Physics 211 Sample Questions for Exam IV Spring 2013

Transcription

1 Each Exam usually consists of 10 Multiple choice questions which are conceptual in nature. They are often based upon the assigned thought questions from the homework. There are also 4 problems in each exam, based upon the assigned homework problems. Partial credit may be awarded for the problems. Physical constants and equation sheets are provided for the exam. Part I Multiple Choice (4 points. ea.) Two children sit on a merry-go-round as it spins about. One child sits all the way out on the outer edge of the merry-go-round. The second one sits closer to the center, halfway between the axis of rotation and the outer edge. In meters per second, which child travels fastest? (A) Both travel at the same speed in m/s since their angular frequency is the same. (B) The speed is greater for the child on the outside edge. (C) The speed is greater for the child on the inner edge. (D) The speed of the child on the outside is half the speed of the child seated on the inside. (E) both (C) and (D). Which of the following rotational quantities are the same for all points on a solid rotating disk? (A) Angular velocity. (B) Tangential velocity. (C) Tangential acceleration. (D) Centripetal acceleration. (E) all of the above.. When would it be possible for a rotating object to have increasing angular speed and have negative angular acceleration? (A) when angular velocity and angular acceleration are both negative. (B) when angular velocity is positive and angular acceleration is negative. (C) when the angular velocity and the angular acceleration are perpendicular. (D) never. Torque ( ), is the (A) rotational analogue of Force (F) in linear motion. (B) rotational analogue of Momentum (p) in linear motion. (C) rotational analogue of Work (W) in linear motion.. (D) linear analogue of Angular Velocity ( ) in rotational motion. (E) linear analogue of Angular Momentum (L) in rotational motion. Angular momentum (L) (A) is the rotational analogue of momentum (m) in linear motion. (B) is conserved in the absence of external torques. (C) allows a skater or gymnast to change rotation rates by repositioning limbs. (D) all of the above. (E) none of the above The moment of inertia I is the rotational analogue of (A) mass, from linear motion. (B) force, from linear motion. (C) kinetic energy, from linear motion. (D) orbital period, from circular motion. (E) none of the above. 1

2 . In rotational motion, the analog of mass in linear motion is (A) angular momentum. (B) moment of inertia. (C) torque. (D) radian. (E) weight.. When a flywheel rotates with constant angular velocity, then a point on its rim will have (A) a constant magnitude tangential acceleration but no radial acceleration. (B) a constant magnitude radial acceleration but no tangential acceleration. (C) a constant magnitude tangential acceleration and constant magnitude tangential acceleration. (D) no acceleration. (E) impossible to tell since tangential and radial acceleration are not related to angular velocity.. For a point on the rim of a flywheel that rotates at constant angular velocity (A) The tangential acceleration and the radial acceleration are zero. (B) The tangential acceleration is zero and the radial acceleration is constant. (C) The radial acceleration is zero and the tangential acceleration is constant. (D) The tangential acceleration is zero and the radial acceleration is constant in magnitude but changing in direction. (E) The tangential and radial acceleration are both changing in both magnitude and direction. A force is applied to a to a wrench as shown. The distance which best indicates the moment arm l of the applied force as it creates a torque about the nut as shown is (A) distance (A) from the diagram. (B) distance (B) from the diagram. (C) distance (C) from the diagram. (D) 0, since the force does not act parallel to the radial line. (E) 0, since the force does not act perpendicular to the radial line. A B C F When a flywheel rotates with constant angular velocity, then a point on its rim will have (A) no acceleration. (B) a constant magnitude tangential acceleration but no radial acceleration. (C) a constant magnitude radial acceleration but no tangential acceleration. (D) a constant magnitude tangential acceleration and constant magnitude tangential acceleration. (E) impossible to tell since tangential and radial acceleration are not related to angular velocity. The moment of inertia of an object (A) will be greater if the mass is distributed away from the axis of rotation. (B) will be greater if the mass is distributed closer to the axis of rotation. (C) will not depend on how the mass is distributed from the axis of rotation, but will depend on how the mass is distributed along the axis of rotation. (D) will not depend on how the mass is distributed at all. (E) does not depend on mass in any way shape or form. 2

