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1.A figure skater goes into a spin with arms fully extended. Which of the following describes the changes in therotational kinetic energy and angular momentum of the skater as the skater’s arms are brought toward the body?(A)Rotational Kinetic EnergyAngular MomentumRemains the sameIncreases(B)Rotational Kinetic EnergyAngular MomentumRemains the sameRemains the same(C)Rotational Kinetic EnergyAngular MomentumIncreasesRemains the same(D)Rotational Kinetic EnergyAngular MomentumDecreasesIncreases(E)Rotational Kinetic EnergyAngular MomentumDecreasesRemains the same2.A hoop of massmand radiusrrolls with constant speed on a horizontal surface without slipping. What is thehoop’s translational kinetic energy divided by its rotational kinetic energy?(A)4(B)2(C)1(D)1/2(E)1/4AP PHYSICS C: MECHANICSScoring Guide2AP Physics C: MechanicsPage 1 of 37
A skater on ice begins to spin in the camel position, as shown in the figure to the left below. As she continues to spin, shestraightens up, pulls her arms to her chest, and crosses her legs to spin in the corkscrew position, as shown in the figure tothe right below. Air resistance and friction are negligible as the skater spins.3.Which of the following statements about the skater’s motion is true?(A)Her angular momentum increases when she moves to the corkscrew position because her angular speedincreases.(B)Her angular momentum increases when she moves to the corkscrew position because her rotationalinertia increases.(C)Her angular momentum remains the same when she moves to the corkscrew position because there isnegligible frictional torque.(D)Her angular momentum decreases when she moves to the corkscrew position because her angular speedincreases.(E)Her angular momentum decreases when she moves to the corkscrew position because her rotationalinertia decreases.Answer CCorrect. Even though the skater’s angular speed increases, her rotational inertia decreases so her angularmomentum stays the same because there are no external torques associated with her change in positionA wheel with rotational inertiaIis mounted on a fixed, frictionless axle. The angular speed ωof the wheel is increasedfrom zero to ω?in a time intervalT.4.What is the average power input to the wheel during this time interval?Scoring Guide2Page 2 of 37AP Physics C: Mechanics
(A)(B)(C)(D)(E)5.A wheel with rotational inertia 0.04 kg•m2and radius 0.02 m is turning at the rate of 10 revolutions per secondwhen a frictional torque is applied to stop it. How much work is done by the torque in stopping the wheel?(A)–0.0008 J(B)–0.4πJ(C)–2 J(D)–2π2J(E)–8π2J6.The disk on the bottom has a rotational inertia ofand is spinning clockwise on a horizontal surface havingnegligible friction at an angular speed of. The top disk is then dropped directly on top of the bottom disk. Thetop disk has a rotational inertia ofand was initially spinning counterclockwise at an angular speed of. If theinitial total kinetic energy of the two-disk system was, what will the kinetic energy of the system be once theobjects reach a common angular speed?Scoring Guide2AP Physics C: MechanicsPage 3 of 37
(A)(B)(C)(D)(E)Answer BCorrect. Using conservation of angular momentum yields. The kinetic energy before thecollision is. The kinetic energy after thecollision is7.A sphere starts from rest at the top of a ramp, as shown above. It rolls without slipping down the ramp. Thepotential energy of the sphere-Earth system is zero at the bottom of the ramp. Which of the following is true of thesphere when it reaches the bottom of the ramp?(A)Its rotational kinetic energy equals the initial potential energy of the sphere-Earth system.(B)Its translational kinetic energy equals the initial potential energy of the sphere-Earth system.(C)Its translational kinetic energy and rotational kinetic energy are equal.(D)The sum of its translational kinetic energy and rotational kinetic energy equals the initial potentialenergy of the sphere-Earth system.(E)The sum of its translational kinetic energy and rotational kinetic energy equals the energy lost becauseof friction.Answer DCorrect. The mechanical energy of the sphere-Earth system will be conserved as the sphere rolls downthe incline; thus, the kinetic energy of the sphere at the bottom of the ramp must be equal to the potentialScoring Guide2Page 4 of 37AP Physics C: Mechanics
energy of the sphere-Earth system when the sphere is at the top of the ramp, and the kinetic energy of thesphere is the sum of its translational kinetic energy and rotational kinetic energy.8.In an experiment, students roll several hoops down the same incline plane. Each hoop has the same mass but adifferent radius. Each hoop rolls down the incline without slipping. Which of the following graphs best shows thelinear speedof the hoops at the bottom of the incline as a function of the radiusof the hoop?Scoring Guide2AP Physics C: MechanicsPage 5 of 37
(A)(B)(C)(D)(E)Answer ACorrect. Using conservation of energy to calculate the linear speed,Scoring Guide2Page 6 of 37AP Physics C: Mechanics
