Mastering Physics Solutions Chapter 11 Rotational Dynamics and Static Equilibrium
Chapter 11 Rotational Dynamics and Static Equilibrium Q.1CQ
Two forces produce the same torque Does it follow that they have the same magnitude? Explain
Solution:
No, we know that the torque exerted by a tangential force a distance r from the axis of rotation
is t=rF
Here, the torque depends on both the magnitude of force and on the distance from the axis of rotation at which it is applied However because the forces are the same, the torque depends
on the axis of rotationS A small force can produce the same torque as a large force, if it is applied farther from the axis of rotation
Chapter 11 Rotational Dynamics and Static Equilibrium Q.1P
To tighten a spark plug, it is recommended that a torque of 15 N · m 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?
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.2CQ
A car pitches down in front when the brakes are applied sharply. Explain this observation in terms of torques.
Solution:
The torque is given by,t = rF
Here, r is the distance of the axis of rotation to the force and F is the tangential force.
When the brakes are applied, it causes the wheels to lock and then the friction plays the role to stop the car. The total force on the car acts on the center of mass but the friction does not be applied on the center of mass of the vehicle and it is applied to the tires. The friction force opposes the motion and its direction is negative and this causes negative torque to be applied to the vehicle which leads to the clockwise rotation of the center of the mass as given by the expression of the torque. This clockwise rotation of the center of the mass causes the front of the car to pitch downward.
The torque by gravity which is in the opposite direction to the torque by friction acts as the restoring torque because the first law of angular motion states that the body will maintain constant angular motion unless the outside torque is acted upon it. Hence, the car pitches down in front when the brakes are applied sharply.
Chapter 11 Rotational Dynamics and Static Equilibrium Q.2P
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.3CQ
A tightrope walker uses a long pole to aid in balancing. Why?
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.3P
A 1.61-kg bowling trophy is held at arm’s 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?
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.4CQ
When a motorcycle accelerates rapidly from a stop it sometimes “pops a wheelie”; that is, its front wheel may lift off the ground. Explain this behavior in terms of torques.
Solution:
The moment of inertia is greatest when more mass is at a greater distance from the axis of rotation. Therefore, rotating the body about an axis through the hips results in the larger moment of inertia. This is true since the angular acceleration is inversely proportional to the moment of inertia. It follows that a given torque produces greater angular acceleration when the body rotates about an axis through the spine.
Chapter 11 Rotational Dynamics and Static Equilibrium Q.4P
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.5CQ
Give an example of a system in which the net torque is zero but the net force is nonzero.
Solution:
A force applied radially to a wheel produces zero torque, though the net force is not zero.
Chapter 11 Rotational Dynamics and Static Equilibrium Q.5P
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.6CQ
Give an example of a system in which the net force is zero but the net torqueis nonzero.
Solution:
Consider an airplane propeller or a ceiling fan that is just starting to rotate. In both these cases the net force is zero. Here, the center of mass is not accelerating, but the net torque is non-zero.
Chapter 11 Rotational Dynamics and Static Equilibrium Q.6P
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. Tn 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?
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.7CQ
Is the normal force exerted by the ground the same for all four tires on yourcar? Explain.
Solution:
No, because the engine is situated in front in most of the cars. Thus, most of the car’s mass is located in the front of the car, and the center of mass of the car is not located at the center of the
car. It is closer to the front end. This means that more force is exerted on the front tires than on the back tires. Thus, the normal force applied is equal for all four tires.
Chapter 11 Rotational Dynamics and Static Equilibrium Q.7P
Consider the pulley-block systems shown in Conceptual Checkpoint 11-1. (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.
Solution:
a) The mass moves in downward directions. So the tension in the string is less than the weight of the mass.
b) The best option is the mass accelerates downwards.
Chapter 11 Rotational Dynamics and Static Equilibrium Q.8CQ
Give two everyday examples of objects that are not in static equilibrium.
Solution:
(i) A truck accelerating from rest is not in static equilibrium because its center of mass is accelerating.
(ii) An airplane propeller that is just starting up is not in static equilibrium because it has an angular acceleration.