3 The moment of inertia of an object (A) will not depend on how the mass is distributed from the axis of rotation, but will depend on how the mass is distributed along the axis of rotation. (B) will be greater if the mass is distributed away from the axis of rotation. (C) will be greater if the mass is distributed closer to the axis of rotation. (D) will not depend on how the mass is distributed at all. (E) does not depend on mass in any way shape or form. A B C D. A uniform disk is to be rotated about any of 4 axis as shown. The axis corresponding to the greatest moment of inertia is (A) the axis 2 radii from the center of the disk. (B) the axis right at the edge of the disk (C) the axis halfway between the center of the disk and the edge of the disk. (D) the axis through the center of the disk. (E) none of the above, since the moment of inertia does not depend upon the location of the axis of rotation.. A uniform disk is to be rotated about any of 5 axis as shown. If the axis are ranked from highest to lowest in terms of the moment of inertia of the disk about that axis, then the most correct ranking would be: (A) A=B=C=D=E. (B) A>B>C>D>E. (C) A=E>B=D>C. (D) C>B=D>A=E. (E) E>D>C>B>A. A B C D E To maximize the moment of inertia of a flywheel while minimizing its weight, the flywheel should be (A) a uniform disk. (B) a thin walled hollow cylinder (i.e. a hoop). (C) a uniform solid sphere. (D) a thin walled hollow sphere. (E) a slender rotated about an axis through one end. For the next two questions: A thin hoop and uniform disk of the same mass and radius are released from rest from the same height at the top of an inclined plane. If the objects roll without slipping, then (A) the hoop will reach the bottom first because it has a greater moment of inertia. (B) the hoop will reach the bottom first because it has a smaller moment of inertia. (C) the disk will reach the bottom first because it has a greater moment of inertia. (D) the disk will reach the bottom first because it has a smaller moment of inertia. (E) any of the above depending upon the initial height, and the details of construction of the hoop and disk. If the slope is frictionless, (A) the hoop will reach the bottom first because it has a greater moment of inertia. (B) the hoop will reach the bottom first because it has a smaller moment of inertia. (C) the disk will reach the bottom first because it has a greater moment of inertia. (D) the disk will reach the bottom first because it has a smaller moment of inertia. (E) the disks will reach the bottom at the same time since there is no torque, making the moment of inertia irrelevant. 3

4 A solid disk and a hoop of the same radius and mass are released from rest at the top of an inclined plane (as in the lecture demonstration). The object which has the greatest kinetic energy at the bottom of the plane is the (A) hoop. (B) disk. (C) both have the same final KE. (D) more information would be required to answer this question. (E) neither, since an interdimensional wormhole will spontaneously open to annihilate both objects before they reach the bottom. A thin hoop and uniform disk of the same mass and radius are released from rest from the same height at the top of an inclined plane. If the slope is frictionless, (A) the hoop will reach the bottom first because it has a greater moment of inertia. (B) the hoop will reach the bottom first because it has a smaller moment of inertia. (C) the disk will reach the bottom first because it has a greater moment of inertia. (D) the disk will reach the bottom first because it has a smaller moment of inertia. (E) the disks will reach the bottom at the same time since there is no torque, making the moment of inertia irrelevant. A force is applied to a to a wrench as shown. The distance which best indicates the moment arm l of the applied force as it creates a torque about the nut as shown is (A) distance (A) from the diagram. (B) distance (B) from the diagram. (C) distance (C) from the diagram. (D) 0, since the force does not act parallel to the radial line. (E) 0, since the force does not act perpendicular to the radial line. A C B F. A force F is applied to a wrench as shown. The distance which best indicates the moment arm l of the applied force as it creates a torque about the nut as shown is (A) distance (A) from the diagram. (B) distance (B) from the diagram. (C) distance (C) from the diagram. (D) 0, since the force does not act parallel to the radial line. (E) 0, since the force does not act perpendicular to the radial line. A B C F 4

