Determine the torques about the axis of rotation $P$ produced by each of the four forces shown in Figure $P 7.1$ . All forces have magnitudes of 120 N and are exerted a distance of 2.0 $\mathrm{m}$ from $P$ on some unshown object O.

Eric M.

Numerade Educator

$*$ EST Your hand holds a liter of milk 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.

Eric M.

Numerade Educator

$*$ EST Body torque 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.

Eric M.

Numerade Educator

Three $200-\mathrm{N}$ forces are exerted on the beam shown in Figure $\mathrm{P} 7.4 .$ (a) Determine the torques about the axis of rotation on the left produced by forces $\vec{F}_{1 \mathrm{lon} \mathrm{B}}$ and $\vec{F}_{2 \mathrm{on} \mathrm{B}}$ (b) At what distance from the axis of rotation must $\vec{F}_{3}$ on be exerted to cause a torque that balances those produced by $\vec{F}_{1 \text { on } \mathrm{B}}$ and $\vec{F}_{2 \text { on } \mathrm{B}} ?$

Eric M.

Numerade Educator

$* \mathrm{A} 2.0$ -m-long, $15-\mathrm{kg}$ ladder is resting against a house wall, making a $30^{\circ}$ angle with the vertical wall. The coefficient of static friction between the ladder feet and the ground is 0.40 , and between the top of the ladder and the wall the coefficient is 0.00 . Make a list of the physical quantities you can determine or estimate using this information and calculate them.

Eric M.

Numerade Educator

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 $\operatorname{Jim}(\text { rope } 2)$ exerts a $40-\mathrm{N}$ 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 $F_{\mathrm{L} \text { on } \mathrm{K} x}$ and $F_{\mathrm{L} \text { on } \mathrm{K} y}$ (c) Use equilibrium conditions to write equations that can be used to determine the magnitude and direction of $\vec{F}_{\mathrm{L} \text { on } \mathrm{K}}$

Eric M.

Numerade Educator

Adrienne from Problem 6 now exerts a $100-\mathrm{N}$ force $\vec{F}_{\mathrm{A} \text { on } \mathrm{K}}$ 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?

Eric M.

Numerade Educator

$*$ 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: $\vec{T}_{1 \text { on }}$ (50 $\mathrm{N}$ in the positive $y$ -direction $), \vec{T}_{2 \text { on }}\left(20 \mathrm{N}, 25^{\circ} \text { above the negative }\right.$ $x$ -axis), and $\vec{T}_{3 \text { on } \mathrm{R}}\left(70 \mathrm{N}, 70^{\circ} \text { below the negative } x \text { -axis). Kate }\right.$ pulls rope $4,$ exerting a force $\vec{T}_{4 \text { on } R}$ so that the ring remains in equilibrium. (a) Construct a force diagram for the ring. (b) Use the first condition of equilibrium to write two equations that can be used to determine $T_{4 \text { on } R x \text { and } T_{4 \text { on } R_{y}}(c)}$ Solve these equations and determine the magnitude and direction of $\vec{T}_{4 \text { on } \mathrm{R}}$ .

Eric M.

Numerade Educator

You hang a light in front of your house using an elaborate system to keep the $1.2-\mathrm{kg}$ light in static equilibrium (see Figure $\mathbf{P} 7.9$ ). What are the magnitudes of the forces that the ropes must exert on the knot connecting the three ropes if $\theta_{2}=37^{\circ}$ and $\theta_{3}=0^{\circ} ?$

Eric M.

Numerade Educator

"Find the values of the forces the ropes exert on the knot if you replace the light in Problem 7.9 with a heavier $12-\mathrm{kg}$ object and the ropes make angles of $\theta_{2}=63^{\circ}$ and $\theta_{3}=45^{\circ}$ (see Figure $\mathrm{P} 7.9 )$

Eric M.

Numerade Educator

Redraw Figure $\mathrm{P} 7.9$ 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 that the other two ropes exert on the knot.

Eric M.

Numerade Educator

$*$ Determine the masses $m_{1}$ and $m_{2}$ of the two objects shown in Figure $\mathbf{P} 7.12$ if the force exerted by the horizontal cable on the knot is 64 $\mathrm{N} .$

Eric M.

Numerade Educator

$*$ Lifting an engine You work in a machine shop and need to move a huge 640 -kg engine up and to the left in order to slide a cart under it. You use the system shown in Figure $P 7.13$ . How hard and in what direction do you need to pull on rope 2 if the angle between rope 1 and the horizontal is $\theta_{1}=60^{\circ} ?$

Eric M.