Chapter 11 Rotational Dynamics and Static Equilibrium Q.8P
Consider the pulley-block systems shown in Conceptual Checkpoint 11–1. (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.
Solution:
a) The mass in the left hand system drops with small acceleration than the mass in the right hand system. The tension in the left hand string is greater than the tension in the right hand string.
b) The mass in the left hand mass has smaller acceleration. Option (a) is correct.
Chapter 11 Rotational Dynamics and Static Equilibrium Q.9CQ
Give two everyday examples of objects that are in static equilibrium.
Solution:
Conditions for static equilibrium
(i) The net force acting on the object must be zero.
(ii) The net torque acting on the object must be zero.
The examples that meet these conditions are
(1) A physics text book on the table
(2) A person sitting on the chair
Chapter 11 Rotational Dynamics and Static Equilibrium Q.9P
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 yorrr 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.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.10CQ
Can an object have zero translational acceleration and, at the same time, have nonzero angular acceleration? if your answer is no, explain why not. If your answer is yes, give a specific example.
Solution:
Yes, an object can have zero translational acceleration with nonzero angular acceleration. An example is a stationary exercise bike, here the wheels do not transport the user anywhere and translational acceleration of the wheels/bike is zero, but through the use of chemical energy the wheels can be rotated faster or slower due to a nonzero angular acceleration.
Chapter 11 Rotational Dynamics and Static Equilibrium Q.10P
A torque of 0.97 N m is applied to a bicycle wheel of radius 35 cm and mass 0.75 kg. Treating the wheel as a hoop, find its angular acceleration.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.11CQ
Stars form when a large rotating cloud of gas collapses. What happens to the angular speed of the gas cloud as it collapses?
Solution:
When stars are forming, the rotating cloud of gas collapses.
Thus, the radius of rotation decreases.
Therefore, the moment of inertia also decreases. However, we have conservation of angular momentum (L)=lω = constant
Therefore, as I decreases, the angular velocity of the gas cloud increases.
Chapter 11 Rotational Dynamics and Static Equilibrium Q.11P
When a ceiling fan rotating with an angular speed of 2.75 rad/s is turned off, a frictional torque of 0.120 N · m slows it to a stop in 22.5 s. What is the moment of inertia of the fan?
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.12CQ
What purpose does the tail rotor on a helicopter serve?
Solution:
The tail rotor on a helicopter has a horizontal axis of rotationi, as opposed to the vertical axis of the main rotor, therefore, the tail rotor produces a horizontal thrust
that tends to rotate the helicopter about a vertical axis. As a result, if the angular speed of the main rotor is increased or decreased. The tail rotor can exert an opposing torque that prevents the entire helicopter from rotating in the opposite direction
Chapter 11 Rotational Dynamics and Static Equilibrium Q.12
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?
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.13CQ
Is it possible to change the angular momentum of an object without changing its linear momentum? If your answer is no, explain why not. If your answer is yes, give a specific example.
Solution:
Yes, by keeping its velocity constant. If we change the distances from the axis of rotation r, then we change the angular momentum.
Chapter 11 Rotational Dynamics and Static Equilibrium Q.13P
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, find the torque the person must exert on the ladder to give it an angular acceleration of 0.302 rad/s2.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.14CQ
Suppose a diver springs into the air with no initial angular velocity. Can the diver begin to rotate by folding into atucked position? Explain.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.14P
Awheel on a game show is given aninitiai 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.)
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.15P
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.16P
The L-shaped object described in Problem 15 can be rotated in one of the following three ways: case A, about the x axis; case B, about the y axis; and case C, about the z axis (which passes through the origin perpendicular’ to the plane of the figure). If the same torque r is applied in the of these cases, rank them in increasing order of the resulting angular acceleration. Indicate ties where appropriate.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.17P
A motorcycle 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 motorcycle’s acceleration increase, decrease, or stay the same? Explain.
Solution:
(a) The rear tire rolls forwards such that the force of static friction, opposing motion, points backwards. By Newton’s 3rd law, there is an equal and opposite force, exerted by the ground on the rear tire, pointing forward that counteracts this frictional force.