5 Which of the following equations requires that angles be measured in radians. (A). = d dt (B) = 0 0 t 1 2. t2 (C) v= r. (D) all of the above. (E) none of the above.. Which of the following equations requires that angles be measured in radians. (A) a tan = r (B) v= r (C) s= r (D) all of the above. (E) none of the above. When calculating the work done by a torque acting through an angular displacement via the equation W =, the units for and are (A) Newton. meters and degrees. (B) Newton. meters and rotations. (C) Watts and degrees. (D) Newtons and radians. (E) Newton. meters and radians. The pivoting motion (constant steady rotation) of the gyroscope/bicycle wheel when suspended by one side only (as demonstrated in class) is called (A) percussion. (B) deflection. (C) precession. (D) inflation. (E) desertion. The pivoting motion (constant steady rotation) of the gyroscope when mass is suspended from the axis of rotation (as demonstrated in class) is called, while the repeated bouncing motion the gyroscope exhibited when the rotation axis was tapped is called. (A) nutation, desertion. (B) percussion, nutation. (C) perturbation, deflection. (D) inflation, recession. (E) precession, nutation. 5

6 . In a classroom demonstration, the professor sat on a spinning chair with weights held out in his extended arms. As he brought his arms in to his torso, his rotation rate increased. This was because (A) his angular momentum was constant, and so by decreasing his moment of inertia his angular velocity necessarily increased. (B) his angular momentum was constant, and so by increasing his moment of inertia his angular velocity necessarily increased. (C) he was creating an external torque with the motion of his arms. (D) by changing his moment of inertia, he was able to use the principle of nutation to increase his rotation rate. (E) by signaling extraterrestrials his desire for increased angular kinetic energy, he was able to initiate an alien insurrection.. In an equilibrium problem, the point about which torques are calculated (A) must pass through one end of the object. (B) must pass through the objects center of mass. (C) must intersect the line of action of at least one force acting on the object. (D) may be located anywhere.. A wrench is suspended by a nail through a hole in its handle as shown, and is free to rotate. The wrench is in (A) stable equilibrium. (B) neutral equilibrium. (C) unstable equilibrium. (D) non-equilibrium. (E) suspended animation. Pivot point Center of gravity. A meter stick is balanced on end as shown. The meter stick is in (A) stable equilibrium. (B) neutral equilibrium. (C) unstable equilibrium. (D) non-equilibrium. (E) suspended animation.. The figure at right shows a rod divided into five equal parts. The rod has negligible weight and a fixed pivot at point c. An upward force of magnitude F is applied at point b, as shown. At what point on the rod could you apply a second force with magnitude half of F, also upward and perpendicular to the rod so that the net torque on the rod about the pivot is zero? Use answer G if it is not possible to create a net torque of zero under these conditions. F A B C D E F. The figure at right shows a rod divided into five equal parts. The rod has negligible weight and a fixed pivot at point c. An upward force of magnitude F is applied at point a, and an identical force is applied at point f, as shown. At what point on the rod could you apply a third force of the same magnitude F, but downward and perpendicular to the rod so that the net torque on the rod about the pivot is zero? Use answer g if it is not possible to create a net torque of zero under these conditions. F a b c d e f F 6

7 . If the sum of the torques on an object in equilibrium is zero about a certain point, it is (A) zero about all other points. (B) zero about some other points (but not all other points). (C) zero about no other points. (D) any of the above, depending upon the situation.. An object in equilibrium must not have (A) acceleration. (B) any forces acting on it. (C) any torques acting on it (D) all of the above. (E) none of the above.. The type of matter that can support (i.e. resist and spring back against) a shear stress is (A) liquid. (B) gas. (C) solid. (D) all of the above. (E) none of the above.. When the stresses on an object exceed the material s Elastic Limit, then (A) the material fails (i.e. breaks). (B) the material is permanently deformed. (C) the material snaps back to its original size and shape when the stress is removed. (D) the material s strain is proportional to the stress. (E) the object undergoes hysteria.. When the stresses on an object exceed the material s Ultimate Strength, then (A) the material is permanently deformed. (B) the material fails (i.e. breaks). (C) the material snaps back to its original size and shape when the stress is removed. (D) the material s strain is proportional to the stress. (E) the object collapses into a screaming mass, requiring therapy for the rest of its earthly existence.. When a force acts on an object producing a stress on the object, the resulting strain is the (A) relative change in its dimensions. (B) the applied force per unit area. (C) the bulk modulus. (D) the ultimate strength of the material.. When a force acts on an object producing a stress on the object resulting in strain, the stress on the object is the (A) relative change in its dimensions. (B) the applied force per unit area. (C) the bulk modulus. (D) the ultimate strength of the material. 7