Numerade Educator

$* /$ More lifting You exert a $630-\mathrm{N}$ force on rope 2 in the previous problem (Figure $\mathrm{P} 7.13 )$ . Write the two equations $(x \text { and } y)$ 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 $(\sin \theta)^{2}+(\cos \theta)^{2}=1$

Eric M.

Numerade Educator

$* /$ Even more lifting A pulley system shown in Figure $P 7.15$ 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 for each pulley. (b) Use the equations of equilibrium and the force diagrams to determine $T_{1}, T_{2}, T_{3},$ and $T_{4}$

Eric M.

Numerade Educator

Tightrope walking 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 avoid breaking?

Eric M.

Numerade Educator

Lifting patients An apparatus to lift hospital patients sitting at the sides of their beds is shown in Figure $\mathbf{P} 7.17$ . At what angle above the horizontal does the rope going under the pulley bend while supporting the 78 -kg person hanging from the pulley?

Eric M.

Numerade Educator

A mutineer on Captain Bligh's ship is made to "walk the plank." The plank, which extends 3.0 $\mathrm{m}$ beyond its support, will break if subjected to a torque greater than 3300 $\mathrm{N} \cdot \mathrm{m} .$ Will the sailor break the plank before stepping off its end? Explain. What assumptions did you make?

Eric M.

Numerade Educator

Brett (mass 70 $\mathrm{kg} )$ sits 1.2 $\mathrm{m}$ from the fulcrum of a uniform seesaw. (a) Determine the magnitude of the torque exerted by him on the seesaw. (b) At what distance from the fulcrum on the other side should $54-\mathrm{kg}$ Dawn sit so that the seesaw is horizontal?

Eric M.

Numerade Educator

$*$ You stand at the end of a uniform diving board a distance $d$ from support 2$(\text { similar to that shown in Figure } P 7.20) .$ Your mass is $m$ . What can you determine from this information? Make a list of physical quantities and show how you will determine them.

Eric M.

Numerade Educator

$*$ You place a $3.0-\mathrm{m}$ -long board across a 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 on the right end of the board, where should $54-\mathrm{kg}$ Komila sit to keep the board stable? What assumptions did you make?

Ohana B.

Numerade Educator

After dinner (see Problem $7.21 ),$ two guests decide to use the same $3.0-\mathrm{m}$ -long 5.0 $\mathrm{kg}$ board as a seesaw, using a small bench as a fulcrum. An 82 -kg man sits on one end and a 64 -kg woman sits on the other end. Where should the bench be located so that the board balances?

Eric M.

Numerade Educator

Car jack You've got a flat tire. To lift your car, you make a homemade lever (see Figure $P 7.23$ ). A very light $1.6-\mathrm{m}$ -long handle part is pushed down on the right side of the fulcrum and a $0.050-\mathrm{m}$ -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?

Nathan S.

Numerade Educator

$*$ Mobile You are building a toy mo- bile, copying the design shown in Figure $P 7.24 . \mathrm{Object}$ A has a $1.0-\mathrm{kg}$ mass. What should be the mass of object $\mathrm{B}$ ? The numbers in Figure $\mathrm{P} 7.24$ indicate the relative lengths of the rods on each side of their

supporting cords.

Eric M.

Numerade Educator

$*$ Another mobile You are building a toy mobile similar to that shown in Figure $\mathrm{P} 7.24 \mathrm{but}$ 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 bottom left 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?

Eric M.

Numerade Educator

$*$ EST Kate is sitting on a 1.5 - -wide porch swing. Because of the rain the night before, the left side of the swing is wet, and Kate sits close to the right side. The mass of the swing seat is 10 $\mathrm{kg}$ and Kate's mass is 55 $\mathrm{kg}$ . Estimate the magnitudes of the forces that the two supporting cables exert on the swing.

Eric M.

Numerade Educator

* Ray decides to paint the outside of his uncle's house. He uses a 4.0 -m-long board supported by 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?

Eric M.

Numerade Educator

$*$ The fulcrum of a uniform 20 -kg seesaw that is 4.0 $\mathrm{m}$ long is located 2.5 $\mathrm{m}$ from one end. A 30 -kg child sits on the long end. Determine the mass a person at the other end would

have to be in order to balance the seesaw.

Eric M.

Numerade Educator

A 2.0 -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?

Eric M.

Numerade Educator

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?

Eric M.

Numerade Educator

$*$ An 80 -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?

Eric M.

