Solved: A cylinder of mass m and radius r has a string

To tighten a spark plug, it is recommended that a torque of be applied. If a mechanic tightens the spark plug with a wrench that is 25 cm long, what is the minimum force necessary to create the desired torque?

Pulling a Weed The gardening tool shown in Figure 1121 is used to pull weeds. If a torque is required to pull a given weed, what force did the weed exert on the tool?

A1.61-kg bowling trophy is held at arms length, a distance of 0.605 m from the shoulder joint. What torque does the trophy exert about the shoulder if the arm is (a) horizontal, or (b) at an angle of 22.5 below the horizontal?

A person slowly lowers a 3.6-kg crab trap over the side of a dock, as shown in Figure 1122. What torque does the trap exert about the persons shoulder?

orce to Hold a BaseballAperson holds a 1.42-N baseball in his hand, a distance of 34.0 cm from the elbow joint, as shown in Figure 1123. The biceps, attached at a distance of 2.75 cm from the elbow, exerts an upward force of 12.6 N on the forearm. Consider the forearm and hand to be a uniform rod with a mass of 1.20 kg. (a)Calculate the net torque acting on the forearm and hand. Use the elbow joint as the axis of rotation. (b)If the net torque obtained in part (a) is nonzero, in which direction will the forearm and hand rotate? (c) Would the torque exerted on the forearm by the biceps increase or decrease if the biceps were attached farther from the elbow joint?

At the local playground, a 16-kg child sits on the end of a horizontal teeter-totter, 1.5 m from the pivot point. On the other side of the pivot an adult pushes straight down on the teeter-totter with a force of 95 N. In which direction does the teeter-totter rotate if the adult applies the force at a distance of (a) 3.0 m, (b) 2.5 m, or (c) 2.0 m from the pivot?

Consider the pulleyblock systems shown in Conceptual Checkpoint 111. (a) Is the tension in the string on the left-hand rotating system greater than, less than, or equal to the weight of the mass attached to that string? (b) Choose the best explanation from among the following:I. The mass is in free fall once it is released. II. The string rotates the pulley in addition to supporting the mass. III. The mass accelerates downward.

Predict/Explain Consider the pulleyblock systems shown in Conceptual Checkpoint 111. (a) Is the tension in the string on the left-hand rotating system greater than, less than, or equal to the tension in the string on the right-hand rotating system? (b) Choose the best explanation from among the following: I. The mass in the right-hand system has the greater downward acceleration. II. The masses are equal.III. The mass in the left-hand system has the greater downward acceleration.

Suppose a torque rotates your body about one of three different axes of rotation: case A, an axis through your spine; case B, an axis through your hips; and case C, an axis through your ankles. Rank these three axes of rotation in increasing order of the angular acceleration produced by the torque. Indicate ties where appropriate.

Atorque of is applied to a bicycle wheel of radius 35 cm and mass 0.75 kg. Treating the wheel as a hoop, nd its angular acceleration.

When a ceiling fan rotating with an angular speed of 2.75 rad/s is turned off, a frictional torque of slows it to a stop in 22.5 s. What is the moment of inertia of the fan?

When the play button is pressed, a CD accelerates uniformly from rest to 450 rev/min in 3.0 revolutions. If the CD has a radius of 6.0 cm and a mass of 17 g, what is the torque exerted on it?

A person holds a ladder horizontally at its center. Treating the ladder as a uniform rod of length 3.15 m and mass 8.42 kg, nd the torque the person must exert on the ladder to give it an angular acceleration of .

Awheel on a game show is given an initial angular speed of 1.22 rad/s. It comes to rest after rotating through 0.75 of a turn. (a) Find the average torque exerted on the wheel given that it is a disk of radius 0.71 m and mass 6.4 kg. (b) If the mass of the wheel is doubled and its radius is halved, will the angle through which it rotates before coming to rest increase, decrease, or stay the same? Explain. (Assume that the average torque exerted on the wheel is unchanged.)

The L-shaped object in Figure 1124 consists of three masses connected by light rods. What torque must be applied to this object to give it an angular acceleration of if it is rotated about (a) the x axis, (b) the y axis, or (c) the z axis (which is through the origin and perpendicular to the page)?