8 . When a force acts on an object producing a stress on the object resulting in strain, the elastic modulus is the (A) relative change in the object's dimensions. (B) applied force per unit area. (C) ultimate strength. (D) stress divided by the strain.. When the stresses on an object create a permanent deformation, then (A) the stress has exceeded the material's Elastic Limit. (B) the object undergoes hysteresis. (C) the deformations may also be referred to as a plastic deformation. (D) all of the above. (E) none of the above.. A solid is the only type of material that can support (i.e. resist and spring back against) (A) a linear compression (change in length). (B) an elongation. (C) a shear stress. (D) a change in volume. (E) an acoustical stress. 8

9 Part II Problems Show all work. No work = no credit! (15 points each) A torque of 100 N. m is applied to a disk which rotates on a fixed axle and which is initially at rest. After the disk has made 4 full rotations in 8 seconds, determine (A) the rotation of the disk in radians, (B) the work done on the disk, (C the final kinetic energy of the disk, (D) the angular acceleration of the disk, (E) the final angular velocity of the disk in rad/s, (F) the moment of inertia of the disk. A torque of 100 N. m is applied to a disk initially at rest. After 30.0 s the speed of the disk has reached 150 rad/s. (A) Determine the angular acceleration of the disk. (B) Determine the number of rotations made by the disk in the 30.0 s (C) Determine the moment of inertia of the disk. (D) Calculate the final rotational kinetic energy of the disk. (E) Determine the work done on the disk by the external torque. A grinding wheel starts from rest and reaches its final angular velocity of 5000 rpm in 8.00 s. (A) What is the final angular velocity in rad/s? (B) What is the angular acceleration of the wheel (in rad/s 2 )? (C) How much rotation (in radians) does the wheel undergo in this time? (D) How many revolutions does the disk make in this time? Consider the rod pictured at right. It is set to rotate about its center. A) What is the net torque on the rod pictured below? B) If the angular acceleration of the rod is 4.20 rad/s2, what is the moment of inertia of the rod? 5.00 N m m N 9

10 A string is wrapped around the inner radius a yoyo which has an outer radius of 4.00 cm and an inner radius of 1.50 cm. The yoyo has a mass of.150 kg and a moment of inertia of 1.00E 4 kg m 2. A constant tension of 1.6 N is applied horizontally to the string as shown. After the yoyo has moved (by rolling without slipping) a distance of.4 m, determine (A) How many revolutions the yoyo has made, (B) How much string has unwound from the inner radius of the yoyo, (C) How far the end of the string (from which the external force maintaining the tension is being applied) has moved, (D) The work done by this external force, (E) the final kinetic energy of the yoyo, (F) the final linear and angular velocities of the yoyo. A 2.00 Kg mass is suspended from a string wrapped around the inner disk of a pulley. The Pulley is composed of two uniform disks, a smaller disk of mass.5 kg and radius.25 meters and a larger disk of mass 1.00 kg and radius 1.00 m. (A) Draw a free-body diagram for each object (pulley and weight). (B) Calculate the total moment of inertia of the pulley. (C) Determine the acceleration of the mass. (D) Determine the angular acceleration of the disk. (E) Determine the tension in the string. A uniform disk (mass 2.50 kg, radius.250 m) is rotating without friction at an initial angular velocity of 12.0 rad/s. A second uniform disk (mass 5 kg, radius. 200 m) is initially at rest and is dropped gently onto the first disk so that they eventually rotate together about the common axis. (A) What are the moments of inertia of the two disks? (B) What is the initial angular momentum of the first disk? (C) What is the final angular momentum of the two disks moving together? (D) What is the final angular velocity of the two disks moving together? (E) What is the initial and final kinetic energies of the system? (F) Can this event be viewed as an elastic collision? Why or why not? at rest moving together A.500 kg lump of clay moving at 24.0 m/s hits the edge of a 2.00 kg turntable tangent to the turntable s rim and sticks to the edge of turntable, which subsequently rotates. The turntable can be treated as a uniform disk of radius m, and the lump of clay can be treated as a point mass. (A) What is the initial angular momentum of the lump of clay just before impact? (B) What is the total moment of inertia of the disk and clay after the clay becomes stuck to the edge of the turntable? (C) What is the angular velocity of the turntable after the impact? (D) How much kinetic energy is lost in this collision? 10