(b) The front tire rolls forwards such that the force of static friction, opposing motion upon ground contact, points backwards. By Newton’s 3rd law, there is an equal and opposite force, exerted by the ground on the front tire, pointing forward, that counteracts the frictional force.
Chapter 11 Rotational Dynamics and Static Equilibrium Q.18P
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.19P
A fish takes the bait and pulls on the line with a force of 2.2 N. The fishing 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 fishing reel? (b) How much line does the fish pull from the reel in 0.25 s?
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.20P
Repeat the previous problem, only now assume the reel has a friction clutch that exerts a restraining torque of 0.047 N · m.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.21P
Suppose the person in Active Example 11-3 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 person’s position.
II. The magnitude of f 2 must increase as the person moves upward.
III. When the person is higher, the ladder presses down harder on the floor.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.22P
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?
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.23P
To loosen the lid on a jar of jam 8.9 cm in diameter, a torque of 8.5 N · m 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?
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.24P
Consider the system in Active Example 11-1, 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.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.25P
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.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.26P
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.27P
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.28P
A schoolyard teeter-totter with a total length of 5.2 m and a mass of 38 kg is pivoted at its center. A 19-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.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.29P
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.30P
A 0.16-kg meterstick is held perpendicular to avertical 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.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.31P
Repeat Example 11-4, this time with a uniform diving board that weighs 225 N.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.32P
Babe Ruth steps to the plate and casually points to left center field 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, find the magnitude and direction of the force exerted by (a) the hand and (b) the shoulder.
Solution:
The force exerted by the shoulder on the bat is pointed to the upward direction.
Chapter 11 Rotational Dynamics and Static Equilibrium Q.33P
A uniform 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.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.34P
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.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.35P
Repeat Active Example 11-3, this time with a uniform 7.2-kg ladder that is 4.0 m long.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.36P
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.37P
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.38P
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.39P
A uniform crate with a mass of 16.2 kg rests on a floor with a coefficient 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.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.40P
In the previous problem, (a) what is the minimum height where the force F can be applied so that the crate begins to tip before sliding? (b) What is the magnitude of the force in this case?
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.41P
A hand-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?
Solution:
Hence, the orange juice container should be placed from center, on the side of cereal carton, at a distance of.18.5cm
Chapter 11 Rotational Dynamics and Static Equilibrium Q.42P
If the cat in Active Example 11-2 has a mass of 2.8 kg, how close to the right end of the two-by-four can it walle before the board begins to tip?
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.43P
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.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.44P
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.45P
A baseball 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.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.46P
A 2.85-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?
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.47P
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.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.48P
A child exerts a tangential 42.2-N force on the rim of a disk-shaped merry-go-round with a radius of 2.40 m. If the merry-go-round starts at rest and acquires an angular speed of 0.0860 rev/s in 3.50 s, what is its mass?
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.49P
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.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.50P
Referring to the previous problem, find the linear acceleration of the 0.67-kg mass.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.51P
A uniform meterstick of mass M has an empty paint can of mass m hangingfrom 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 stick-can 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.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.52P
Atwood’s Machine An Atwood’s machine consists of two masses, m\ and m-i, connected by a string that passes over a pulley. If the pulley is a disk of radius R and mass M, find the acceleration of the masses.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.53P
Calculate the angular momentum of the Earth about its own axis, due to its daily rotation. Assume that the Earth is a uniform sphere.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.54P
A 0.015-kg record with aradiusof!5 cm rotates with an angular speed of rpm. Find the angular momentum of the record.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.55P
Tn the previous problem, a 1.1-g fly lands on the rim of the record. What is the fly’s angular momentum?