The L-shaped object described in Problem 15 can be rotated in one of the following three ways: case A, about the xaxis; case B, about the y axis; and case C, about the z axis (which passes through the origin perpendicular to the plane of the gure). If the same torque is applied in each of these cases, rank them in increasing order of the resulting angular acceleration. Indicate ties where appropriate.

Amotorcycle accelerates from rest, and both the front and rear tires roll without slipping. (a) Is the force exerted by the ground on the rear tire in the forward or in the backward direction? Explain. (b) Is the force exerted by the ground on the front tire in the forward or in the backward direction? Explain. (c)If the moment of inertia of the front tire is increased, will the motorcycles acceleration increase, decrease, or stay the same? Explain.

The torque can act about the x axis, the y axis, or the z axis, which passes through the origin and points out of the page. (a)In which case does the object experience the greatest angular acceleration? The least angular acceleration? Explain. Find the angular acceleration when the torque acts about (b) the x axis, (c) the y axis, and (d) the z axis.

Ash takes the bait and pulls on the line with a force of 2.2 N. The shing reel, which rotates without friction, is a cylinder of radius 0.055 m and mass 0.99 kg. (a) What is the angular acceleration of the shing reel? (b)How much line does the sh pull from the reel in 0.25 s?

Repeat the previous problem, only now assume the reel has a friction clutch that exerts a restraining torque of .

Suppose the person in Active Example 113 climbs higher on the ladder. (a) As a result, is the ladder more likely, less likely, or equally likely to slip? (b) Choose the best explanation from among the following: I. The forces are the same regardless of the persons position. II. The magnitude of f2 must increase as the person moves upward. III. When the person is higher, the ladder presses down harder on the oor.

A string that passes over a pulley has a 0.321-kg mass attached to one end and a 0.635-kg mass attached to the other end. The pulley, which is a disk of radius 9.40 cm, has friction in its axle. What is the magnitude of the frictional torque that must be exerted by the axle if the system is to be in static equilibrium?

To loosen the lid on a jar of jam 8.9 cm in diameter, a torque of must be applied to the circumference of the lid. If a jar wrench whose handle extends 15 cm from the center of the jar is attached to the lid, what is the minimum force required to open the jar?

Consider the system in Active Example 111, this time with the axis of rotation at the location of the child. Write out both the condition for zero net force and the condition for zero net torque. Solve for the two forces.

Referring to the person holding a baseball in Problem 5, suppose the biceps exert just enough upward force to keep the system in static equilibrium. (a)Is the force exerted by the biceps more than, less than, or equal to the combined weight of the forearm, hand, and baseball? Explain. (b) Determine the force exerted by the biceps.

A Persons Center of Mass To determine the location of her center of mass, a physics student lies on a lightweight plank supported by two scales 2.50 m apart, as indicated in Figure 1126. If the left scale reads 290 N, and the right scale reads 122 N, nd (a) the students mass and (b) the distance from the students head to her center of mass.

TriceratopsAset of fossilized triceratops footprints discovered in Texas show that the front and rear feet were 3.2 m apart, as shown in Figure 1127. The rear footprints were observed to be twice as deep as the front footprints. Assuming that the rear feet pressed down on the ground with twice the force exerted by the front feet, nd the horizontal distance from the rear feet to the triceratopss center of mass.

Aschoolyard teeter-totter with a total length of 5.2 m and a mass of 38 kg is pivoted at its center. A19-kg child sits on one end of the teeter-totter. (a) Where should a parent push vertically downward with a force of 210 N in order to hold the teeter-totter level? (b) Where should the parent push with a force of 310 N? (c)How would your answers to parts (a) and (b) change if the mass of the teeter-totter were doubled? Explain.

A 0.122-kg remote control 23.0 cm long rests on a table, as shown in Figure 1128, with a length Loverhanging its edge. To operate the power button on this remote requires a force of 0.365 N. How far can the remote control extend beyond the edge of the table and still not tip over when you press the power button? Assume the mass of the remote is distributed uniformly, and that the power button is 1.41 cm from the overhanging end of the remote.