Numerade Educator

EST Two people $(50 \mathrm{kg} \text { and } 75 \mathrm{kg})$ holding hands stand on rollerblades 1.0 $\mathrm{m}$ apart. (a) Estimate the location of their center of mass. ( b) The two people push off each other and roll apart so the distance between them is now 4.0 $\mathrm{m} .$ Estimate the new location of the center of mass. What assumptions did you make?

Eric M.

Numerade Educator

$*$ You have 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's center of mass?

Eric M.

Numerade Educator

*A person whose height is 1.88 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 kg and scale 2 reads 39.3 kg. Where is the center of mass of the person?

Eric M.

Numerade Educator

EST Estimate the location of the center of mass of the person described in the previous problem when he bends over and touches the floor with his hands.

Eric M.

Numerade Educator

A seesaw has a mass of 30 kg, a length of 3.0 m, and fulcrum beneath its midpoint. It is balanced when a 60-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 m long and has a mass of 30 kg if two people of masses 60 kg and 75 kg sit on its ends?

Eric M.

Numerade Educator

Find the center of mass of an L-shaped object. The vertical leg has a mass of $m_{a}$ of length $a$ and the horizontal leg has a mass of $m_{b}$ of length $b$ . Both legs have the same width $w,$ which is much smaller than $a$ or $b$ .

Eric M.

Numerade Educator

$* *$ You have a $10-\mathrm{kg}$ table with each leg of mass $1.0 \mathrm{kg}-$ total mass 14 $\mathrm{kg} .$ If you place a $5.0-\mathrm{kg}$ pot of soup in the back right corner of the table, where is the table's center of mass?

Eric M.

Numerade Educator

$*$ Using biceps to hold a child 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.

Eric M.

Numerade Educator

$* \mathrm{BlO}$ Using triceps to push a table A man pushes on a table exerting a $20-\mathrm{N}$ 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 arm. If you did not ignore the mass of the arm, would the force you determined be smaller or larger? Explain.

Eric M.

Numerade Educator

$* \mathrm{B} 10$ Using biceps to hold a barbell Find the force that the biceps muscle shown in Figure $\mathrm{P} 7.26$ exerts on the forearm when you lift a $16-\mathrm{kg}$ barbell 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.

Eric M.

Numerade Educator

Leg support A person's broken leg is kept in place by the ap- paratus shown in Figure $\mathbf{P} 7.42 .$ If the rope pulling on the leg exerts a $120-\mathrm{N}$ force on it, how massive should be the block

hanging from the rope that passes over the pulley?

Eric M.

Numerade Educator

$*$ Blo Hamstring 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 $\mathbf{P} 7.43$ ). Determine the magnitude of the tension force $\vec{T}_{\mathrm{H} \text { on } L}$ exerted by the hamstring muscles on the leg and the magnitude of compression force $\vec{F}_{\mathrm{F} \text { on } \mathrm{B}}$ at the knee joint that the femur exerts on the calf bone.

The cord exerts a $20-1 \mathrm{b}$ force $\vec{F}_{\mathrm{C} \text { on } \mathrm{F}}$ on the foot.

Eric M.

Numerade Educator

$*$ You decide to hang a new 10 -kg flowerpot using the arrangement shown in Figure $P 7.44$ . 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.

Eric M.

Numerade Educator

"You decide to hang another plant from a $1.5-\mathrm{m}-\mathrm{long} 2.0 \mathrm{-kg}$

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 -N pot from the beam 1.4 $\mathrm{m}$ away from the wall. What is the force that the cable exerts on the beam?

Eric M.

Numerade Educator

$* *$ The plant in the hanging pot described in Problem 45 grows, and the pot and plant now have mass 12 $\mathrm{kg}$ . Determine the new 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).

Eric M.

Numerade Educator

$* *$ Now you decide to change the way you hang the pot described in Problems 45 and $46 .$ 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 $y-\mathrm{components}$ and the magnitude and direction of that force).

Eric M.

Numerade Educator

$*$ * Diving board The diving board shown in Figure $\mathrm{P} 7.20$ 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 -kg person stands on the end of the board 2.8 $\mathrm{m}$ from support post 2 .

Eric M.

Numerade Educator

$* *$ A uniform cubical $200-\mathrm{kg}$ box sits on the floor with its bottom left edge pressing against a ridge. The length $L$ of a side of the box is 1.2 $\mathrm{m}$ . Determine 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 $\mathbf{P} 7.49$ (Note: With the right edge slightly off the floor, the ground and ridge exert their forces on the bottom left edge of the box.)

Eric M.

Numerade Educator

"If the force $F$ shown in Figure $\mathrm{P} 7.49$ is 840 $\mathrm{N}$ 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 $\mathrm{m} ?$

Eric M.

