1986MI. The figure above shows an 80-kilogram person standing on a 20-kilogram platform suspencled by a rope passing over a stationa that is free to rotate. The other end of the rope is held by the person. The ‘masses of the rope and pulley are negligible. You may use g~ 10 nv s’. Assume that friction is negligible, and the parts of the rope shown remain vertical. Ifthe platform and the person are at rest, what isthe tension in the rope? ‘The person now pulls on the rope so that the acceleration of the person and the platform is 2 nvs? upward. 'b. Whats the tension inthe rope under these new conditions? ©. Under these conditions, what is the force exerted by the platform on the person? 351986MI. The figure above shows an 80-kilogram person standing on a 20-kilogram platform suspencled by a rope passing over a stationa that is free to rotate. The other end of the rope is held by the person. The ‘masses of the rope and pulley are negligible. You may use g~ 10 nv s’. Assume that friction is negligible, and the parts of the rope shown remain vertical. Ifthe platform and the person are at rest, what isthe tension in the rope? ‘The person now pulls on the rope so that the acceleration of the person and the platform is 2 nvs? upward. 'b. Whats the tension inthe rope under these new conditions? ©. Under these conditions, what is the force exerted by the platform on the person? 351998M3. Block 1 of mass m, is placed on block 2 of mass m; which is then placed on a tale. A string connecting a ‘block 2 to « hanging mass M passes over a Pilly attached to one end of the table, as shown above. The mass and fiction ofthe pulley are negligible. The coefficients of fiction between blocks 1 and 2 and between block 2 and the tabletop are nonzero and are given in the following table. Cocficient Between Coofcient Between Blocks 1 and 2 Block 2 andthe Tabletop = Ba Ba Kinetic oa Me Express your answers in terms of the masses, coefficients of friction, and g, the acceleration du to gravity. ‘Suppose thatthe value of M is small enough thatthe blocks remain at rest when released. For each of the following forces, determine the magnitude ofthe force and draw a vector on the block provided to indicate the diteetion ofthe force if itis nonzero, i. Thenormal foree N, exerted on block 1 by block 2 ii, The fiction force f; exerted on block I by block 2 iii, The force T exerted on block 2 by the sting cm iv. The normal foree N; exerted on block 2 by the tabletop ‘¥. The friction force f; exerted on block 2 by the tabletop ‘Determine the largest value of M for which the blocks can remain at rest. "Now suppose that M is large enough that the hanging block descends when the blocks are released. Assume that blocks I and 2 are moving asa unit (no slippage). Determine the magnitude a oftheir acceleration. [Now suppose that M is large enough that asthe hanging block descends, block 1 is slipping on block 2. ‘Determine each of the following. i, The magnitude a, of te acceleration of block 1 ii, The magnitude a of the acceleration of block 22009n3 ‘A Dock of mass M/2 rests ona frictionless horizontal table, as shown above. It is connected to one end of a String tha passes over a massless MY and has another block of mass M/2 hanging from its other end, The apparatus is released from rest a Derive an expression forthe speed v, ofthe hanging block as a function ofthe distance dit descends ‘Now the block and pulley system is replaced by a uniform rope of length L and mass M, with one end of the rope hanging slighlly over the edge ofthe frictionless table. The rope is released rom ret, and at sometime Jntr theres length yof rope hanging over the edge, a shown below. Express your answers to parts (b),(), and () in terms oy, , M, and fundamental constants 7 i Determine an expression for the force of gravity onthe hanging part ofthe rope asa function of. Derive an expression for the work done by gravity on the rope as a function of y, assuming y is initially zero. Derive an expression forthe speed, ofthe rope asa finction of ‘The hanging block and the right end ofthe rope are each allowed to fall a distance L(the length of the rope). The string is long enough thatthe sliding block doesnot ht the pulley. Tndicate whether vy ftom pat (2) or fiom part (is greater afer the block and the end of the rope have traveled this distance we vei greater. ys greater, The speeds are equal. Justify your answer,a2 1991M2. Two masses. my and m; are connected by light cables tothe perimeters of two cylinders of rir, and >, respectively. as shown in the diagram above. The cylinders are rigidly connected to each other but are free to rotate without friction on a common axle. The moment of inertia of the pair of eylinders is T= 45 kgern’ Also, =05 meter,r; = 1.5 meters, and m= 20 kilograms. a. Determine m, such thatthe system will remain in equilibrium. ‘The mass m; is removed and the system is released from rest. bb, Determine the angular acceleration ofthe eylinders, ce. Determine the tension i the cable supporting m, 4d. Determine the linear speed of m ja the ime ithas descended 1.0 meter.passing over tho pulley has two blocks of mass m