A 0.16-kg meterstick is held perpendicular to a vertical wall by a 2.5-m string going from the wall to the far end of the stick. (a) Find the tension in the string. (b) If a shorter string is used, will its tension be greater than, less than, or the same as that found in part (a)? (c) Find the tension in a 2.0-m string.

Repeat Example 114, this time with a uniform diving board that weighs 225 N.

Babe Ruth steps to the plate and casually points to left center eld to indicate the location of his next home run. The mighty Babe holds his bat across his shoulder, with one hand holding the small end of the bat. The bat is horizontal, and the distance from the small end of the bat to the shoulder is 22.5 cm. If the bat has a mass of 1.10 kg and has a center of mass that is 67.0 cm from the small end of the bat, nd the magnitude and direction of the force exerted by (a) the hand and (b) the shoulder.

Auniform metal rod, with a mass of 3.7 kg and a length of 1.2 m, is attached to a wall by a hinge at its base. A horizontal wire bolted to the wall 0.51 m above the base of the rod holds the rod at an angle of 25 above the horizontal. The wire is attached to the top of the rod. (a) Find the tension in the wire. Find (b) the horizontal and (c) the vertical components of the force exerted on the rod by the hinge.

In the previous problem, suppose the wire is shortened, so that the rod now makes an angle of 35 with the horizontal. The wire is horizontal, as before. (a) Do you expect the tension in the wire to increase, decrease, or stay the same as a result of its new length? Explain. (b) Calculate the tension in the wire.

Repeat Active Example 113, this time with a uniform 7.2-kg ladder that is 4.0 m long.

A rigid, vertical rod of negligible mass is connected to the oor by a bolt through its lower end, as shown in Figure 1129. The rod also has a wire connected between its top end and the oor. If a horizontal force Fis applied at the midpoint of the rod, nd(a)the tension in the wire, and (b)the horizontal and (c)the vertical components of force exerted by the bolt on the rod.

Forces in the Foot Figure 1130 shows the forces acting on a sprinters foot just before she takes off at the start of the race. Find the magnitude of the force exerted on the heel by the Achilles tendon, , and the magnitude of the force exerted on the foot at the ankle joint, .

Astick with a mass of 0.214 kg and a length of 0.436 m rests in contact with a bowling ball and a rough oor, as shown in Figure 1131. The bowling ball has a diameter of 21.6 cm, and the angle the stick makes with the horizontal is 30.0. You may assume there is no friction between the stick and the bowling ball, though friction with the oor must be taken into account. (a)Find the magnitude of the force exerted on the stick by the bowling ball.(b)Find the horizontal component of the force exerted on the stick by the oor. (c)Repeat part (b) for the vertical component of the force.

Auniform crate with a mass of 16.2 kg rests on a oor with a coefcient of static friction equal to 0.571. The crate is a uniform cube with sides 1.21 m in length. (a) What horizontal force applied to the top of the crate will initiate tipping? (b) If the horizontal force is applied halfway to the top of the crate, it will begin to slip before it tips. Explain.

In the previous problem, (a) what is the minimum height where the force Fcan be applied so that the crate begins to tip before sliding? (b) What is the magnitude of the force in this case?

Ahand-held shopping basket 62.0 cm long has a 1.81-kg carton of milk at one end, and a 0.722-kg box of cereal at the other end. Where should a 1.80-kg container of orange juice be placed so that the basket balances at its center?

If the cat in Active Example 112 has a mass of 2.8 kg, how close to the right end of the two-by-four can it walk before the board begins to tip?

A0.34-kg meterstick balances at its center. If a necklace is suspended from one end of the stick, the balance point moves 9.5 cm toward that end. (a) Is the mass of the necklace more than, less than, or the same as that of the meterstick? Explain. (b) Find the mass of the necklace.

Maximum Overhang Three identical, uniform books of length L are stacked one on top the other. Find the maximum overhang distance d in Figure 1132 such that the books do not fall over.

Abaseball bat balances 71.1 cm from one end. If a 0.560-kg glove is attached to that end, the balance point moves 24.7 cm toward the glove. Find the mass of the bat.

