Axis. Axis. Axis. Solid cylinder (or disk) about. Hoop about. Annular cylinder (or ring) about central axis. central axis.

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1 Instructor(s): Acosta, inzler PHYSICS DEPATMENT PHY 048, Spring 04 Final Exa March 4, 04 Nae (print, last first): Signature: On y honor, I have neither given nor received unauthorized aid on this exaination. YOU TEST NUMBE IS THE 5-DIGIT NUMBE AT THE TOP OF EACH PAGE. () Code your test nuber on your answer sheet (use lines on the answer sheet for the 5-digit nuber). Code your nae on your answer sheet. DAKEN CICLES COMPLETELY. Code your UFID nuber on your answer sheet. () Print your nae on this sheet and sign it also. (3) Do all scratch work anywhere on this exa that you like. Circle your answers on the test for. At the end of the test, this exa printout is to be turned in. No credit will be given without both answer sheet and printout. (4) Blacken the circle of your intended answer copletely, using a # pencil or blue or black ink. Do not ake any stray arks or soe answers ay be counted as incorrect. (5) The answers are rounded off. Choose the closest to exact. There is no penalty for guessing. If you believe that no listed answer is correct, leave the for blank. (6) Hand in the answer sheet separately. Where needed use g = 9.80 /s Hoop about central axis Annular cylinder (or ring) about central axis L Solid cylinder (or disk) about central axis I = M I = M( + ) I = M L Solid cylinder (or disk) about central diaeter L Thin rod about axis through center perpendicular to length Solid sphere about any diaeter I = M 4 + ML I = ML I = 5 M Thin spherical shell about any diaeter Hoop about any diaeter a b Slab about perpendicular axis through center I = 3 M I = M I = M(a + b )

2 . Four identical soda cans initially at rest have a firecracker explode between the such that one can recoils against the other three which are still stuck together. With V and V 3 the velocities of the single and 3 cans, respectively, what is the ratio of the agnitude of the speed of the single can to the agnitude of the speed of the other three (V /V 3 )? V V 3 () 3 () (3) 4 (4) /3 (5) /. A cylindrical bar aligned along the x-axis has a ass density that increases linearly along its length: λ(x) = (4 kg/ )x. If the bar extends fro x = 0 to x =.5, at what position along the x-axis (in ) is the center-of-ass of the bar? () () / (3) /3 (4) 4/3 (5) 3/ 3. A cylindrical bar aligned along the x-axis has a ass density that increases linearly along its length: λ(x) = (4 kg/ )x. If the bar extends fro x = 0 to x = 0.5, at what position along the x-axis (in ) is the center-of-ass of the bar? () /3 () / (3) (4) 4/3 (5) 3/ 4. A cylindrical bar aligned along the x-axis has a ass density that increases linearly along its length: λ(x) = (4 kg/ )x. If the bar extends fro x = 0 to x =, at what position along the x-axis (in ) is the center-of-ass of the bar? () 4/3 () / (3) /3 (4) (5) 3/ 5. Consider a ball of ass 0.5 kg that travels only in one diension along the x axis. Its initial velocity at tie t = 0 is v = 5.0 /s. A force is then applied to the ball in the x direction as a function of tie as shown by the graph. What is the velocity of the ball along the x-axis, including sign, at a tie of 0.4 s? F (N) t (s) () 3 /s () 8 /s (3) 3 /s (4) /s (5) 0 /s 6. In the figure, a horizontal force F a of agnitude 0 N is applied to a.5 kg book as the book slides a distance d =.0 up a frictionless rap at angle θ = 30. What is the speed of the book at the end of the displaceent if it starts initially at rest? d θ Psychology F a ().0 /s () 4.0 /s (3) 7.5 /s (4) 6.0 /s (5).0 /s 7. In the figure, a horizontal force F a of agnitude 0 N is applied to a.5 kg book as the book slides a distance d = 4.0 up a frictionless rap at angle θ = 30. What is the speed of the book at the end of the displaceent if it starts initially at rest? d θ Psychology F a () 4.0 /s ().0 /s (3) 7.5 /s (4) 6.0 /s (5).0 /s