attached to either end, as shown above. Assume thatthe cord does not slip on the pulley. Determine the answers to parts a. and b. in terms of m, Ry, and fundamental constants. ‘a. Determine the tension T inthe cord. One block is now removed from the right and hung on the left, When the system is eleased from res, the three blocks on the left accelerate downward with an acceleration g/3 . Determine the following. i. The tension T; in the section of cord supporting the three blocks on the left ii, The tension; inthe section of cord supporting the single block on the right Sil, The rotational inertia, ofthe pulley ¢. The blocks are now removed and the cor is ied into a loop, whichis passed around the original pulley and second ply of radius 2; an rotational inertia 16. Te ai ofthe orginal lly altace oa mcr Gta oats it ot angular speedo, which intr cases the larger Bll orotate. The oop doe not lip on the ploy. Determine the following in terms of, Ry and {The angle speed ofthe lp ley ii, The angular momentum L. of the larger pulley ii, Te toul Kinai energy othe systeSECTION A ear Dynamics L-coaterwmiht ls 1976B1. The two guide rails for the elevator shown above each exert a constant fiction force of 100 newtons on the ‘car when the clevator car is moving upward with an acceleration of 2 meters per second squared. The has negligible friction and mass. Assume g = 10 m/sec’ On the diagram below, draw and label all frees acting on the elevator car. Identify the source of each fore. '. Caleulate the tension in the cable lifting the 400-kilogram elevator car dusing an upward acceleration of 2 asec’. (Assume g 10 msec") «Calculate the mass M the counterweight must have to raise the elevator car with an acceleration of 2 misec* ots eet yoo perry Eee'l amt 197982. A 10ilogrm bios tently ona table a shown in eases and T above. The coticient of sing friction between the block and the table is 0.2. The block is connected to a cord of negligible mass, which tangs over a masses fictions pull). tn ese Ta fore of 30 nevions is applied tothe cord. Incase Tl cbjectof mass ilgrams is ung on the bottom ofthe cord Use g = 10 meters per scond saad Calculate the acceleration of the 10-kilogram block in case I. 1b. On the diagrams below, draw and label all the forces acting on each block in case kg Ske Calculate the ucceleration ofthe 10-kilogram block in ease I.1985B2 (modified) Two 10-kilogram boxes are connected by 2 massless string that passes over a massless b Peed ain nga iene er bengc l Se the one onthe left 2.0 meters frm te bottom of an inclined plane that makes an angle of 60° with he horizontal, The coecients of kinetic fiction and static friction between te left-hand box and the plane ae 015 and 030, respectively. You may use g= 10 ms, sin 60° 0-87, and cos 60° ~ 0.50 What is the tension T in the string? (nthe diagram below, draw and label all he frees acting onthe box thats on the plane Determine the magnitude of the frictional force aeting on the box on the plane.20ke 40kg rox) 1986B1, Three blocks of masses 1.0, 2.0, and 4.0 kilograms are connected by massless strings, one of which passes. ‘overa fictiones pulley of negligible mast, as shown above. Calculate each ofthe following, The acceleration ofthe kilogram bloc 1 The tension inthe suing supporting the 4-kilogram block ©) The tension inthe sting connected tothe kilogram block FI M 1987B1, In the system shown above, the block of mass M, is on a rough horizontal table. The string that attaches it to the block of mass M; passes over a trictionless pile of negligible mass. The coefficient of kinetic friction Hy between M, and the table is less than the eoefTicient of stati friction i, On the diagram below, draw and identify al the forees acting on the block of mass My. be] ._ Interms of M, and M; determine the minimum value of p, that will prevent the blocks from moving ‘The blocks are set in motion by giving M, a momentary downward push, In terms of M,, Ms, jl, and 2, determine each ofthe following: ‘¢ The magnitude ofthe acceleration of My dd, The tension inthe string.200082. Block 1 and2 of masses m and ms respectively, reconnected by alight string as shown above. These ‘locks are farther connected to block of mass M by another ight string that passes over a ple of neil imass and fiction. Blocks | and ? move with «constant velocity v dawn the inclined plane, which makes oh angle 6 withthe horizontal. The kinetic frictional force on block 1 is nd that on block 282. 4. Onthe figure below, draw and abe all dhe foes on block mn, Express your answers to each ofthe following in terms of my ma, 2,8, and f. 