. So the radius of the hoop is not a factor, and the graph is a horizontal line.9.The ring and the disk shown above have identical masses, radii, and velocities, and are not attached to each other. Ifthe ring and the disk each roll without slipping up an inclined plane, how will the distances that they move up theplane before coming to rest compare?(A)The ring will move farther than will the disk.(B)The disk will move farther than will the ring.(C)The ring and the disk will move equal distances.(D)The relative distances depend on the angle of elevation of the plane.(E)The relative distances depend on the length of the plane.A sphere of massM,radiusr, and rotational inertiaIis released from rest at the top of an inclined plane of heighthasshown above.10.If the plane has friction so that the sphere rolls without slipping, what is the speedvcmof the center of mass at thebottom of the incline?Scoring Guide2AP Physics C: MechanicsPage 7 of 37
(A)(B)(C)(D)(E)11.Two uniform spheresandroll without slipping along a horizontal surface, as shown above. Spherehastimes the radius,times the mass, andthe translational speed of sphere. The rotational inertia of a sphere is. Ifis the kinetic energy of sphere, andin the kinetic energy of, what is the ratio?(A)(B)(C)(D)(E)Answer ECorrect. Using the equation for kinetic energy that includes both linear and rotational kinetic energy:Scoring Guide2Page 8 of 37AP Physics C: Mechanics
12.A ball, rolling without slipping on a horizontal surface, encounters a frictionless, downwardsloping ramp, as shownabove. Which of the following correctly describes the motion of the ball on the ramp?(A)Constant translational speed with no angular speed(B)Increasing translational speed with no angular speed(C)Increasing translational speed with constant nonzero angular speed(D)Increasing translational speed with decreasing angular speed(E)Increasing translational speed with increasing angular speed13.Four round objects of equal mass and radius roll without slipping along a horizontal surface that then bends upwardand backward into an arc of half a circle. The objects all have the same linear speed initially. The objects are ahollow cylinder, a solid cylinder, a solid sphere, and a hollow sphere. The objects to go up the arc and exit the arcgoing in the opposite direction they entered without falling off the arc. Now, several trials are run for each object.For each trial, the initial speed of the object is reduced until the object does not make it through the full arc. Thespeed needed for each object to just make it through the arc is recorded. Which of the following correctly lists theobjects in order from fastest to slowest speed needed to make it through the arc?(A)Hollow cylinder, hollow sphere, solid cylinder, solid sphere(B)Hollow cylinder, solid sphere, solid cylinder, hollow sphere(C)Solid sphere, solid cylinder, hollow sphere, hollow cylinder(D)Solid sphere, hollow sphere, solid cylinder, hollow cylinder(E)Hollow sphere, solid cylinder, hollow cylinder, solid sphereAnswer CCorrect. The objects with the lowest rotational inertia take the least effort to slow down and thus neededScoring Guide2AP Physics C: MechanicsPage 9 of 37
the fastest initial speed to make it through the loop. The solid sphere would have the lowest rotationalinertia, followed by the solid cylinder, hollow sphere, and hollow cylinder.14.A solid sphere, a hollow sphere, and a ring all have the same mass and radius. Each is released from rest at the topof the same incline and rolls without slipping down the incline. Which of the following indicates the object that willreach the bottom of the incline first and provides correct reasoning?(A)The ring, because it has the greatest rotational inertia.(B)The ring, because it has the smallest rotational inertia.(C)The hollow sphere, because it has the greatest rotational inertia.(D)The solid sphere, because it has the greatest rotational inertia.(E)The solid sphere, because it has the smallest rotational inertia.Answer ECorrect. The solid sphere has the smallest rotational inertiaabout its center, since its mass isdistributed closer to its center compared to the other objects. Using the parallel axis theoremand the fact thatis the same for all three objects, the solid sphere therefore alsohas the smallest rotational inertia about its point of contact with the incline. The three objects experienceequal torquesabout the point of contact. By the relationship, equal torquesmean that the angular acceleration is greatest when the rotational inertia is the smallest. The solid spherehas the greatest angular acceleration, and therefore the greatest linear acceleration and reaches the bottomof the incline first. Alternatively, the object with the smallest rotational inertia has the greatest fraction ofits kinetic energy in the form of translational (versus rotational) kinetic energy, and so has the greatestspeed of the three objects at any position on the incline and reaches the bottom first.Scoring Guide2Page 10 of 37AP Physics C: Mechanics
15.A projectile of massand an initial velocity ofcollides with the end of a blade attached to a turbine.The rotational inertia of the turbine is. Assume the loss of energy of the projectile in the collision iscompletely transferred to the blades, causing them to spin. If the final rebound velocity of the projectile after hittinga turbine blade is, which of the following is most nearly the rotational velocity of the turbine after thecollision?(A)(B)(C)(D)(E)Answer BCorrect. Using conservation of kinetic energy,.Scoring Guide2AP Physics C: MechanicsPage 11 of 37