3 8. The figure gives the force acting on a 5.0 kg particle as it oves fro rest along an x axis fro x = 0 to x = 8.0. What is the particle s speed and direction (give a positive answer if the particle oves along the x axis in the positive direction and negative otherwise) of travel when it reaches x = 5.0? F (N) x () () +7. /s () +5.5 /s (3) 5.5 /s (4) 4.5 /s (5) 0 /s 9. The figure gives the force acting on a 5.0 kg particle as it oves fro rest along an x axis fro x = 0 to x = 8.0. What is the particle s speed and direction (give a positive answer if the particle oves along the x axis in the positive direction and negative otherwise) of travel when it reaches x = 6.0? F (N) x () () +5.5 /s () +7. /s (3) 5.5 /s (4) 4.5 /s (5) 0 /s 0. A high juper of ass 70 kg runs toward a high jup bar with a speed of 6.7 /s. What would be the axiu bar height the center-of-ass of the juper could clear if he/she were able to convert all of his/her initial kinetic energy into a vertical jup? ().3 () 4.6 (3).4 (4) 0.4 (5).5. A 0.5 kg ass is attached to the end of a relaxed, vertical hanging spring and the ass let go fro rest. What is the axiu distance the spring stretches downward after the ass is released if the spring constant is k = 50 N/? () 0 c () 30 c (3) 5 c (4) 0 c (5) 40 c. Arollercoasteris togothroughaverticalloopasshown. What isthe iniu speed at the botto of the loop (V b ) necessary so that the cars do not fall off of the track at the top of the loop if the radius of the loop is = 0? Treat a roller coaster car as a point ass going through the loop. V t V b () /s () 0 /s (3) 0 /s (4) 7 /s (5) 500 /s 3. Two sticky equal ass air hockey pucks collide and stick together at the origin of the frictionless x-y plane shown in the figure. The shown angles are θ = 30 and θ = 60. The initial speed of both pucks are equal: v = v = /s. Deterine the outgoing angle θ 3 for the cobined syste of two pucks, easured counter-clockwise with respect to the x-axis. y v v 3 θ θ 3 θ x v () 5 () 60 (3) 0 (4) 75 (5) 30

4 4. Thesystedepictedinthefigureisinequilibriu. AblockofassM = 300kg hangs fro the end of a assless strut which is fixed to the ground at the other end and akes an angle θ = 30 with respect to the horizontal. Find the tension T in the assless, horizontal cable attaching the strut to the wall. T θ M () 500 N () 900 N (3) 700 N (4) 5900 N (5) 3400 N 5. Thesystedepictedinthefigureisinequilibriu. AblockofassM = 300kg hangs fro the end of a assless strut which is fixed to the ground at the other end and akes an angle θ = 60 with respect to the horizontal. Find the tension T in the assless, horizontal cable attaching the strut to the wall. T θ M () 700 N () 900 N (3) 500 N (4) 5900 N (5) 3400 N 6. A ass is attached to, and hangs fro, two ropes which are fixed at their other ends in the ceiling, as shown in the figure. The left rope akes an angle of o 30 o with respect to the horizontal, and the right rope 30. What is the ratio of the tension in the left rope to that of the right (T L /T )? T L T () 3 () / 3 (3) (4) 3/ (5) 7. A ass is attached to, and hangs fro, two ropes which are fixed at their other ends in the ceiling, as shown in the figure. The left rope akes an angle of 60 o 30 o 60 with respect to the horizontal, and the right rope 30. What is the ratio of the tension in the right rope to that of the left (T /T L )? T L T () / 3 () 3 (3) (4) 3/ (5) 8. A 5 kg object is suspended fro the ceiling by a cylindrical copper wire with a diaeter of. If the wire initially has a length of, what is the increase in its length in illieters () after the object is attached if the Young s odulus of copper is N/? ().0 () (3) 0.5 (4) (5) The graph showsthe potential energyof an object acted upon by a conservative force as a function of its position x. The potential values indicated are: U A = 0 J, U B = 5 J, U C = 40 J, and U D = 50 J. At which of the listed positions in x is the force largest in the +x direction? () () 7.5 (3) 0.5 (4) 5 (5) 6.5 U (J) U D U C U B U A x ()