Numerade Educator

We know from the second condition of equilibrium that if two different magnitude forces are exerted on the same object, their rotational effects can cancel if their torques are the same magnitude but opposite sign. For example, you can lift a heavy boulder by exerting a force much smaller than the weight of the boulder. Design an experiment where you can lift a $100-\mathrm{kg}$ rock by exerting a downward $100-\mathrm{N}$ push (it is much easier to push than to pull).

Eric M.

Numerade Educator

$*$ 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.]

Eric M.

Numerade Educator

$* / A 70$ -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.

Eric M.

Numerade Educator

$* *$ You are trying to tilt a very tall refrigerator $(2.0 \mathrm{m} \text { high, }$ 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.

Eric M.

Numerade Educator

$*$ You have an Atwood machine 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.

Eric M.

Numerade Educator

$*$ EST 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.

Eric M.

Numerade Educator

$* * B 10$ Lift with bent legs You injure your back at work lifting a $420-\mathrm{N}$ radiator. To understand how it happened, you model your back as a weightless beam (Figure $\mathrm{P} 7.57 )$ , analogous

a)

b)

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?

Mary Jo O.

Numerade Educator

$* *$ Determine the tension force that the horizontal cable exerts on the beam in Figure $\mathrm{P} 7.57,$ but with the beam tilted at $30^{\circ}$ rather than $15^{\circ} .$ The cable remains horizontal.

Eric M.

Numerade Educator

$\# *$ Determine the tension force that the horizontal cable exerts on the beam in Figure $P 7.59$ if the horizontal cable is moved down (compared to Figure (compared to Figure P7.57) so that its right

end attaches to the beam 0.30 $\mathrm{m}$ from its top right end. The cable remains horizontal and the beam is tilted at $15^{\circ} .$

Eric M.

Numerade Educator

$*$ Bl0 Barbell lift I A woman lifts a $3.6-\mathrm{kg}$ barbell in each hand with her arm in a horizontal position at the side of her body and holds it there for 3 $\mathrm{s}(\text { see Figure } \mathrm{P} 7.60) .$ What force does the deltoid muscle in her shoulder exert on the humerus bone while holding the barbell? The deltoid attaches 13 $\mathrm{cm}$ from the shoulder joint and makes a $13^{\circ}$ angle with the humerus. The barbell 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.

Eric M.

Numerade Educator

$* * B 10$ Barbell lift II Repeat the previous problem with a $7.2-\mathrm{kg}$ barbell. Determine both the force that the deltoid exerts on the humerus and the force that the lifter's shoulder joint exerts on her humerus.

Eric M.

Numerade Educator

$* B 10$ Facemask penalty The head of a football running back (see Figure $P 7.62 )$ 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 $180-\mathrm{N}(40-1 \mathrm{b})$ force on the facemask, what muscle force would these neck muscles now need to exert to keep the head in equilibrium?

Eric M.

Numerade Educator

$* *$ 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.

Eric M.

Numerade Educator

$*$ A board of mass $m$ and length $l$ is placed on a horizontal tabletop. The coefficient of static friction between the board and the table is $\mu$ . How far from the edge of the tabletop can one extend the board before it falls off?

Eric M.

Numerade Educator

$* *$ Tightrope walker $\mathrm{A} 60-\mathrm{kg} 1.6-\mathrm{m}$ -tall tightrope walker stands on a tightrope. (a) In his hands he holds a $10-\mathrm{kg} 2.0-\mathrm{m}$ -long horizontal bar. At each end of the bar are two $5.0-\mathrm{kg}$ balls hanging from $0.50-\mathrm{m}$ -long strings. How much does this apparatus lower his center of mass? (b) How long should the strings be so that the center of mass of the walker-bar hanging ball system is at the level of the rope? Indicate all assumptions made for each part of the problem.

Eric M.

Numerade Educator

$*$ " Lecturing on a beam A 70 -kg professor sits on a $20-\mathrm{kg}$ beam while lecturing (Figure $P 7.66 ) .$ A rope attached to the end of the beam passes over a pulley and down to a harness that wraps around the professor (the professor is supported partly by the beam and partly by the harness). The professor sits in the middle of the beam. Determine the force that the rope exerts on the harness and professor and the force that the beam exerts on the professor.

Eric M.

Numerade Educator

$* *$ A 70 -kg person stands on a $6.0-\mathrm{m}$ -long 50 -kg ladder. The ladder is tilted $60^{\circ}$ above the horizontal. The coefficient of friction between the floor and the ladder is $0.40 .$ How high can the person climb without the ladder slipping?

Eric M.