16.A solid cylinder and hollow cylinder have the same massand radius. Both cylinders have a string wrappedaround a shallow groove along their circumference so that the string can be pulled without interfering with thecylinder’s motion as they roll across the table. The strings attached to each cylinder are pulled with the same force, as shown in the side view figure. The rotational inertia of the solid and hollow cylinder about their centers isgiven byand, respectively. Each cylinder rolls and accelerates without slipping whilethe string is pulled. What is the ratio of the translational acceleration of the center of mass of the solid cylinder tothat of the hollow cylinder?(A)(B)(C)(D)Answer BCorrect. For each cylinder, there are two horizontal forces exerted on it: the string forcetoward theright at the top of the cylinder, and a friction forcewhere the bottom of the cylinder contacts thesurface. By choosing the rotation axis to be the (instantaneously) stationary contact point, only the torquedue toneeds to be considered, sincedoes not result in a torque about this point. ApplyingNewton’s second law in rotational form, and using the parallel axis theorem for the rotational inertiaabout the contact point, the angular acceleration of a cylinder is:. The translational and rotational accelerations are related via, so forequal radii, the ratio of translational accelerations is the same as the ratio of angular accelerations. Sincethe angular acceleration is inversely proportional to the quantity, the ratio of thetranslational accelerations is,which is.Scoring Guide2Page 12 of 37AP Physics C: Mechanics
17.Wheelsandhave the same radius and roll without slipping on a horizontal surface. The translational speed androtational kinetic energy of both wheels is the same. However, thetotal kinetic energy of Wheelis greater thanthetotal kinetic energy of Wheel. Which of the following correctly describes Wheelin comparison to Wheel?(A)Wheelis more massive and on average, its mass is distributed closer to the wheel’s center.(B)Wheelis more massive and on average, its mass is distributed farther from the wheel’s center.(C)Wheelis less massive and on average, its mass is distributed closer to the wheel’s center.(D)Wheelis less massive and on average, its mass is distributed farther from the wheel’s center.Answer ACorrect. To have more total kinetic energyand the same rotational kineticenergy, Wheelmust have more translational kinetic energy. To have more translational kinetic energywith the same center-of-mass (translational) speed, Wheelmust have more mass.The two wheels have the same angular speedbecause both the center-of-mass speed and theradius are the same for the two wheels. Because the angular speed and the rotational kinetic energyare the same for the two wheels, the rotational inertias of the wheels must be equal aswell. The only way for Wheelto have more mass but equal rotational inertiais if itsmass is distributed closer to the center than Wheel. For example, Wheelcould be a solid diskwhereas Wheelcould be a hoop or ring.Scoring Guide2AP Physics C: MechanicsPage 13 of 37
18.A hollow cylinder is initially sliding without rotating up a smooth section of a ramp that makes an anglewith thehorizontal, as shown in the figure. The cylinder then reaches a rough section of the ramp where the coefficient ofkinetic friction between the cylinder and the ramp is. The cylinder, which has mass, radius, androtational inertia about its center of, starts to rotate while slipping on the rough section of the ramp.When the cylinder is rolling upward while slipping on the rough section, what is the magnitudeof the cylinder’stranslational acceleration in terms of the magnitudeof its angular acceleration and the given quantities?(A)(B)(C)(D)Answer DCorrect. For translational motion, the two forces contributing to the net force along the ramp are thegravitational force and the frictional force. The translational acceleration is directed down the ramp andhas magnitude, which simplifies to. For rotational motion, the only force resulting in a torque about the cylinder’scenter is the frictional force. The rotational acceleration is clockwise and has magnitude, which simplifies to. Using these expressions, theratiois then. Solving this equation forresults in this answer.Scoring Guide2Page 14 of 37AP Physics C: Mechanics
19.A sphere is held at rest at the top of a ramp, as shown. The sphere is released and rolls without slipping down theramp. The gravitational potential energy of the sphere-Earth system is zero at the bottom of the ramp. Which of thefollowing is true of the sphere when it reaches the bottom of the ramp?(A)The sphere’s rotational kinetic energy equals the initial potential energy of the sphere-Earth system.(B)The sphere’s translational kinetic energy equals the initial potential energy of the sphere-Earth system.(C)The sphere’s translational kinetic energy and rotational kinetic energy are equal.(D)The sum of the sphere’s translational kinetic energy and rotational kinetic energy equals the initialpotential energy of the sphere-Earth system.Answer DCorrect. The mechanical energy of the sphere-Earth system will be conserved as the sphere rolls downthe incline; thus, the kinetic energy of the sphere at the bottom of the ramp must be equal to the potentialenergy of the sphere-Earth system when the sphere is at the top of the ramp, and the kinetic energy of thesphere is the sum of its translational kinetic energy and rotational kinetic energy.Scoring Guide2AP Physics C: MechanicsPage 15 of 37