5 0. The following objects (i) a solid sphere, (ii) a spherical shell and (iii) a cube all have the sae ass. The objects all start to ove, fro rest, fro the sae elevation on an inclined surface (sae incline), at the sae tie. The spheres each roll without slipping while the cube slides without friction. The order in which they get to the botto of the incline, fastest first, is: () (iii), (i), (ii) () (iii), (ii), (i) (3) (i), (ii), (iii) (4) (ii), (i), (iii) (5) (ii), (iii), (i). A wheel initially has an angular velocity of 30 rad/s but is slowing at a rate of rad/s. By the tie it stops, the nuber of revolutions it will have turned through is: () 36 () 49 (3) 64 (4) 8 (5) 5. A wheel initially has an angular velocity of 35 rad/s but is slowing at a rate of rad/s. By the tie it stops, the nuber of revolutions it will have turned through is: () 49 () 36 (3) 64 (4) 8 (5) 5 3. A wheel initially has an angular velocity of 40 rad/s but is slowing at a rate of rad/s. By the tie it stops, the nuber of revolutions it will have turned through is: () 64 () 36 (3) 49 (4) 8 (5) 5 4. A thin hoop of radius and ass M is welded to a thin unifor disk of the sae ass and the sae outer radius such that the hoop and disk are coaxial (dotted line). Calculate the rotational inertia for this object about an axis that is d = (3/4) fro the dotted axis, as shown in the sketch. (Hint: Make sure to consider the total ass of the syste after welding.) d ().63M () 3.03M (3).06M (4).7M (5).63M 5. A thin hoop of radius and ass M is welded to a thin unifor disk of the sae ass and the sae outer radius such that the hoop and disk are coaxial (dotted line). Calculate the rotational inertia for this object about an axis that is d = (7/8) fro the dotted axis, as shown in the sketch. (Hint: Make sure to consider the total ass of the syste after welding.) d () 3.03M ().63M (3).06M (4).7M (5).63M 6. A solid unifor sphere of ass 0.4 kg and diaeter 0 c rolls fro rest without slipping down a 30 incline. After rolling 8.0, easured along the incline, its rotational kinetic energy is: () 4.48 J () 5.60 J (3) 6.7 J (4) 7.84 J (5) 8.96 J 7. A solid unifor sphere of ass 0.4 kg and diaeter 0 c rolls fro rest without slipping down a 30 incline. After rolling 0.0, easured along the incline, its rotational kinetic energy is: () 5.60 J () 4.48 J (3) 6.7 J (4) 7.84 J (5) 8.96 J 8. A solid unifor sphere of ass 0.4 kg and diaeter 0 c rolls fro rest without slipping down a 30 incline. After rolling.0, easured along the incline, its rotational kinetic energy is: () 6.7 J () 5.60 J (3) 4.48 J (4) 7.84 J (5) 8.96 J