11 Block A (mass = 3.00 kg) and Block B (mass = 4.00 kg) are attached by a string which goes over a pulley (mass = 6.00 kg, uniform disk of radius.2m) as shown. Block B rests on a horizontal frictionless surface. (A) Draw a free body diagram for each block and for the pulley, indicating the forces and torques acting on each object. (B) Determine the acceleration of the masses. (C) Determine the angular acceleration of the pulley (D) Determine the tension in the string where it is attached to mass A. (E) Determine the tension in the string where it is attached to mass B. A B Block A (mass = 4.00 kg) and Block B (mass = 8.00 kg) are suspended by a string on either side of a friction-less pulley (mass = 4.00 kg, uniform disk of radius.2m). (a) Draw a free body diagram for each block and for the pulley, indicating the forces and torques acting on each object. (b) Determine the acceleration of the masses. (c) Determine the angular acceleration of the pulley (d) Determine the tensions in the string on the left side. (e) Determine the tensions in the string on the right side. A B Space Yo-Yo: A yoyo is operated in space, so that there are no other forces acting on it but the tension in the string. The yoyo has a mass of.150 kg and a moment of inertia of 1.00E 4 kg m 2. A string is wrapped around the inner radius the yoyo which has an outer radius of 4.00 cm and an inner radius of 1.50 cm. If a constant tension of N is applied to the string, determine (A) The acceleration of the yo-yo s center of mass, (B) the angular acceleration of the yo-yo about its center of mass. If the yo-yo starts from rest, determine the following after 2.00 s have elapsed: (C) the displacement of the yo-yo, (D) the rotation (in rad) of the yo-yo, (E) how much string has unwound from the yo-yo, (F) the displacement of the other end of the string (where the operator is applying the N) (G) the work done by the yo-yo operator (H) the final kinetic energy of the yo-yo A 2.00 Kg mass is suspended from a string wrapped around the inner disk of a pulley. The Pulley is composed of two uniform disks, a smaller disk of mass 1.50 kg and radius.150 meters and a larger disk of mass 5.00 kg and radius.600 m. (A) Draw a free-body diagram for each object (pulley and weight). (B) Calculate the total moment of inertia of the pulley. (C) Determine the acceleration of the mass. (D) Determine the angular acceleration of the disk. (E) Determine the tension in the string. 11

12 A cord is wrapped around a hollow 25.0 kg sphere which has a diameter of m, and which rotates (frictionlessly) about an axis through its center. A 15.0 N force is applied to the end of the cable, causing the cable to unwind and the sphere (initially at rest) to rotate. After the cable has unwound a distance 1.25 m, use work and energy methods to determine (A) the work done by the force, (B) the kinetic energy of the sphere, (C) the final rotational velocity of the sphere A box of weight w 1 is suspended from the end of a (hinged) horizontal uniform beam of length L and weight w 2, as shown. Note that the hinged connection of the beam to the wall does not exert any torque on the beam about the hinge. (A freebody diagram will be essential to the successful completion of this problem.) (A) Find expressions for tension in the cable and the x and y components of the reaction force exerted on the strut by the hinge. 30 o Suppose that L is 2.00 m, w 1 is 500. N and w 2 is 100. N. (B) What is the tension in the cable and the x and y components of the reaction w 2 w 1 force exerted on the strut by the hinge? (C) What is the maximum weight w 1 that can be suspended from this system as shown if the cable is not to permanently stretch? The cable has a cross section of 1.6E-5 m 2, and is made of a material with Young s modulus 11 x N/m 2, Elastic Limit 1.5 x 10 8 N/m 2, and Ultimate Strength 3.4 x 10 8 N/m 2. A box of weight w 1 is suspended from the end of a (hinged) uniform beam of length L weight w 2, as shown. Note that the hinged connection of the beam to the wall does not any torque on the beam about the hinge. The top cable connecting the beam to the wall horizontal. (A free-body diagram will be essential to the successful completion of this problem.) (A) Find expressions for tension in the cable and the x and y components of the reaction force exerted on the strut by the hinge in terms of w 1 and w 2. Suppose that L is 2.00 m, w 1 is 500. N and w 2 is 100. N. and exert is (B) What is the tension in the cable and the x and y components of the reaction force exerted on the strut by the hinge? (C) What is the minimum cross sectional area of the horizontal cable if the cable is not to permanently stretch? The cable is made of a material with Young s modulus 11 x N/m 2, Elastic Limit 1.5 x 10 8 N/m 2, and Ultimate Strength 3.4 x 10 8 N/m o w 2 w 1 12