20.Two uniform, solid cylinders,and, each roll without slipping along a horizontal surface with the same initialtranslational velocityto the right, as shown in the figure. They each then roll without slipping up a ramp and riseto a maximum heightfor Cylinderandfor Cylinder. Cylinderhas four times the mass and twice theradius of Cylinder, and the rotational inertia about the center of a uniform, solid cylinder of massand radiusis. What is the ratioof the heights reached by the cylinders?(A)(B)(C)(D)Answer ACorrect. Using conservation of energy:, or. Making the substitutiondue to the cylinders notslipping yields. Canceling the terms(common factor) andresults in an equation that is independent of cylinder mass and radius, therefore both disks reach the sameheight.21.A sphere is held at the top of a straight ramp. When released, the sphere rolls down the ramp without slipping.Which of the following claims about the force of friction, if any, is correct?Scoring Guide2Page 16 of 37AP Physics C: Mechanics
(A)There is a nonzero force of static friction exerted on the sphere that increases in magnitude as therotation of the sphere increases.(B)There is a nonzero force of static friction exerted on the sphere that does not dissipate any mechanicalenergy.(C)There is a nonzero force of kinetic friction exerted on the sphere because the sphere is moving.(D)There is a nonzero force of kinetic friction exerted on the sphere that dissipates mechanical energy.Answer BCorrect. Rolling is caused by a force of friction exerted on the sphere by the ramp. Because the sphere isnot slipping across the surface of the ramp, the friction is static, and will not dissipate any mechanicalenergy.22.At one time, bicycles were commonly made with unequal wheel sizes. The bicycle shown in the figure has wheelswith radiiandand is moving along a road with a translational speed. The rotational inertias of the largeand small wheels areand, respectively, about the axes passing through their respective centers. If the smallwheel has a rotational kinetic energy, what is the rotational kinetic energy of the large wheel?Scoring Guide2AP Physics C: MechanicsPage 17 of 37
(A)(B)(C)(D)Answer CCorrect. Rotational kinetic energy is, and each wheel’s angular speed is related to thecenter-of-mass speed by. Substituting forin the equation forgives,which is proportional to.Therefore, the large wheel has a rotational kinetic energy that istimes that of the smallwheel.23.A solid sphere is initially moving at a constant speedon a horizontal surface, as shown in the figure. At first, thesphere is sliding without rotating and moves with negligible friction on a smooth section of the surface. The spherethen reaches a rough section of the surface where the coefficient of kinetic friction is. Sometime later, the sphereis rolling without slipping on the rough section. The sphere has a mass, a radius, and a rotational inertiaabout its center. What is the final angular speed of the sphere?Scoring Guide2Page 18 of 37AP Physics C: Mechanics
(A)Zero(B)(C)(D)Answer CCorrect. When the sphere enters the rough section, the kinetic frictional force causes the translationalspeed to decrease from its initial value ofand also results in a torque about the center of mass, whichcauses the angular speed to increase from its initial value of. The final speed occurs when the sphere isrolling without slipping and the kinetic frictional force becomes zero, at which time the translational andangular speeds match according to the equation. Taking the positive direction to be toward theright for translational motion, the translational acceleration of the sphere on the rough section is. Taking the timeas the instant the sphere enters the roughsection, the sphere’s translational speed as a function of time is. Taking thepositive direction to be clockwise for rotational motion, the angular acceleration of the sphere about itscenter is. The sphere’s angular speed is then. Applyto find the time at which rolling without slipping occurs:;;. The final angularspeed is then.24.A solid cylinder of uniform density has mass, radius, and rotational inertia. The cylinder isrolling without slipping along a horizontal surface as the cylinder’s center of mass moves with a translational speed. What is the total kinetic energy of the cylinder?(A)(B)(C)(D)Scoring Guide2AP Physics C: MechanicsPage 19 of 37
Answer CCorrect. The total kinetic energy is the sum of the translational and rotational kinetic energies. Thetranslational kinetic energy is. The rotational kinetic energy is, which, since, is equal to. Adding the two forms of kinetic energy togethergives this answer.25.A solid cylinder is released from rest and rolls without slipping down an inclined plane. Which of the followingquantities remains constant as the cylinder rolls down the incline?(A)The angular speed of the cylinder(B)The angular acceleration of the cylinder(C)The translational speed of a point on the outside surface of the cylinder(D)The translational acceleration of a point on the outside surface of the cylinderAnswer BCorrect. All forces exerted on the cylinder (gravitational, normal, and frictional) are constant, so thetranslational accelerationmust be constant. Since the cylinder rolls without slipping, theangular acceleration isand is constant as well.26.A hollow cylinder and uniform solid cylinder of the same mass and radius roll along a horizontal surface. Thehollow cylinder and solid cylinder both roll up a ramp and momentarily come to rest at vertical heightsand,respectively, above the horizontal surface. Both cylinders roll without slipping at all times. If the cylinders had thesame initial linear speeds at the bottom of the ramp, which of the following correctly comparestoandprovides reasoning to support the comparison?(A). The hollow cylinder had more mechanical energy at the bottom of the ramp than the solidcylinder.(B). The force of gravity exerted on the hollow cylinder is less than the force of gravity exerted onthe solid cylinder.(C). The solid cylinder had more mechanical energy at the bottom of the ramp than the hollowcylinder.(D). The force of gravity exerted on the solid cylinder is less than the force of gravity exerted onthe hollow cylinder.Scoring Guide2Page 20 of 37AP Physics C: Mechanics