A2.85-kg bucket is attached to a disk-shaped pulley of radius 0.121 m and mass 0.742 kg. If the bucket is allowed to fall, (a)what is its linear acceleration? (b) What is the angular acceleration of the pulley? (c) How far does the bucket drop in 1.50 s?

In the previous problem, (a) is the tension in the rope greater than, less than, or equal to the weight of the bucket? Explain. (b) Calculate the tension in the rope.

Achild exerts a tangential 42.2-N force on the rim of a diskshaped merry-go-round with a radius of 2.40 m. If the merrygo-round starts at rest and acquires an angular speed of 0.0860 rev/s in 3.50 s, what is its mass?

You pull downward with a force of 28 N on a rope that passes over a disk-shaped pulley of mass 1.2 kg and radius 0.075 m. The other end of the rope is attached to a 0.67-kg mass. (a) Is the tension in the rope the same on both sides of the pulley? If not, which side has the largest tension? (b) Find the tension in the rope on both sides of the pulley.

Referring to the previous problem, nd the linear acceleration of the 0.67-kg mass.

Auniform meterstick of mass Mhas an empty paint can of mass m hanging from one end. The meterstick and the can balance at a point 20.0 cm from the end of the stick where the can is attached. When the balanced stickcan system is suspended from a scale, the reading on the scale is 2.54 N. Find the mass of (a) the meterstick and (b) the paint can.

Atwoods MachineAn Atwoods machine consists of two masses, and , connected by a string that passes over a pulley. If the pulley is a disk of radius R and mass M, nd the acceleration of the masses.

Calculate the angular momentum of the Earth about its own axis, due to its daily rotation. Assume that the Earth is a uniform sphere.

A0.015-kg record with a radius of 15 cm rotates with an angular speed of Find the angular momentum of the record.

In the previous problem, a 1.1-g y lands on the rim of the record. What is the ys angular momentum?

Jogger 1 in Figure 1133 has a mass of 65.3 kg and runs in a straight line with a speed of 3.35 m/s. (a)What is the magnitude of the joggers linear momentum? (b) What is the magnitude of the joggers angular momentum with respect to the origin, O?

Repeat the previous problem for the case of jogger 2, whose speed is 2.68 m/s and whose mass is 58.2 kg.

Suppose jogger 3 in Figure 1133has a mass of 62.2 kg and a speed of 5.85 m/s. (a) Is the magnitude of the joggers angular momentum greater with respect to point Aor point B? Explain. (b) Is the magnitude of the joggers angular momentum with respect to point B greater than, less than, or the same as it is with respect to the origin, O? Explain. (c)Calculate the magnitude of the joggers angular momentum with respect to points A, B, and O.

A torque of is applied to an egg beater. (a) If the egg beater starts at rest, what is its angular momentum after 0.65 s? (b) If the moment of inertia of the egg beater is , what is its angular speed after 0.65 s?

A windmill has an initial angular momentum of . The wind picks up, and 5.86 s later the windmills angular momentum is . What was the torque acting on the windmill, assuming it was constant during this time?

Two gerbils run in place with a linear speed of 0.55 m/s on an exercise wheel that is shaped like a hoop. Find the angular momentum of the system if each gerbil has a mass of 0.22 kg and the exercise wheel has a radius of 9.5 cm and a mass of 5.0 g.

Predict/Explain Astudent rotates on a frictionless piano stool with his arms outstretched, a heavy weight in each hand. Suddenly he lets go of the weights, and they fall to the oor. As a result, does the students angular speed increase, decrease, or stay the same? (b) Choose the best explanation from among the following: I. The loss of angular momentum when the weights are dropped causes the student to rotate more slowly. II. The students moment of inertia is decreased by dropping the weights. III. Dropping the weights exerts no torque on the student, but the oor exerts a torque on the weights when they land.

Apuck on a horizontal, frictionless surface is attached to a string that passes through a hole in the surface, as shown in Figure 1134. As the puck rotates about the hole, the string is pulled downward, bringing the puck closer to the hole. During this process, do the pucks (a) linear speed, (b) angular speed, and(c)angular momentum increase, decrease, or stay the same?

Apuck on a horizontal, frictionless surface is attached to a string that wraps around a pole of nite radius, as shown in Figure 1135. (a)As the puck moves along the spiral path, does its decrease, or stay the same? Explain. (b) Does its angular momentum increase, decrease, or stay the same? Explain.