13 Sir Robin makes a mad dash for the drawbridge which is supported at one end by a frictionless hinge and the other end by a vertical cable. The bridge is 8.00 m long and has a mass of 100 kg. Sir Robin has a mass of 60.0 kg (in armor, soaking wet). At the instant Sir Robin has made it three fourths of the way across the bridge (as shown) (A) use the conditions of equilibrium to determine the tension in the cable and the components of the reaction force at the hinge. (B) given the tension you found in (A), what is the minimum cross sectional area of the cable, given that it did not break. The cable is made of a material with Young s modulus 11 x N/m 2, Elastic Limit 1.5 x 10 8 N/m 2, and Ultimate Strength 3.4 x 10 8 N/m 2. Dr Mic rides his motorcycle across a drawbridge as shown at right. The bridge is 8.00 m long and has a mass of 400 kg. Rider and motorcycle have a combined mass of 350 kg. At the instant the motorcycle has made it three fourths of the way across the bridge (as shown) (A) use the conditions of equilibrium to determine the tension in the cable hinge and the components of the reaction force at the hinge. (B) given the tension you found in (A), what is the minimum cross x sectional area of the cable, given that it was not permanently stretched. The cable is made of a material with Young s modulus 11 x N/m 2, Elastic Limit 1.5 x 10 8 N/m 2, and Ultimate Strength 3.4 x 10 8 N/m 2. x Dr Mic rides his motorcycle across a drawbridge as shown at right. The bridge is 6.00 m long and has a mass of 200 kg. Rider and motorcycle have a combined mass of 400 kg. The cable is made of a material with Young s modulus 1.1E11 N/m2, Elastic Limit 1.5E8 N/m 2, and Ultimate Strength 3.0E8 N/m 2. hinge (A) What is the maximum force the cable on the right side can x withstand without breaking, if it has a cross sectional area of 1.00E-5 m 2? (B) use the conditions of equilibrium to relate the actual tension in the cable to the position of the motorcycle at a distance x from the hinge side of the drawbridge. (Hint: look at the torques about the left end of the bridge) (C) How far from the hinge side of the drawbridge does Dr Mic get before he plunges into the abyss because of a snapped cable? Dr Mic rides his motorcycle across a drawbridge as shown at right. The bridge is 6.00 m long and has a mass of 400 kg. Rider and motorcycle have a combined mass of 350 kg. (A) What is the maximum force the cable on the right side can withstand without breaking, if it has a cross sectional area of 1.50x10-5 m 2? hinge (B) use the conditions of equilibrium to relate the actual tension in the cable to the position of the motorcycle at a distance x from the x hinge side of the drawbridge. (Hint: look at the torques about the left end of the bridge) (C) How far from the hinge side of the drawbridge does Dr Mic get before he plunges into the abyss because of a snapped cable? The cable is made of a material with Young s modulus 11 x N/m 2, Elastic Limit 1.5 x 10 8 N/m 2, and Ultimate 13

14 Strength 3.0 x 10 8 N/m 2. A brave and trustworthy (and 90.0 kg) physics professor is being forced to walk (3.00m, 50.0 kg) plank by a horde of angry students. The plank is supported meter from the left end, and another force is applied downwards at the very end. instructor is.500 meters from the right end of the board. (A) Draw a free body diagram, indicating all the forces on the plank. (B) Determine the forces exerted by the two supports. the one The A 4.00 m uniform beam of mass 40.0 kg is suspended by vertical ropes at each end. A 120 mass sits on the beam 1.00 m from the left end of the beam. Determine the tension in each A B Kg rope. 14