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.56P
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.57P
Repeat the previous problem for the case of jogger 2, whose speed is 2.68 m/s and whose mass is 58.2 kg.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.58P
Suppose jogger 3 in Figure 11-33 has a mass of 62.2 kg and a speed of 5.85 m/s. (a) Is the magnitude of the jogger’s angular momentum greater with respect to point A or point B? Explain. (b) Is the magnitude of the jogger’s 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 jogger’s angular momentum with respect to points A, B, and O.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.59P
A torque of 0.12 N m is applied to an egg beater. (a) Tf the egg beater starts at rest, what is its angular momentum after 0.65 s? (b) Tf the moment of inertia of the egg beater is 2.5 × 10−3 kg · m2, what is its angular speed after 0.65 s?
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.60P
A windmill has an initial angular momentum of 8500 kg · m2 /s. The wind picks up, and 5.86 s later the windmill’s angular momentum is 9700 kg · m2 /s. What was the torque acting on the windmill, assuming it was constant during this time?
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.61P
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 the 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.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.62P
A student rotates on a frictionless piano stool with his arms outstretched, a heavy weight in the hand. Suddenly he lets go of the weights, and they fall to the floor. As a result, does the student’s 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 student’s moment of inertia is decreased by dropping the weights.
III. Dropping the weights exerts no torque on the student, but the floor exerts a torque on the weights when they land.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.63P
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.64P
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.65P
As an ice skater begins a spin, his angular speed is 3.17 rad/s. After pulling in Ms arms, his angular speed increases to 5.46 rad/s. Find the ratio of the skater’sfinalmoment of inertia to his initial moment of inertia.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.66P
Calculate both the initial and the final kinetic energies of the system described in Active Example 11-5.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.67P
A diver tucks her body in midnight, decreasing her moment of inertia by a factor of two. By what factor does her angular speed change?
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.68P
In the previous problem, (a) does the diver’s kinetic energy increase, decrease, or stay the same? (b) Calculate the ratio of the final kinetic energy to the initial kinetic energy for the diver.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.69P
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-go-round 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-round’s rim. What is the final angidar speed of the merry-go-round?
Solution:
Mass of the merry-go-round (M) = 155 kg
Radius of the merry-go-round (R) = 2.63 m
Initial angular speed of the merry-go-round
Chapter 11 Rotational Dynamics and Static Equilibrium Q.70P
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 final kinetic energies for this system.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.71P
A student sits at rest on a piano stool tha t can rotate without friction. The moment of inertia of the student-stool system is 4.1 kg · m2. A second 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?
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.72P
Referring to the previous problem, (a) does the kinetic energy of the mass-student-stool system increase, decrease, or stay the same as the mass is caught? (b) Calculate the initial and final kinetic energies of the system.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.73P
A turntable with a moment of inertia of 5.4 × 10 −3 kg-m2 rotates freely with an angular speed of rpm. 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 rthees the center.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.74P
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 the 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 5.43 kg -m2, a value that remains constant. (a) As the student pulls his arms inward, his angular speed increases to 3.54 rcv/s. How farare the masses from the axis of rotation at this time, considering the masses to be points? (b) Calculate the initial and final kinetic energies of the system.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.75P
A child of mass m tands 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-round’s surface. (a) What is the child’s speed with respect to the ground? Check your results in the limits (b) I → 0 and (c) I →∞
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.76P
Two spheres of equal mass and radius are rolling across the floor 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.
Solution:
The mass, radius and speed of the both balls are same. The kinetic energy of the hallow sphere is more than the kinetic energy of the sold sphere. According to work energy theorem, the work done would be greater for ball that posses more kinetic energy. So Work required to stop sphere1 is less than the work done to stop sphere2. The moment of inertia of the of the hallow ball is greater and hence the greater rotational kinetic energy.
Chapter 11 Rotational Dynamics and Static Equilibrium Q.77P
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.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.78P
Turning a doorknob through. 0.25 of a revolution requires 0.14 J of work. What is the torque required to turn the doorknob?
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.79P
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°?
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.80P
To prepare homemade ice cream, a crank must be turned with a torque of 3.95 N· m. How much work is required for the complete turn of the crank?
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.81P
A popular make of dental drill can operate at a speed of 42, 500 rpm while producing a torque of 3.68 oz · in. What is the power output of this drill? Give your answer in watts.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.82P
The L-shaped object in Figure 11-24 consists 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).