As an ice skater begins a spin, his angular speed is 3.17 rad/s. After pulling in his arms, his angular speed increases to 5.46 rad/s. Find the ratio of the skaters nal moment of inertia to his initial moment of inertia.

Calculate both the initial and the nal kinetic energies of the system described in Active Example 115.

Adiver tucks her body in midight, decreasing her moment of inertia by a factor of two. By what factor does her angular speed change?

In the previous problem, (a)does the divers kinetic energy increase, decrease, or stay the same? (b)Calculate the ratio of the nal kinetic energy to the initial kinetic energy for the diver.

A disk-shaped merry-go-round of radius 2.63 m and mass 155 kg rotates freely with an angular speed of 0.641 rev/s. A 59.4-kg person running tangential to the rim of the merry-goround at 3.41 m/s jumps onto its rim and holds on. Before jumping on the merry-go-round, the person was moving in the same direction as the merry-go-rounds rim. What is the nal angular speed of the merry-go-round?

In the previous problem, (a) does the kinetic energy of the system increase, decrease, or stay the same when the person jumps on the merry-go-round? (b)Calculate the initial and nal kinetic energies for this system.

Astudent sits at rest on a piano stool that can rotate without friction. The moment of inertia of the studentstool system is . Asecond student tosses a 1.5-kg mass with a speed of 2.7 m/s to the student on the stool, who catches it at a distance of 0.40 m from the axis of rotation. What is the resulting angular speed of the student and the stool?

Referring to the previous problem, (a) does the kinetic energy of the massstudentstool system increase, decrease, or stay the same as the mass is caught? (b)Calculate the initial and nal kinetic energies of the system.

Aturntable with a moment of inertia of rotates freely with an angular speed of Riding on the rim of the turntable, 15 cm from the center, is a cute, 32-g mouse. (a)If the mouse walks to the center of the turntable, will the turntable rotate faster, slower, or at the same rate? Explain. (b)Calculate the angular speed of the turntable when the mouse reaches the center.

A student on a piano stool rotates freely with an angular speed of 2.95 rev/s. The student holds a 1.25-kg mass in each outstretched arm, 0.759 m from the axis of rotation. The combined moment of inertia of the student and the stool, ignoring the two masses, is , a value that remains constant. (a) As the student pulls his arms inward, his angular speed increases to 3.54 rev/s. How far are the masses from the axis of rotation at this time, considering the masses to be points? (b) Calculate the initial and nal kinetic energies of the system.

Walking on a Merry-Go-Round A child of mass m stands at rest near the rim of a stationary merry-go-round of radius R and moment of inertia I. The child now begins to walk around the circumference of the merry-go-round with a tangential speed v with respect to the merry-go-rounds surface. (a) What is the childs speed with respect to the ground? Check your result in the limits (b) and (c

Predict/Explain Two spheres of equal mass and radius are rolling across the oor with the same speed. Sphere 1 is a uniform solid; sphere 2 is hollow. Is the work required to stop sphere 1 greater than, less than, or equal to the work required to stop sphere 2? (b) Choose the best explanation from among the following: I. Sphere 2 has the greater moment of inertia and hence the greater rotational kinetic energy. II. The spheres have equal mass and speed; therefore, they have the same kinetic energy. III. The hollow sphere has less kinetic energy.

How much work must be done to accelerate a baton from rest to an angular speed of 7.4 rad/s about its center? Consider the baton to be a uniform rod of length 0.53 m and mass 0.44 kg.

Turning a doorknob through 0.25 of a revolution requires 0.14 J of work. What is the torque required to turn the doorknob?

A person exerts a tangential force of 36.1 N on the rim of a disk-shaped merry-go-round of radius 2.74 m and mass 167 kg. If the merry-go-round starts at rest, what is its angular speed after the person has rotated it through an angle of 32.5?

To prepare homemade ice cream, a crank must be turned with a torque of . How much work is required for each complete turn of the crank?

Power of a Dental Drill Apopular make of dental drill can operate at a speed of 42,500 rpm while producing a torque of . What is the power output of this drill? Give your answer in watts.