Answer ACorrect. Due to the conservation of energy, the maximum height reached by both cylinders will dependon the total amount mechanical energy of the cylinder at the bottom of the ramp. Both the hollowcylinder and solid cylinder have the same translational kinetic energy because they travel at the samelinear speed. The mass of the hollow cylinder is on average distributed farther from the axis of rotationthan the solid cylinder, so the rotational inertia of the hollow cylinder is greater than that of the solidcylinder. Both cylinders roll without slipping and have the same radius, so the angular speed of both isthe same. Therefore, the hollow cylinder has greater rotational kinetic energy than the solid cylinder.This means the total mechanical energy of the hollow cylinder is greater than the total mechanical energyof the solid cylinder.27.A disk is spinning about its center at a constant, clockwise angular velocity. After an external torque is exerted onthe disk, the disk spins at a constant,counterclockwise angular velocity. Which of the following claims is trueabout the rotational kinetic energy of the disk?(A)It is greater after the torque is applied.(B)It is greater before the torque is applied.(C)It is the same before and after the torque is applied.(D)It cannot be determined when the rotational kinetic energy is greater without additional information.Answer CCorrect. The rotational kinetic energy depends on the disk’s rotational inertia and angular speed via theequation. Even though the angular velocity changes direction, the angular speed is thesame before and after the torque is applied, so the rotational kinetic energy is the same.28.A cylinder rolling across a horizontal surface travels up a ramp without slipping. The cylinder momentarily comesto rest at a heightabove the horizontal surface before rolling back down. If there was negligible friction betweenthe ramp and the cylinder, how would the maximum height of the cylinder compare to? What reasoning bestsupports the stated comparison?(A)It is the same as. The gravitational force on the cylinder does the same amount of work.(B)It is the same as. The initial total kinetic energy is still converted to gravitational potential energy.(C)It is less than. The initial translational kinetic energy does not get converted to gravitational potentialenergy.(D)It is less than. The initial rotational kinetic energy does not get converted to gravitational potentialenergy.Scoring Guide2AP Physics C: MechanicsPage 21 of 37
Answer DCorrect. The cylinder keeps rotating, even when instantaneously at rest, and only the translational kineticenergy gets converted to potential energy. Since less kinetic energy overall is converted, the finalpotential energy, and therefore height, is less.29.A uniform solid disk can rotate about a fixed pivot point located on its edge, as shown in the figure. The disk hasmass, radius, and a rotational inertia about itscenter of. The disk’s center travels in acircle, also of radius, as the disk rotates about the pivot point with angular speedin the plane of the figure.How does the rotational kinetic energy of the disk compare to the translational kinetic energy of the disk?(A)The rotational kinetic energy is zero.(B)The rotational kinetic energy is not zero and is less than the translational kinetic energy.(C)The rotational kinetic energy equals the translational kinetic energy.(D)The rotational kinetic energy is greater than the translational kinetic energy.Answer DCorrect. The parallel axis theorem can be used to find the rotational inertia of the disk about its edge,which is then used to find the rotational kinetic energy.Scoring Guide2Page 22 of 37AP Physics C: Mechanics
.The translational kinetic energy due to the center-of-mass motion at speedis.Since the center-of-mass speed is, the translational kinetic energy is, thus the rotational kinetic energy is greater than thetranslational kinetic energy.30.A solid, uniform disk is spinning about an axis through its center while its center of mass remains at rest. Which ofthe following correctly describes the disk’s kinetic energy, and provides supporting reasoning?(A)is zero. The velocity of the disk’s center of mass is zero.(B)is zero. For every point on the disk moving with a translational velocity, there is another point onthe disk moving with the opposite velocity.(C)is greater than zero. The disk’s mass and rotational inertia are both greater than zero.(D)is greater than zero. Except for the disk’s center, all parts of the disk are moving.Answer DCorrect. A rigid system can have rotational kinetic energy while its center of mass is at rest due to theindividual points within the rigid system having translational speed and, therefore, kinetic energy.31.Three identical uniform disks,,, and, can each slide along a horizontal surface with negligible friction. Diskis rotating with angular speedwhile its center of mass remains in place. Diskis moving with speedwithout rotating. Diskis rotating with the same angular speedas Diskwhile its center of mass is movingwith the same speedas Disk. Which disk has the greatest total kinetic energy?(A)Diskhas the most total kinetic energy.(B)Diskhas the most total kinetic energy.(C)Diskhas the most total kinetic energy.(D)It cannot be determined which disk has the most total kinetic energy without more information on,, and the radius of the disks.Scoring Guide2AP Physics C: MechanicsPage 23 of 37