6 9. A dubbell rotor consists of a assless rigid rod free to pivot around a vertical axis through the iddle of the rod. Two sall, equal asses, are attached to the rod at a distance fro the rotation axis. The rotor rotates with an angular velocity of.0 rad/s. An internal echanis pulls the asses in syetrically until they are each at fro the axis. The angular velocity of the rotor after this ove (in rad/s) is: () 4 () 8 (3) (4) 4 (5) A dubbell rotor consists of a assless rigid rod free to pivot around a vertical axis through the iddle of the rod. Two sall, equal asses, are attached to the rod at a distance fro the rotation axis. The rotor rotates with an angular velocity of.0 rad/s. An internal echanis pulls the asses in syetrically until they are each at fro the axis. The angular velocity of the rotor after this ove (in rad/s) is: () 8 () 4 (3) (4) 4 (5) 8 3. A dubbell rotor consists of a assless rigid rod free to pivot around a vertical axis through the iddle of the rod. Two sall, equal asses, are attached to the rod at a distance fro the rotation axis. The rotor rotates with an angular velocity of 3.0 rad/s. An internal echanis pulls the asses in syetrically until they are each at fro the axis. The angular velocity of the rotor after this ove (in rad/s) is: () () 4 (3) 8 (4) 4 (5) 8 3. A spherical shell is free to rotate about its vertical axis having rotational inertia I = 40 N about that axis. A otor causes a torque that gives the shell a constant angular acceleration of.4 rad/s. In rotating the shell through 5 revolutions the otor does an aount of work (in J) equal to () () (3) (4) (5). 0 3

7 33. A spherical shell is free to rotate about its vertical axis having rotational inertia I = 40 N about that axis. A otor causes a torque that gives the shell a constant angular acceleration of.6 rad/s. In rotating the shell through 5 revolutions the otor does an aount of work (in J) equal to () () (3) (4) (5) A spherical shell is free to rotate about its vertical axis having rotational inertia I = 40 N about that axis. A otor causes a torque that gives the shell a constant angular acceleration of. rad/s. In rotating the shell through 5 revolutions the otor does an aount of work (in J) equal to () () (3) (4) (5) Tarzan, having a ass of 88.0 kg, swings on a 0.0 long vine such that just before getting to the botto of the swing his angular velocity is.40 rad/s. At the botto of the swing he picks up Cheetah (ass of 0.0 kg) who was waiting there at rest. Just after the pick-up their angular velocity in (rad/s) was (hint: treat both Tarzan and Cheetah as point asses): ().4 ().30 (3).47 (4) 0.98 (5) Tarzan, having a ass of 88.0 kg, swings on a 0.0 long vine such that just before getting to the botto of the swing his angular velocity is.60 rad/s. At the botto of the swing he picks up Cheetah (ass of 0.0 kg) who was waiting there at rest. Just after the pick-up their angular velocity in (rad/s) was (hint: treat both Tarzan and Cheetah as point asses): ().30 ().4 (3).47 (4) 0.98 (5) Tarzan, having a ass of 88.0 kg, swings on a 0.0 long vine such that just before getting to the botto of the swing his angular velocity is.80 rad/s. At the botto of the swing he picks up Cheetah (ass of 0.0 kg) who was waiting there at rest. Just after the pick-up their angular velocity in (rad/s) was (hint: treat both Tarzan and Cheetah as point asses): ().47 ().4 (3).30 (4) 0.98 (5) 0.8 FOLLOWING GOUPS OF QUESTIONS WILL BE SELECTED AS ONE GOUP FOM EACH TYPE TYPE Q# S Q# S 3 Q# S 4 TYPE Q# S 6 Q# S 7 TYPE 3 Q# S 8 Q# S 9 TYPE 4 Q# S 4 Q# S 5 TYPE 5 Q# S 6 Q# S 7 TYPE 6 Q# S Q# S Q# S 3 TYPE 7 Q# S 4 Q# S 5 TYPE 8 Q# S 6

8 Q# S 7 Q# S 8 TYPE 9 Q# S 9 Q# S 30 Q# S 3 TYPE 0 Q# S 3 Q# S 33 Q# S 34 TYPE Q# S 35 Q# S 36 Q# S 37

On my honor, I have neither given nor received unauthorized aid on this examination.

Instructor(s): Profs. D. Reitze, H. Chan PHYSICS DEPARTMENT PHY 2053 Exam 2 April 2, 2009 Name (print, last first): Signature: On my honor, I have neither given nor received unauthorized aid on this examination.