The L-shaped object in Figure 1124consists of three masses connected by light rods. Find the work that must be done on this object to accelerate it from rest to an angular speed of 2.35 rad/s about (a) the x axis, (b) the y axis, and (c) the z axis (which is through the origin and perpendicular to the page).

The rectangular object in Figure 1125 consists of four masses connected by light rods. What power must be applied to this object to accelerate it from rest to an angular speed of 2.5 rad/s in 6.4 s about (a) the x axis, (b) the y axis, and (c) the z axis (which is through the origin and perpendicular to the page)?

A circular saw blade accelerates from rest to an angular speed of 3620 rpm in 6.30 revolutions. (a)Find the torque exerted on the saw blade, assuming it is a disk of radius 15.2 cm and mass 0.755 kg. (b)Is the angular speed of the saw blade after 3.15 revolutions greater than, less than, or equal to 1810 rpm? Explain. (c) Find the angular speed of the blade after 3.15 revolutions.

Auniform disk stands upright on its edge, and rests on a sheet of paper placed on a tabletop. If the paper is pulled horizontally to the right, as in Figure 1136, (a) does the disk rotate clockwise or counterclockwise about its center? Explain. (b)Does the center of the disk move to the right, move to the left, or stay in the same location? Explain.

Consider the two rotating systems shown in Figure 1137, each consisting of a mass mattached to a rod of negligible mass pivoted at one end. On the left, the mass is attached at the midpoint of the rod; to the right, it is attached to the free end of the rod. The rods are released from rest in the horizontal position at the same time. When the rod to the left reaches the vertical position, is the rod to the right not yet vertical (location A), vertical (location B), or past vertical (location C)? Explain.

Predict/Explain A disk and a hoop (bicycle wheel) of equal radius and mass each have a string wrapped around their circumferences. Hanging from the strings, halfway between the disk and the hoop, is a block of mass m, as shown in Figure 1138. The disk and the hoop are free to rotate about their centers. When the block is allowed to fall, does it stay on the center line, move toward the right, or move toward the left? (b)Choose the best explanation from among the following: I. The disk is harder to rotate, and hence its angular acceleration is less than that of the wheel. II. The wheel has the greater moment of inertia and unwinds more slowly than the disk. III. The system is symmetric, with equal mass and radius on either side.

Abeetle sits at the rim of a turntable that is at rest but is free to rotate about a vertical axis. Suppose the beetle now begins to walk around the perimeter of the turntable. Does the beetle move forward, backward, or does it remain in the same location relative to the ground? Answer for two different cases, (a) the turntable is much more massive than the beetle and (b) the turntable is massless.

A beetle sits near the rim of a turntable that is rotating without friction about a vertical axis. The beetle now begins to walk toward the center of the turntable. As a result, does the angular speed of the turntable increase, decrease, or stay the same? Explain.

A beetle sits near the rim of a turntable that is rotating without friction about a vertical axis. The beetle now begins to walk toward the center of the turntable. As a result, does the angular speed of the turntable increase, decrease, or stay the same? Explain.

After getting a drink of water, a hamster jumps onto an exercise wheel for a run. Afew seconds later the hamster is running in place with a speed of 1.3 m/s. Find the work done by the hamster to get the exercise wheel moving, assuming it is a hoop of radius 0.13 m and mass 6.5 g.

A47.0-kg uniform rod 4.25 m long is attached to a wall with a hinge at one end. The rod is held in a horizontal position by a wire attached to its other end. The wire makes an angle of 30.0 with the horizontal, and is bolted to the wall directly above the hinge. If the wire can withstand a maximum tension of 1450 N before breaking, how far from the wall can a 68.0-kg person sit without breaking the wire?

Apuck attached to a string moves in a circular path on a frictionless surface, as shown in Figure 1134. Initially, the speed of the puck is vand the radius of the circle is r. If the string passes through a hole in the surface, and is pulled downward until the radius of the circular path is r/2,(a)does the speed of the puck increase, decrease, or stay the same? (b)Calculate the nal speed of the puck.