Answer CCorrect. Since the disks are identical, they all have the same rotational inertia (about their centers) andthe same mass. Disksandhave the same angular speed and therefore the same rotational kineticenergy. However, Diskhas additional translational kinetic energy and therefore more total kineticenergy than Disk. Similarly, disksandhave the same linear speed and therefore the sametranslational kinetic energy. However, Diskhas additional rotational kinetic energy and therefore moretotal kinetic energy than Disk. So, Diskhas more total kinetic energy than either Diskor Disk.32.Two identical spheresandare connected to rigid rods of negligible mass, as shown in the figure. The rods-spheres system rotates about pointwith a constant angular velocity. With respect to point, how do the spheres’rotational inertiasandcompare, and how do the spheres’ rotational kinetic energiesandcompare?(A)and(B)and(C)and(D)andAnswer ACorrect. Spherehas a greater rotational inertiathan Spherebecause it has the same massat a greater distance from the axis of rotation. The angular speed of each sphere is the same, so, using, since Spherehas a greater rotational inertia and is traveling with the same angularScoring Guide2Page 24 of 37AP Physics C: Mechanics
speed, it has greater rotational inertia as well.33.Three hoops are each rotating about their respective centers, which are fixed in place, as shown in the figure. Themass, radius, and angular speed of each hoop are as follows.•Hoophas mass, radius, and angular speed.•Hoophas mass, radius, and angular speed.•Hoophas mass, radius, and angular speed.How do the total kinetic energies,, andof the three hoops compare to one another?(A)(B)(C)(D)Answer ACorrect. Since every point on a hoop has the same translational speed, the total kinetic energy is. For the purposes of ranking, note that this is proportional to the quantity, and can be ranked according to this quantity, dropping the factor ofin the calculations. Thequantityisfor Hoop,for Hoop, andfor Hoop, resulting in the ranking shown in this answer.Scoring Guide2AP Physics C: MechanicsPage 25 of 37
34.ObjectNameRadiusRotationalInertiaTranslationalSpeedAngularSpeedHoopDiskDiskA circular hoop and two solid disks are placed on a horizontal surface where they can move with negligible friction.The hoop and disks are set into motion such that each moves forward and rotates about its center of mass, as shownin the Top View figure. The hoop and disks have the same mass and angular speeds. The radii, rotational inertia,and translational speeds of the hoop and disks are listed in the table. Which of the following correctly compares thetotal kinetic energies,, andof the hoop, Disk, and Disk, respectively?(A)(B)(C)(D), but it cannot be determined howcompares without more information aboutand.Answer DCorrect. The hoop and Diskhave the same translational kinetic energy, while thehoop has more rotational kinetic energy, because the hoop has a greater rotationalinertia. Therefore, the hoop has more total kinetic energy than Disk. Diskhas greater translationalkinetic energy than both the hoop and Diskbecause of its greater translational speed, but it has lessScoring Guide2Page 26 of 37AP Physics C: Mechanics
rotational kinetic energy because of its smaller rotational inertia. Without more information aboutand, the translational and rotational kinetic energies cannot be directly added together to compare the totalkinetic energy of Diskwith the hoop and Disk. For instance, ifis very small, Diskmay have thegreatest total kinetic energy, but ifis very large, the hoop would have the greatest total kinetic energy.35.A torsion spring is fixed to the end of a rod of rotational inertia. The torsion spring is fixed to a horizontal tablewith negligible friction, as shown in the Top View. When the rod is displaced an anglefrom equilibrium, thetorsion spring exerts a restoring torque of magnitude, whereis a positive constant with appropriateunits. After being displaced by an angle, the rod is released and rotates through its equilibrium position withangular speed. Which of the following is a correct expression for?(A)Zero(B)(C)(D)Answer DCorrect. This answer correctly integrates the torque applied to the rod over the angular displacement tofind the work done in rotating the section, and then correctly equates that work to the change in rotationalkinetic energy of the rod.Scoring Guide2AP Physics C: MechanicsPage 27 of 37
⯑⯑36.A small solid sphere is released from rest at the top of an inclined plane as shown. The sphere rolls down withoutslipping and reaches the bottom of the plane. How would the angular momentum of the sphere at the bottom of theplane change if the coefficient of static friction between the sphere and the plane were increased?(A)It would decrease because the force of friction would be greater.(B)It would decrease because the net torque would be larger.(C)It would increase because the angular acceleration would increase.(D)It would increase because the linear acceleration would increase.(E)It would remain the same because the net torque would remain the same.Answer ECorrect. If the coefficient of friction is large enough to provide for rolling without slipping, increasing itfurther would have no effect on the motion of the sphere. The force of friction would remain the same,and so would the angular acceleration and, therefore, the angular momentum. Note that it will be staticfriction, but it will be below its maximum value of.Scoring Guide2Page 28 of 37AP Physics C: Mechanics