Numerade Educator

$* *$ What is a safe angle between a wall and a ladder for a 60 -kg painter to climb two-thirds of the height of the ladder without the ladder leaving the state of equilibrium? The ladder's mass is 10 $\mathrm{kg}$ and its length is 6.0 $\mathrm{m}$ . The coefficient of static friction between the floor and the feet of the ladder is 0.50 .

Eric M.

Numerade Educator

$*^{*}$ A ladder rests against a wall. The coefficient of static friction between the bottom end of the ladder and the floor is $\mu_{1} ;$ the coefficient between the top end of the ladder and the wall is $\mu_{2}$ . At what angle should the ladder be oriented so it does not slip and both coefficients of friction are 0.50$?$

Eric M.

Numerade Educator

$* *$ Every rope or cord has a maximum tension that it can withstand before breaking. Investigate how a ski lift works and explain how it can safely move a large number of passengers of different mass uphill during peak hours, without the cord that carries the chairs breaking.

Eric M.

Numerade Educator

You hold a 10 -lb ball in your hand with your forearm hori- zontal, forming a $90^{\circ}$ angle with the upper arm (Figure 7.26$)$ Which type of muscle produces the torque that allows you to hold the bell?

(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

Eric M.

Numerade Educator

In Figure 7.26, 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) $0,5,35 \quad$ (b) $35,5,0 \quad$ (c) $35,5,3$

(d) $35,5,-3 \quad$ (e) $30,5,0$

Eric M.

Numerade Educator

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.

Eric M.

Numerade Educator

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.

Eric M.

Numerade Educator

Rank in order the magnitudes of the distances of the four forces exerted on the backbone with respect to the joint (see Figure 7.27c), with the largest distance listed first.

$$

\begin{array}{ll}{\text { (a) } 1>3>2>4} & {\text { (b) } 4>2=3>1} \\ {\text { (c) } 4>3>2>1} & {\text { (d) } 2>3>1>4} \\ {\text { (e) } 1} {>2>2>4}\end{array}

$$

Eric M.

Numerade Educator

Rank in order the magnitudes of the torques caused by the four forces exerted on the backbone (see Figure 7.27c), with the largest torque listed first.

$$

\begin{array}{ll}{\text { (a) } 1>2>3>4} & {\text { (b) } 2=3>1>4} \\ {\text { (c) } 3>2>1>4} & {\text { (d) } 2>1>3>4} \\ {\text { (e) } 1} {=2=3=4}\end{array}

$$

Eric M.

Numerade Educator

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 7.27c?

$$

\begin{array}{ll}{\text { (a) }+,+,+,+} & {\text { (b) }-,+,-, 0} \\ {(\mathrm{c})+,-,+, 0} & {\text { (d) }-,-,-, 0} \\ {\text { (e) }+,-,+,} & {\text { (d) }-,-,-, 0}\end{array}

$$

Eric M.

Numerade Educator

Which expression below best describes the torque caused by force F4 = FDonB, the force that the disk in the lower back exerts on the backbone of length L?

$$

\begin{array}{l}{\text { (a) } 0 \qquad \text { (b) } F_{4}(2 L / 3) \sin 12^{\circ}} \\ {\text { (c) } F_{4}(L) \cos 30^{\circ}} \\ {\text { (e) }-F_{4}(L) \cos 30^{\circ}}\end{array}

$$

Eric M.

Numerade Educator

Which expression below best describes the torque caused by force F3 = FE on B, the force that Earth exerts on the upper body at its center of mass for the backbone of length L?

$$

\begin{array}{ll}{\text { (a) } 0} & {\text { (b) } F_{3}(2 L / 3) \sin 12^{\circ}} \\ {\text { (c) } F_{3}(L / 2) \cos 30^{\circ}} & {\text { (d) }-F_{3}(2 L / 3) \sin 12^{\circ}}\\ {\text { (e) }-F_{3}(L / 2) \cos 30^{\circ}}\end{array}

$$

Eric M.

Numerade Educator

Which expression below best describes the torque caused by force $F_{2}=T_{\text { M on } \mathrm{B}}$ exerted by the muscle on the backbone?

$\begin{array}{ll}{\text { (a) } 0} & {\text { (b) } F_{2}(2 L / 3) \sin 12^{\circ}} \\ {\text { (c) } F_{2}(L) \cos 30^{\circ}} & {\text { (d) }-F_{2}(2 L / 3) \sin 12^{\circ}}\\ {\text { (e) }-F_{2}(L) \cos 30^{\circ}}\end{array}$

$$

$$

Eric M.

Numerade Educator