The Masseter MuscleThe masseter muscle, the principal muscle for chewing, is one of the strongest muscles for its size in the human body. It originates on the lower edge of the zygomatic arch (cheekbone) and inserts in the angle of the mandible. Referring to the lower diagram in Figure 1139, where and , (a) nd the torque produced about the axis of rotation by the masseter muscle. The force exerted by the masseter muscle is . (b)Find the biting force, , exerted on the mandible by the upper teeth. Find (c) the horizontal and (d) the vertical component of the force exerted on the mandible at the joint where it attaches to the skull. Assume that the mandible is in static equilibrium, and that upward is the positive vertical direction.

Exercising the Biceps You are designing exercise equipment to operate as shown in Figure 1140, where a person pulls upward on an elastic cord. The cord behaves like an ideal spring and has an unstretched length of 31 cm. If you would like the torque about the elbow joint to be in the position shown, what force constant, k, is required for the cord?

Horsepower of a Car Auto mechanics use the following formula to calculate the horsepower (HP) of a car engine: In this expression, Torque is the torque produced by the engine in , RPM is the angular speed of the engine in revolutions per minute, and C is a dimensionless constant. (a) Find the numerical value of C. (b) The Shelby Series 1 engine is advertised to generate 320 hp at 6500 rpm. What is the corresponding torque produced by this engine? Give your answer

Balancing a T. rex Paleontologists believe that Tyrannosaurus rex stood and walked with its spine almost horizontal, as indicated in Figure 1141, and that its tail was held off the ground to balance its upper torso about the hip joint. Given that the total mass of T. rex was 5400 kg, and that the placement of the center of mass of the tail and the upper torso was as shown in Figure 1141,nd the mass of the tail required for balance.

You hold a uniform, 28-g pen horizontal with your thumb pushing down on one end and your index nger pushing upward 3.5 cm from your thumb. The pen is 14 cm long. (a)Which of these two forces is greater in magnitude? (b) Find the two forces.

In Active Example 113, suppose the ladder is uniform, 4.0 m long, and weighs 60.0 N. Find the forces exerted on the ladder when the person is (a) halfway up the ladder and (b) three-fourths of the way up the ladder.

When you arrive at Dukes Dude Ranch, you are greeted by the large wooden sign shown in Figure 1142. The left end of the sign is held in place by a bolt, the right end is tied to a rope that makes an angle of 20.0with the horizontal. If the sign is uniform, 3.20 m long, and has a mass of 16.0 kg, what are (a) the tension in the rope, and (b)the horizontal and vertical components of the force, , exerted by the bolt?

A 67.0-kg person stands on a lightweight diving board supported by two pillars, one at the end of the board, the other 1.10 m away. The pillar at the end of the board exerts a downward force of 828 N. (a) How far from that pillar is the person standing? (b) Find the force exerted by the second pillar.

In Example 114, nd and as a function of the distance, x, of the swimmer from the left end of the diving board. Assume that the diving board is uniform and has a mass of 85.0 kg.

Flats Versus Heels A woman might wear a pair of at shoes to work during the day, as in Figure 1143(a), but a pair of high heels, Figure 1143 (b), when going out for the evening. Assume that each foot supports half her weight, , and that the forces exerted by the oor on her feet occur at the points A and B in both gures. Find the forces (point A) and (point B) for (a) at shoes and (b) high heels. (c)How have the high heels changed the weight distribution between the womans heels and toes?

A young girl sits at the edge of a dock by the bay, dipping her feet in the water. At the instant shown in Figure 1144, she holds her lower leg stationary with her quadriceps muscle at an angle of 39with respect to the horizontal. Use the information given in the gure, plus the fact that her lower leg has a mass of 3.4 kg, to determine the magnitude of the force, exerted on the lower leg by the quadriceps.

Deltoid MuscleAcrossing guard holds a STOPsign at arms length, as shown in Figure 1145. Her arm is horizontal, and we assume that the deltoid muscle is the only muscle supporting her arm. The weight of her upper arm is , the weight of her lower arm is , the weight of her hand is , and the weight of the sign is . The location where each of these forces acts on the arm is indicated in the gure. A force of magnitude is exerted on the humerus by the deltoid, and the shoulder joint exerts a force on the humerus with horizontal and vertical components given by and respectively. (a) Is the magnitude of greater than, less than, or equal to the magnitude of Explain. Find (b) (c) and(d) (The weights in Figure 1145 are drawn to scale; the unknown forces are to be determined. If a force is found to be negative, its direction is opposite to that shown.)

