Evaluate the expression, analyzing the limiting cases.
Design two experiments to determine the mass of a ruler, using different methods. Your available materials are the ruler, a spring, and a set of objects of standard mass: $50 \mathrm{g}, 100 \mathrm{g},$ and $200 \mathrm{g}$. One of the methods should involve your knowledge of static equilibrium. After you design and perform the experiment, decide whether the two methods give you the same or different results.
An $80-\mathrm{kg}$ person stands at one end of a $130-\mathrm{kg}$ boat. He then walks to the other end of the boat so that the boat moves $80 \mathrm{cm}$ with respect to the bottom of the lake. (a) What is the length of the boat?(b) How much did the center of mass of the person-boat system move when the person walked from one end to the other? (Hint: Note that the total momentum of the person-boat system remains constant.)
Two people $(50 \mathrm{kg}$ and $75 \mathrm{kg}$ ) holding hands stand on Rollerblades $1.0 \mathrm{m}$ apart. $(\mathrm{a})$ Estimate the location of their center of mass. (b) The 66 two people push each other and roll apart. Estimate the new location of the center of mass when they are $4.0 \mathrm{m}$ apart. What assumptions did you make? (Hint: Note that the total momentum of the two people remains constant.)
Find the center of mass of an L-shaped object. The vertical leg has a mass of $m_{\mathrm{a}}$ of length $a$ and the horizontal leg has a mass of $m_{\mathrm{b}}$ of length $b$. Both legs have the same width $w,$ which is much smaller than $a$ or $b$.
You hold a 10 -lb ball in your hand with your forearm horizontal, forming a $90^{\circ}$ angle with the upper arm (Figure 8.25 ). Which type of muscle produces the torque that allows you to hold the ball?(a) Flexor muscle in the upper arm(b) Extensor muscle in the upper arm(c) Flexor muscle in the forearm(d) Extensor muscle in the forearm
In Figure $8.25,$ how far in centimeters from the axis of rotation are the forces that the ball exerts on the hand, that the biceps exerts on your forearm, and that the upper arm exerts on your forearm at the elbow joint?(a) 0,5,35(b) 35,5,0(c) 35,5,3(d) 35,5,-3(e) 30,5,0
Why is it easier to hold a heavy object using a bent arm than a straight arm?(a) More flexor muscles are involved.(b) The distance from the joint to the place where gravitational force is exerted by Earth on the object is smaller.(c) The distance from the joint to the place where force is exerted by the object on the hand is smaller.(d) There are two possible axes of rotation instead of one.
Why are muscles arranged in pairs at joints?(a) Two muscles can produce a bigger torque than one.(b) One can produce a positive torque and the other a negative torque.(c) One muscle can pull on the bone and the other can push.(d) Both a and b are true.
Rank in order the magnitudes of the torques caused by the four forces exerted on the backbone (see Figure $8.26 \mathrm{b}$ ), with the largest torque listed first.(a) $1>2>3>4$(b) $2=3>1>4$(c) $3>2>1>4$(d) $2>1>3>4$(e) $1=2=3=4$
What are the signs of the torques caused by forces $1,2,3,$ and $4,$ respectively, about the origin of the coordinate system shown in Figure $8.26 \mathrm{b}$ ?(a) $+,+,+,+$(b) $-,+,-, 0$(c) $+,-,+, 0$(d) $-,-,-, 0$(e) $+,-,+,-$
Which expression below best describes the torque caused by force $F_{3}=F_{\mathrm{E} \text { on } \mathrm{B}},$ the force that Earth exerts on the upper body at its center of mass for the backbone of length $L ?$(a) 0(b) $F_{3}(2 L / 3) \sin 12^{\circ}$(c) $\quad F_{3}(L / 2) \cos 30^{\circ}$(d) $-F_{3}(2 L / 3) \sin 12^{\circ}$(e) $-F_{3}(L / 2) \cos 30^{\circ}$
Which expression below best describes the torque caused by force $F_{2}=T_{\mathrm{M} \text { on } \mathrm{B}}$ exerted by the muscle on the backbone?(a) 0(b) $F_{2}(2 L / 3) \sin 12^{\circ}$(c) $\quad F_{2}(L) \cos 30^{\circ}$(d) $-F_{2}(2 L / 3) \sin 12^{\circ}$(e) $-F_{2}(L) \cos 30^{\circ}$