'b. Determine the coefficient of kinetic friction between the inclined plane and block | cc. Determine the value of the suspended mass M that allows blocks | and 2 to move with constant velocity down, the plane 4d, The string between blocks 1 and 2is now eut, Determine the accoleration of block | while itis onthe inclined planeStudent B Oke Student A "Tokg 200381 A rope of negligible mass pases overall of negligible mass attached tothe citing as shown above. ‘One end ofthe rope isheld by Stdent A of ass 7D kg, whois atest on the floor. The opposite end ofthe rope is held by Student B of mass 60 kg, who is suspended at rest above the floor. Use g = 10 mis? ‘4 Onte dot below that represent the students, drow and label fee-body diagrams showing the forces on Student ‘$end on Student B eB oA 'b. Calculate the magnitude of the force exerted by the floor on Student A. Student B now climbs up the ope at a constant acceleration of 0.25 mus with respect to the floor. Calculate the tension in the rope while Student B is accelerating. |As Student Bis accelerating, is Student A pulled upward off the floor? Justify your answer, & With what minimum acceleration must Student B climb up the rope to lft Student A upward off the floor?1998M3. Block 1 of mass m, is placed on block 2 of mass m; which is then placed on a tale. A string connecting a ‘block 2 to « hanging mass M passes over a Pilly attached to one end of the table, as shown above. The mass and fiction ofthe pulley are negligible. The coefficients of fiction between blocks 1 and 2 and between block 2 and the tabletop are nonzero and are given in the following table. Cocficient Between Coofcient Between Blocks 1 and 2 Block 2 andthe Tabletop = Ba Ba Kinetic oa Me Express your answers in terms of the masses, coefficients of friction, and g, the acceleration du to gravity. ‘Suppose thatthe value of M is small enough thatthe blocks remain at rest when released. For each of the following forces, determine the magnitude ofthe force and draw a vector on the block provided to indicate the diteetion ofthe force if itis nonzero, i. Thenormal foree N, exerted on block 1 by block 2 ii, The fiction force f; exerted on block I by block 2 iii, The force T exerted on block 2 by the sting cm iv. The normal foree N; exerted on block 2 by the tabletop ‘¥. The friction force f; exerted on block 2 by the tabletop ‘Determine the largest value of M for which the blocks can remain at rest. "Now suppose that M is large enough that the hanging block descends when the blocks are released. Assume that blocks I and 2 are moving asa unit (no slippage). Determine the magnitude a oftheir acceleration. [Now suppose that M is large enough that asthe hanging block descends, block 1 is slipping on block 2. ‘Determine each of the following. i, The magnitude a, of te acceleration of block 1 ii, The magnitude a of the acceleration of block 22009n3 ‘A Dock of mass M/2 rests ona frictionless horizontal table, as shown above. It is connected to one end of a String tha passes over a massless MY and has another block of mass M/2 hanging from its other end, The apparatus is released from rest a Derive an expression forthe speed v, ofthe hanging block as a function ofthe distance dit descends ‘Now the block and pulley system is replaced by a uniform rope of length L and mass M, with one end of the rope hanging slighlly over the edge ofthe frictionless table. The rope is released rom ret, and at sometime Jntr theres length yof rope hanging over the edge, a shown below. Express your answers to parts (b),(), and () in terms oy, , M, and fundamental constants 7 i Determine an expression for the force of gravity onthe hanging part ofthe rope asa function of. Derive an expression for the work done by gravity on the rope as a function of y, assuming y is initially zero. Derive an expression forthe speed, ofthe rope asa finction of ‘The hanging block and the right end ofthe rope are each allowed to fall a distance L(the length of the rope). The string is long enough thatthe sliding block doesnot ht the pulley. Tndicate whether vy ftom pat (2) or fiom part (is greater afer the block and the end of the rope have traveled this distance we vei greater. ys greater, The speeds are equal. Justify your answer,a2 1991M2. Two masses. my and m; are connected by light cables tothe perimeters of two cylinders of rir, and >, respectively. as shown in the diagram above. The cylinders are rigidly connected to each other but are free to rotate without friction on a common axle. The moment of inertia of the pair of eylinders is T= 45 kgern’ Also, =05 meter,r; = 1.5 meters, and m= 20 kilograms. a. Determine m, such thatthe system will remain in equilibrium. ‘The mass m; is removed and the system is released from rest. bb, Determine the angular acceleration ofthe eylinders, ce. Determine the tension i the cable supporting m, 4d. Determine the linear speed of m ja the ime ithas descended 1.0 meter.passing over tho pulley has two blocks of mass m attached to either end, as shown above. Assume thatthe cord does not slip on the pulley. Determine the answers to parts a. and b. in terms of m, Ry, and fundamental constants. ‘a. Determine the tension T inthe cord. One block is now removed from the right and hung on the left, When the system is eleased from res, the three blocks on the left accelerate downward with an acceleration g/3 . Determine the following. i. The tension T; in the section of cord supporting the three blocks on the left ii, The tension; inthe section of cord supporting the single block on the right Sil, The rotational inertia, ofthe pulley ¢. The blocks are now removed and the cor is ied into a loop, whichis passed