37.A solid sphere is placed on a frictionless floor in a very long corridor and is given a quick push so that it begins toslide, without rotating, along the corridor. How would the angular speed of the sphere be changing if the floor werenot frictionless?(A)It would be increasing until the slipping between the sphere and the floor stops.(B)It would be increasing until the translational motion stops.(C)It would be increasing until the linear and the angular speeds become equal.(D)It would remain zero because the net torque is still zero.(E)It would remain zero because the angular momentum is conserved.Answer ACorrect. If there was friction with the floor, this would apply a torque to the sphere and increase itsangular speed until the sphere stops slipping. At that point, there would no longer be slipping on thefloor, so the angular speed would then stay constant.38.A spinning disk is brought to rest by a constant torque. The disk undergoes ten complete rotations before coming torest. How does the magnitude of the change in the kinetic energy of the disk during the first five rotations compareto the magnitude of the change in the kinetic energy of the disk during the last five rotations?(A)It is greater during the first five rotations.(B)It is greater during the last five rotations.(C)It is the same during the first five and last five rotations.(D)The kinetic energy transfer cannot be compared without knowing the rotational inertia and initialangular speed of the disk.Answer CCorrect. The applied torque is constant, and the work done is simply torque times the angulardisplacement. As long as the angular displacements are the same, the work will be the same.Scoring Guide2AP Physics C: MechanicsPage 29 of 37
39.A time-varying counterclockwise torque is applied to the outer edge of a platform that is initially rotating clockwiseabout its center. The torque continuously increases in magnitude from an initial value of zero. The platform comesto a stop afterrotations, at which time the torque is removed. How does the decrease in the rotational kineticenergy of the platform during the first rotation compare to the decrease in rotational kinetic energy during the finalrotation?(A)The rotational kinetic energy decreases more during the first rotation.(B)The rotational kinetic energy decreases more during the last rotation.(C)The rotational kinetic energy decreases by the same amount during the first and the last rotation.(D)The decrease in rotational kinetic energy cannot be compared without knowing whether the torqueincreases at a greater rate during the first or the last rotationAnswer BCorrect. The change in rotational kinetic energy is the same as the work done by the torque, which is. Since the angular interval is the same (rotation) in both cases, the higher torqueduring the last rotation means that the magnitude of the work done, and therefore the decrease in kineticenergy, is grater for the last rotation.40.A constant net torqueis exerted on a sphere that can rotate about an axis through its center. Thegraph shows the sphere’s angular velocityas a function of time. Which of the following correctly describes amethod for determining the work done on the sphere by the torque during thetime interval shown in the‐graph?Scoring Guide2Page 30 of 37AP Physics C: Mechanics
(A)Multiplyby.(B)Divideby the slope of the‐graph.(C)Determine the area under the‐graph.(D)Multiply the area under the‐graph by.Answer DCorrect. The area under the‐graph is the sphere’s angular displacement. The work is, which for a constant torqueis simply.41.A varying net torque is exerted on a disk that is rotating about a fixed axis. The disk's rotational kinetic energy as afunction of angular positionis shown in the graph. Which of the following options correctly ranks the magnitudes,, andof the torque applied during the three intervals,, and, respectively?(A)(B)(C)(D)Answer BCorrect. The change in energy over an interval is the work done on the disk over that interval. The workScoring Guide2AP Physics C: MechanicsPage 31 of 37
done over a given interval is the area under the curve of a torque-vs-angle graph, so the torque is theslope of a graph of either work done or energy as a function of angle. The torque magnitudes aretherefore ranked according to the slopes in the graph, which are equal forand, and half as muchfor.42.The four graphs show the net torqueexerted on four identical rotating disks as a function of each disk’s angularposition. Which of the following ranks the magnitudeof the work done on each disk as the disks rotate fromradians toradians?(A)(B)(C)(D)Answer DCorrect. The work done on each disk is found from the area under the graph of torque as a function ofangular position. The area is equivalent to the average torque multiplied by the angular displacement.Scoring Guide2Page 32 of 37AP Physics C: Mechanics
Disks,, andhave the same average torque ofnewton-meters and the same angular displacementofradians, so the area and therefore work done are equal for these three disks. For Disk, the grapharea is easily compared to that of Diskand is greater, so the work done on Diskis the greatest.43.A wheel is spinning freely on an axle passing through its center. A constant torque is exerted on the wheel,increasing the wheel’s rotational kinetic energy fromtoas the wheel rotates throughrevolutions.What is the magnitude of the torque?(A)(B)(C)(D)Answer DCorrect. The work done on the wheel is equal to the change in kinetic energy:. The work is related to the torque by the equation.Since the torque is constant, this integral is simply. Solving forgives, which is. (Note that radians are dimensionless and may be omitted in the finalanswer.)Scoring Guide2AP Physics C: MechanicsPage 33 of 37
44.An object can rotate about an axis passing through its center of mass. At, the object is spinning in the positivedirection and has an initial angular position of zero. The graph represents the net torqueexerted on the object asa function of angular position. Which of the following claims is true about the total work done on the object andthe rotational kinetic energy of the object as it rotates fromradians toradians?(A)The total work is zero. The rotational kinetic energy is a maximum when.(B)The total work is zero. The rotational kinetic energy is a maximum when.(C)The total work is negative. The rotational kinetic energy is a maximum when.(D)The total work is negative. The rotational kinetic energy is a maximum when.Answer BCorrect. The work done on the object is found from the area under the graph of torque as a function ofangular position. The positive area between zero andradian is equal in magnitude to the negative areabetweenandradians, resulting in zero area. The torque transfers energy into and out of the object.The initially positive torque increases the object’s initially positive angular velocity, and therefore itsrotational kinetic energy, fromtoradian. For greater angles, the negative torque decreases theangular speed, so the speed and rotational kinetic energy have maximum values atradian.45.A uniform disk is initially at rest at an angular position of zero. An electric motor rotates the disk by exerting aconstant torqueon the disk. During the disk’s first and second complete revolutions, the motor does amounts ofworkandon the disk, respectively. The changes in angular speed of the disk during the first and secondrevolution areand, respectively. Which of the following correctly comparesto, andto?Scoring Guide2Page 34 of 37AP Physics C: Mechanics
(A)and(B)and(C)and(D)andAnswer BCorrect. The amount of work done during each revolution is the same because the angular displacementis the same. The change in angular speed during the second revolution will be less than that of the firstbecause as the speed of the disk increases, more work is required to increase the disk’s speed by the sameamount, as shown by the work-energy theorem.46.A wheel can rotate about an axis that passes through its center and is perpendicular to the page. The edge of thewheel is attached to one end of a wire that is connected to a motor, which is fixed in place. The motor exerts a forceon the wire such that the wire exerts a counterclockwise torque of magnitudeabout the center of the wheel whenthe wheel is at an angular position. As the wheel rotates, the torque exerted on the wheel changes as afunction ofthat is given by, whereis a positive constant. If the positive direction foriscounterclockwise, how much work is done by the wire on the wheel as the wheel rotates fromto an angularposition?(A)(B)(C)(D)Scoring Guide2AP Physics C: MechanicsPage 35 of 37
Answer DCorrect. The work done by a variable torque is given by. Substituting the expression fortorque and applying the correct limits gives the following.,which evaluates to.47.At time, the wheel shown in the diagram has clockwise rotation with an angular speed of. Thecenter of mass of the wheel is initially moving to the right at. The wheel rolls on a flat rough surface and hasa radius of. Which of the following is correct about the type of motion, the angular speed, and the linearspeed of the wheel at time?(A)The type of motion is rolling with slipping, the angular speed remains the same, and the linear speed isdecreasing.(B)The type of motion is rolling with slipping, the angular speed is decreasing, and the linear speed isincreasing.(C)The type of motion is rolling with slipping, the angular speed is decreasing, and the linear speed isdecreasing.(D)The type of motion is rolling without slipping, the angular speed is increasing, and the linear speed isincreasing.(E)The type of motion is rolling without slipping, the angular speed is decreasing, and the linear speed isdecreasing.Scoring Guide2Page 36 of 37AP Physics C: Mechanics
Answer BCorrect. The value ofis greater than the speed of the center of mass, so the wheel is rolling and slipping. If there was rolling without slipping,andwould have thesame value. For rolling without slipping to occur, friction must be applied at the bottom of the wheel anddirected to the right to increaseand decrease. This produces a rightward net force that increasesthe rightwardand applies a counterclockwise net torque to decrease the clockwise.48.A wheel of radiushas a rotational inertia of. The wheel is spinning at a rate ofrevolutionsper second. A frictional force is applied tangentially to the wheel to bring it to a stop. The work done by the torqueto stop the wheel is most nearly(A)zero(B)(C)(D)(E)Scoring Guide2AP Physics C: MechanicsPage 37 of 37