Triceps To determine the force a persons triceps muscle can exert, a doctor uses the procedure shown in Figure 1146, where the patient pushes down with the palm of his hand on a force meter. Given that the weight of the lower arm is and that the force meter reads what is the force exerted vertically upward by the triceps?

Suppose a fourth book, the same as the other three, is added to the stack of books shown in Figure 1132. (a) What is the maximum overhang distance, d, in this case? (b) If the mass of each book is increased by the same amount, does your answer to part (a) increase, decrease, or stay the same? Explain.

Suppose partial melting of the polar ice caps increases the moment of inertia of the Earth from to (a) Would the length of a day (the time required for the Earth to complete one revolution about its axis) increase or decrease? Explain. (b) Calculate the change in the length of a day. Give your answer in seconds.

Abicycle wheel of radius R and mass M is at rest against a step of height 3R/4, as illustrated in Figure 1147. Find the minimum horizontal force Fthat must be applied to the axle to make the wheel start to rise up over the step.

A0.101-kg yo-yo has an outer radius R that is 5.60 times greater than the radius r of its axle. The yo-yo is in equilibrium if a mass m is suspended from its outer edge, as shown in Figure 1148. Find the tension in the two strings, and , and the mass m.

In Problem 36, assume that the rod has a mass of M and that its bottom end simply rests on the oor, held in place by static friction. If the coefcient of static friction is , nd the maximum force Fthat can be applied to the rod at its midpoint before it slips.

In the previous problem, suppose the rod has a mass of 2.3 kg and the coefcient of static friction is 1/7. (a) Find the greatest force F that can be applied at the midpoint of the rod without causing it to slip. (b)Show that if Fis applied 1/8 of the way down from the top of the rod, it will never slip at all, no matter how large the force F.

A cylinder of mass m and radius r has a string wrapped around its circumference. The upper end of the string is held xed, and the cylinder is allowed to fall. Show that its linear acceleration is (2/3)g.

Repeat the previous problem, replacing the cylinder with a solid sphere. Show that its linear acceleration is (5/7)g.

A mass M is attached to a rope that passes over a diskshaped pulley of mass m and radius r. The mass hangs to the left side of the pulley. On the right side of the pulley, the rope is pulled downward with a force F. Find (a) the acceleration of the mass, (b) the tension in the rope on the left side of the pulley, and (c) the tension in the rope on the right side of the pulley. (d) Check your results in the limits

Bricks in Equilibrium Consider a system of four uniform bricks of length L stacked as shown in Figure 1149. What is the maximum distance, x, that the middle bricks can be displaced outward before they begin to tip?

The two, solid straight lines in Figure 1150(b) represent the two forces applied to the tooth. Which line corresponds to which force? A. B. I = F2y, II = F1y

What is the value of the torque that corresponds to one of the forces being equal to zero? A. B. C. D

Find the values of and required to give zero net torque. A. B. C. D

Find the values of and required to give a net torque of This is a torque that would be effective at rotating the tooth. A. B. C. D

Referring to Example 117Suppose the mass of the pulley is doubled, to 0.160 kg, and that everything else in the system remains the same. (a) Do you expect the value of to increase, decrease, or stay the same? Explain. (b) Calculate the value of for this case.

Referring to Example 117Suppose the mass of the cart is doubled, to 0.62 kg, and that everything else in the system remains the same. (a)Do you expect the value of to increase, decrease, or stay the same? Explain. (b) Calculate the value of for this case.

Referring to Active Example 115 Suppose the child runs with a different initial speed, but that everything else in the system remains the same. What initial speed does the child have if the angular speed of the system after the collision is

Referring to Active Example 115Suppose everything in the system is as described in Active Example 115 except that the child approaches the merry-go-round in a direction that is not tangential. Find the angle between the direction of motion and the outward radial direction (as in Example 118) that is required if the nal angular speed of the system is to be 0.272 rad/s.