around the original pulley and second ply of radius 2; an rotational inertia 16. Te ai ofthe orginal lly altace oa mcr Gta oats it ot angular speedo, which intr cases the larger Bll orotate. The oop doe not lip on the ploy. Determine the following in terms of, Ry and {The angle speed ofthe lp ley ii, The angular momentum L. of the larger pulley ii, Te toul Kinai energy othe systeSECTION A ear Dynamics L-coaterwmiht ls 1976B1. The two guide rails for the elevator shown above each exert a constant fiction force of 100 newtons on the ‘car when the clevator car is moving upward with an acceleration of 2 meters per second squared. The has negligible friction and mass. Assume g = 10 m/sec’ On the diagram below, draw and label all frees acting on the elevator car. Identify the source of each fore. '. Caleulate the tension in the cable lifting the 400-kilogram elevator car dusing an upward acceleration of 2 asec’. (Assume g 10 msec") «Calculate the mass M the counterweight must have to raise the elevator car with an acceleration of 2 misec* ots eet yoo perry Eee'l amt 197982. A 10ilogrm bios tently ona table a shown in eases and T above. The coticient of sing friction between the block and the table is 0.2. The block is connected to a cord of negligible mass, which tangs over a masses fictions pull). tn ese Ta fore of 30 nevions is applied tothe cord. Incase Tl cbjectof mass ilgrams is ung on the bottom ofthe cord Use g = 10 meters per scond saad Calculate the acceleration of the 10-kilogram block in case I. 1b. On the diagrams below, draw and label all the forces acting on each block in case kg Ske Calculate the ucceleration ofthe 10-kilogram block in ease I.1985B2 (modified) Two 10-kilogram boxes are connected by 2 massless string that passes over a massless b Peed ain nga iene er bengc l Se the one onthe left 2.0 meters frm te bottom of an inclined plane that makes an angle of 60° with he horizontal, The coecients of kinetic fiction and static friction between te left-hand box and the plane ae 015 and 030, respectively. You may use g= 10 ms, sin 60° 0-87, and cos 60° ~ 0.50 What is the tension T in the string? (nthe diagram below, draw and label all he frees acting onthe box thats on the plane Determine the magnitude of the frictional force aeting on the box on the plane.20ke 40kg rox) 1986B1, Three blocks of masses 1.0, 2.0, and 4.0 kilograms are connected by massless strings, one of which passes. ‘overa fictiones pulley of negligible mast, as shown above. Calculate each ofthe following, The acceleration ofthe kilogram bloc 1 The tension inthe suing supporting the 4-kilogram block ©) The tension inthe sting connected tothe kilogram block FI M 1987B1, In the system shown above, the block of mass M, is on a rough horizontal table. The string that attaches it to the block of mass M; passes over a trictionless pile of negligible mass. The coefficient of kinetic friction Hy between M, and the table is less than the eoefTicient of stati friction i, On the diagram below, draw and identify al the forees acting on the block of mass My. be] ._ Interms of M, and M; determine the minimum value of p, that will prevent the blocks from moving ‘The blocks are set in motion by giving M, a momentary downward push, In terms of M,, Ms, jl, and 2, determine each ofthe following: ‘¢ The magnitude ofthe acceleration of My dd, The tension inthe string.200082. Block 1 and2 of masses m and ms respectively, reconnected by alight string as shown above. These ‘locks are farther connected to block of mass M by another ight string that passes over a ple of neil imass and fiction. Blocks | and ? move with «constant velocity v dawn the inclined plane, which makes oh angle 6 withthe horizontal. The kinetic frictional force on block 1 is nd that on block 282. 4. Onthe figure below, draw and abe all dhe foes on block mn, Express your answers to each ofthe following in terms of my ma, 2,8, and f. 'b. Determine the coefficient of kinetic friction between the inclined plane and block | cc. Determine the value of the suspended mass M that allows blocks | and 2 to move with constant velocity down, the plane 4d, The string between blocks 1 and 2is now eut, Determine the accoleration of block | while itis onthe inclined planeStudent B Oke Student A "Tokg 200381 A rope of negligible mass pases overall of negligible mass attached tothe citing as shown above. ‘One end ofthe rope isheld by Stdent A of ass 7D kg, whois atest on the floor. The opposite end ofthe rope is held by Student B of mass 60 kg, who is suspended at rest above the floor. Use g = 10 mis? ‘4 Onte dot below that represent the students, drow and label fee-body diagrams showing the forces on Student ‘$end on Student B eB oA 'b. Calculate the magnitude of the force exerted by the floor on Student A. Student B now climbs up the ope at a constant acceleration of 0.25 mus with respect to the floor. Calculate the tension in the rope while Student B is accelerating. |As Student Bis accelerating, is Student A pulled upward off the floor? Justify your answer, & With what minimum acceleration must Student B climb up the rope to lft Student A upward off the floor?