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.83P
The rectangular object in Figure 11-25 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)?
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.84P
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.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.85GP
Solution:
a) From the given figure, if the paper is pulled horizontally to the right, then the disk rotates in a counterclockwise direction because the force is exerted to the right on the bottom of the disk.
b) The center of the disk moves toward the right because the paper is pulled to the right. This is the direction of the net force exerted on the disk.
Chapter 11 Rotational Dynamics and Static Equilibrium Q.86GP
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.87GP
Solution:
(a) When the block is allowed to fall, the block moves towards the left.
(b) The wheel has larger moment of inertia and smaller angular acceleration. Therefore the string unwinds from the wheel more slowly than the disk, and so the block moves to the left.
Option (II) is correct.
Chapter 11 Rotational Dynamics and Static Equilibrium Q.88GP
A beetle 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.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.89GP
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.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.90GP
Suppose the Earth were to magically expand, doubling its radius while keeping its mass the same. Would the length of the day increase, decrease, or stay the same? Explain.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.91GP
After getting a drink of water, a hamster jumps onto an exercise wheel for a run. A few 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.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.92GP
A 47.0-kg unif orm 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?
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.93GP
A puck attached to a string moves in a circular path on a frictionless surface, as shown in Figure 11–34. Initially, the speed of the puck is v and the radius of the circle is r. If the string passes through ahole in the surface, and is pulled downward until the radius of the eircularpath is r /2, (a) does the speed of the puck increase, decrease, or stay the same? (b) Calculate the final speed of the puck.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.94GP
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.95GP
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.96GP
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 ft · lb, 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 in ft · lb.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.97GP
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.98GP
You hold a uniform, 28-g pen horizontal with your thumb pushing down on one end and your index finger 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.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.99GP
In Active Example 11-3, 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.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.100GP
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.101GP
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.
Solution:
Therefore, the force exerted by the second pillar is.1.49KN
Chapter 11 Rotational Dynamics and Static Equilibrium Q.102GP
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.103GP
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.104GP
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.105GP
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.106GP
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.107GP
Suppose a fourth book, the same as the other three, is added to the stack of books shown in Figure 11-32. (a) What is the maximum overhang distance, d, in this case? (b) If the mass of the book is increased by the same amount, does your answer to part (a) increase, decrease, or stay the same? Explain.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.108GP
Suppose partial melting of the polar ice caps increases the moment of inertia of the Earth from 0.331 M E R E 2 to 0.332 M E R E 2. (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.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.109GP
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.110GP
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.111GP
In Problem, assume that the rod has a mass of M and that its bottom end simply rests on the floor, held in place by static friction. If the coefficient of static friction is μs, find the maximum force F that can be applied to the rod at its midpoint before it slips.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.112GP
In the previous problem, suppose the rod has a mass of 2.3 kg and the coefficient 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 F is 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.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.113GP
A cylinder of mass m and radius r has a string wrapped around its circumference. The upper end of the string is held fixed, and the cylinder is allowed to fall. Show that its linear acceleration is (2/3)g.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.114GP
Repeat the previous problem, replacing the cylinder with a solid sphere. Show that its linear acceleration is (5/7)g.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.115GP
A mass M is attached to a rope that passes over a disk-shaped 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 m → 0 and m →∞.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.116GP
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.117GP
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.118PP
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.119PP
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.120PP
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.121IP
Suppose 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 T 2 to increase, decrease, or stay the same? Explain, (b) Calculate the value of T 2 for this case.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.122IP
Suppose 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 T 2 to increase decrease, or stay the same? Explain. (b) Calculate the value of T 2 for this case.
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.123IP
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 0.425 rad/s?
Solution:
Chapter 11 Rotational Dynamics and Static Equilibrium Q.124IP
Suppose everything in the system is as described in Active Example 11-5 except that the child approaches the merry-go-round in a direction that is not tangential Find the angle θ between the direction of motion and the outward radial direction (as in Example 11-8) that is required if the final angular speed of the system is to be 0.272 rad/s.
Solution:
