# EQUILIBRIUM AND ELASTICITY

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1 Chapter 12: EQUILIBRIUM AND ELASTICITY 1 A net torque applied to a rigid object always tends to produce: A linear acceleration B rotational equilibrium C angular acceleration D rotational inertia E none of these Section: 12{2; Di±culty: E 2 The conditions that the net force and the net torque both vanish: A hold for every rigid body in equilibrium B hold only for elastic solid bodies in equilibrium C hold for every solid body D are always su±cient to calculate the forces on a solid object in equilibrium E are su±cient to calculate the forces on a solid object in equilibrium only if the object is elastic ans: A Section: 12{3; Di±culty: E 3 For an object in equilibrium the net torque acting on it vanishes only if each torque is calculated about: A the center of mass B the center of gravity C the geometrical center D the point of application of the force E the same point ans: E Section: 12{3; Di±culty: E 4 For a body to be in equilibrium under the combined action of several forces: A all the forces must be applied at the same point B all of the forces form pairs of equal and opposite forces C thesumofthecomponentsofalltheforcesinanydirectionmustequalzero D any two of these forces must be balanced by a third force E the lines of action of all the forces must pass through the center of gravity of the body Section: 12{3; Di±culty: E 5 For a body to be in equilibrium under the combined action of several forces: A all the forces must be applied at the same point B all of the forces form pairs of equal and opposite forces C any two of these forces must be balanced by a third force D thesumofthetorquesaboutanypointmustequalzero E the lines of action of all the forces must pass through the center of gravity of the body Section: 12{3; Di±culty: E Chapter 12: EQUILIBRIUM AND ELASTICITY 199

2 6 A meter stick on a horizontal frictionless table top is pivoted at the 80-cm mark A force ~ F 1 is applied perpendicularly to the end of the stick at 0 cm, as shown A second force ~ F 2 (not shown) is applied perpendicularly at the 100-cm end of the stick The forces are horizontal If the stick does not move, the force exerted by the pivot on the stick: ~F 1 pivot 0cm 20 cm 40 cm 60 cm 80 cm 100 cm A must be zero B must be in the same direction as F ~ 1 and have magnitude j F ~ 2 j j F ~ 1 j C must be directed opposite to F ~ 1 and have magnitude j F ~ 2 j j F ~ 1 j D must be in the same direction as F ~ 1 and have magnitude j F ~ 2 j + j F ~ 1 j E must be directed opposite to F ~ 1 and have magnitude j F ~ 2 j + j F ~ 1 j ans: E Section: 12{3; Di±culty: M 7 A meter stick on a horizontal frictionless table top is pivoted at the 80-cm mark A force ~ F 1 is applied perpendicularly to the end of the stick at 0 cm, as shown A second force ~ F 2 (not shown) is applied perpendicularly at the 60-cm mark The forces are horizontal If the stick does not move, the force exerted by the pivot on the stick: ~F 1 pivot 0cm 20 cm 40 cm 60 cm 80 cm 100 cm A must be zero B must be in the same direction as F ~ 1 and have magnitude j F ~ 2 j j F ~ 1 j C must be directed opposite to F ~ 1 and have magnitude j F ~ 2 j j F ~ 1 j D must be in the same direction as F ~ 1 and have magnitude j F ~ 2 j + j F ~ 1 j E must be directed opposite to F ~ 1 and have magnitude j F ~ 2 j + j F ~ 1 j Section: 12{3; Di±culty: M 200 Chapter 12: EQUILIBRIUM AND ELASTICITY

3 8 Three identical uniform rods are each acted on by two or more forces, all perpendicular to the rods and all equal in magnitude Which of the rods could be in static equilibrium if an additional force is applied at the center of mass of the rod? A Only 1 B Only 2 C Only 3 D Only 1 and 2 E All three Section: 12{3; Di±culty: M A 160-N child sits on a light swing and is pulled back and held with a horizontal force of 100 N Themagnitudeofthetensionforceofeachofthetwosupportingropesis: A 60 N B 94 N C 120 N D 190 N E 260 N Section: 12{3; Di±culty: M 10 To determine if a rigid body is in equilibrium the vector sum of the gravitational forces acting on the particles of the body can be replaced by a single force acting at: A the center of mass B the geometrical center C the center of gravity D a point on the boundary E none of the above Section: 12{4; Di±culty: E 11 The center of gravity coincides with the center of mass: A always B never C if the center of mass is at the geometrical center of the body D if the acceleration due to gravity is uniform over the body E if the body has a uniform distribution of mass Section: 12{4; Di±culty: E Chapter 12: EQUILIBRIUM AND ELASTICITY 201

4 12 The location of which of the following points within an object might depend on the orientation of the object? A Its center of mass B Its center of gravity C Its geometrical center D Its center of momentum E None of the above Section: 12{4; Di±culty: E 13 A cylinder placed so it can roll on a horizontal table top, with its center of gravity above its geometrical center, is: A in stable equilibrium B in unstable equilibrium C in neutral equilibrium D not in equilibrium E none of the above Section: 12{2, 4; Di±culty: E 14 A cylinder placed so it can roll on a horizontal table top, with its center of gravity below its geometrical center, is: A in stable equilibrium B in unstable equilibrium C in neutral equilibrium D not in equilibrium E none of the above ans: A Section: 12{2, 4; Di±culty: E 15 A cube balanced with one edge in contact with a table top and with its center of gravity directly above the edge is in equilibrium with respect to rotation about the edge and in equilibrium with respect to rotation about a horizontal axis that is perpendicular to the edge A stable, stable B stable, unstable C unstable, stable D unstable, unstable E unstable, neutral Section: 12{2, 4; Di±culty: E 202 Chapter 12: EQUILIBRIUM AND ELASTICITY

5 16 The diagram shows a stationary 5-kg uniform rod (AC), 1 m long, held against a wall by a rope (AE) and friction between the rod and the wall To use a single equation to nd the forceexertedontherodbytheropeatwhichpointshouldyouplacethereferencepointfor computing torque? A B C D E Section: 12{5; Di±culty: E 17 A picture P of weight W is hung by two strings as shown The magnitude of the tension force of each string is T The total upward pull of the strings on the picture is: P µ µ T T A 2W cos µ B T sin µ C T cos µ D 2T sin µ E 2T cos µ Chapter 12: EQUILIBRIUM AND ELASTICITY 203

6 18 A picture can be hung on a wall with string in three di erent ways, as shown The magnitude of the tension force of the string is: I II III A least in I B greatest in I C greatest in II D least in III E greatest in III ans: E 19 A uniform plank is supported by two equal 120-N forces at X and Y, as shown The support at X is then moved to Z (half-way to the plank center) The supporting forces at Y and Z are then: X Z Y A F Y =240N,F Z =120N B F Y =200N,F Z =40N C F Y =40N,F Z =200N D F Y =80N,F Z =160N E F Y =160N,F Z =80N 204 Chapter 12: EQUILIBRIUM AND ELASTICITY

7 20 A uniform rod AB is 1:2mlongandweighs16N ItissuspendedbystringsACandBDas shown A block P weighing 96 N is attached at E, 0:30mfromAThemagnitudeofthetension force of the string BD is: C D A E B P A 8:0N B 24 N C 32 N D 48 N E 80 N 21 A 5:0-m weightless strut, hinged to a wall, is used to support an 800-N block as shown The horizontal and vertical components of the force of the hinge on the strut are: hinge 3m 800 N A F H =800N,F Y =800N B F H =600N,F Y =800N C F H =800N,F Y =600N D F H =1200N,F Y =800N E F H =0,F Y = 800 N Chapter 12: EQUILIBRIUM AND ELASTICITY 205

8 22 A uniform plank is 6:0 m long and weighs 80 N It is balanced on a sawhorse at its center An additional 160 N weight is now placed on the left end of the plank To keep the plank balanced, it must be moved what distance to the left? A 6:0m B 2:0m C 1:5m D 1:0m E 0:50 m 23 A uniform 240-g meter stick can be balanced by a 240-g weight placed at the 100-cm mark if the fulcrum is placed at the point marked: A 75 cm B 60 cm C 50 cm D 40 cm E 80 cm ans: A 24 A ladder leans against a wall If the ladder is not to slip, which one of the following must be true? A The coe±cient of friction between the ladder and the wall must not be zero B The coe±cient of friction between the ladder and the oor must not be zero C Both A and B D Either A or B E Neither A nor B 206 Chapter 12: EQUILIBRIUM AND ELASTICITY

9 25 An 80-N uniform plank leans against a frictionless wall as shown The magnitude of the torque (about point P) applied to the plank by the wall is: A 40 N m B 60 N m C 120 N m D 160 N m E 240 N m P Ã 3m! 26 An 800-N man stands halfway up a 5:0-m long ladder of negligible weight The base of the ladder is 3:0 m from the wall as shown Assuming that the wall-ladder contact is frictionless, thewallpushesagainsttheladderwithaforceofmagnitude: " jjj 4m j # j 3m A 150 N B 300 N C 400 N D 600 N E 800 N Chapter 12: EQUILIBRIUM AND ELASTICITY 207

11 29 The 600-N ball shown is suspended on a string AB and rests against a frictionless vertical wall The string makes an angle of 30 ± withthewall Themagnitudeofthetensionforceofthe string is: 600 N 30 ± B A A 690 N B 1200 N C 2100 N D 2400 N E none of these ans: A 30 The 600-N ball shown is suspended on a string AB and rests against a frictionless vertical wall The string makes an angle of 30 ± with the wall The ball presses against the wall with a force of mangitude: 600 N 30 ± B A A 120 N B 300 N C 350 N D 600 N E 690 N Chapter 12: EQUILIBRIUM AND ELASTICITY 209

12 31 The uniform rod shown below is held in place by the rope and wall Suppose you know the weight of the rod and all dimensions Then you can solve a single equation for the force of the rope on the rod, provided you write expressions for the torques about the point: A 1 B 2 C 3 D 4 E 1, 2, or 3 32 A 240-N weight is hung from two ropes as shown The tension force of the horizontal rope has magnitude: 30 ± 240 N A 0 B 656 N C 480 N D 416 N E 176 N 210 Chapter 12: EQUILIBRIUM AND ELASTICITY

13 33 A 960-N block is suspended as shown The beam AB is weightless and is hinged to the wall at A The tension force of the cable BC has magnitude: " j 3m j # A C B Ã 4m! 960 N A 720 N B 1200 N C 1280 N D 1600 N E none of these 34 A horizontal beam of weight W is supported by a hinge and cable as shown The force exerted on the beam by the hinge has a vertical component that must be: hinge cable W A nonzero and up B nonzero and down C nonzero but not enough information given to know whether up or down D zero E equal to W ans: A Chapter 12: EQUILIBRIUM AND ELASTICITY 211

14 35 A 400-N uniform vertical boom is attached to the ceiling by a hinge, as shown An 800-N weight W and a horizontal guy wire are attached to the lower end of the boom as indicated The pulley is massless and frictionless The tension force T of the horizontal guy wire has magnitude: pulley hinge boom (400 N) W =800N 60 ± guy wire A 340 N B 400 N C 690 N D 800 N E 1200 N 36 A picture is to be hung from the ceiling by means of two wires Order the following arrangements of the wires according to the tension force of wire B, from least to greatest A B A B A B cm cm cm A I, II, III B III, II, I C I and II tie, then III D II, I, III E all tie Section: 12{5; Di±culty: E I II III 212 Chapter 12: EQUILIBRIUM AND ELASTICITY

15 37 The pull P is just su±cient to keep the 14-N block and the weightless pulleys in equilibrium as shown The magnitude T of the tension force of the upper cable is: T 14 N P A 14 N B 28 N C 16 N D 9:33 N E 18:7N Section: 12{5; Di±culty: E 38 The ideal mechanical advantage (ie the ratio of the weight W to the pull P for equilibrium) of the combination of pulleys shown is: P A 1 B 2 C 3 D 4 E 5 Section: 12{5; Di±culty: E W Chapter 12: EQUILIBRIUM AND ELASTICITY 213

16 39 Stress can be measured in: A N/m 2 B N m 2 C N/m D N m E none of these (it is unitless) ans: A Section: 12{7; Di±culty: E 40 Strain can be measured in: A N/m 2 B N m 2 C N/m D N m E none of these (it is unitless) ans: E Section: 12{7; Di±culty: E 41 Young's modulus can be correctly given in: A N m B N/m 2 C N m/s D N/m E joules Section: 12{7; Di±culty: E 42 Young's modulus is a proportionality constant that relates the force per unit area applied perpendicularly at the surface of an object to: A the shear B the fractional change in volume C the fractional change in length D the pressure E the spring constant Section: 12{7; Di±culty: E 43 Young's modulus can be used to calculate the strain for a stress that is: A just below the ultimate strength B just above the ultimate strength C well below the yield strength D well above the yield strength E none of the above Section: 12{7; Di±culty: E 214 Chapter 12: EQUILIBRIUM AND ELASTICITY

17 44 The ultimate strength of a sample is the stress at which the sample: A returns to its original shape when the stress is removed B remains underwater C breaks D bends 180 ± E does none of these Section: 12{7; Di±culty: E 45 A certain wire stretches 0:90 cm when outward forces with magnitude F are applied to each end The same forces are applied to a wire of the same material but with three times the diameter and three times the length The second wire stretches: A 0:10 cm B 0:30 cm C 0:90 cm D 2:7cm E 8:1cm Section: 12{7; Di±culty: M 46 A force of 5000 N is applied outwardly to each end of a 5:0-m long rod with a radius of 34:0cm and a Young's modulus of N=m 2 The elongation of the rod is: A 0:0020 mm B 0:0040 mm C 0:14 mm D 0:55 mm E 1:42 mm Section: 12{7; Di±culty: M 47 A 4:0-m long steel beam with a cross-sectional area of 1: m 2 and a Young's modulus of 2: N=m 2 is wedged horizontally between two vertical walls In order to wedge the beam, it is compressed by 0:020 mm If the coe±cient of static friction between the beam and the walls is 0:70 the maximum mass (including its own) it can bear without slipping is: A 0 B 3:6kg C 36 kg D 71 kg E 710 kg ans: E Section: 12{7; Di±culty: M Chapter 12: EQUILIBRIUM AND ELASTICITY 215

18 48 Two supports, made of the same material and initially of equal length, are 2:0 m apart A sti boardwithalengthof4:0m and a mass of 10 kg is placed on the supports, with one support at the left end and the other at the midpoint A block is placed on the board a distance of 0:50 m from the left end As a result the board is horizontal The mass of the block is: A zero B 2:3kg C 6:6kg D 10 kg E 20 kg ans: E Section: 12{7; Di±culty: H 49 The bulk modulus is a proportionality constant that relates the pressure acting on an object to: A the shear B the fractional change in volume C the fractional change in length D Young's modulus E the spring constant Section: 12{7; Di±culty: E 50 A cube with edges exactly 2 cm long is made of material with a bulk modulus of 3: N=m 2 When it is subjected to a pressure of 3: Pa its volume is: A 7:31 cm 3 B 7:99931 cm 3 C 8:00069 cm 3 D 8:69 cm 3 E none of these Section: 12{7; Di±culty: E 51 A cube with 2:0-cm sides is made of material with a bulk modulus of 4: N=m 2 When it is subjected to a pressure of 2: Pa the length of its any of its sides is: A 0:85 cm B 1:15 cm C 1:66 cm D 2:0cm E none of these Section: 12{7; Di±culty: M 216 Chapter 12: EQUILIBRIUM AND ELASTICITY

19 52 To shear a cube-shaped object, forces of equal magnitude and opposite directions might be applied: A to opposite faces, perpendicular to the faces B to opposite faces, parallel to the faces C to adjacent faces, perpendicular to the faces D to adjacent faces, neither parallel or perpendicular to the faces E to a single face, in any direction Section: 12{7; Di±culty: E 53 A shearing force of 50 N is applied to an aluminum rod with a length of 10 m, a cross-sectional area of 1: m, and a shear modulus of 2: N=m 2 As a result the rod is sheared through a distance of: A zero B 1:9mm C 1:9cm D 19 cm E 1:9m Section: 12{7; Di±culty: M Chapter 12: EQUILIBRIUM AND ELASTICITY 217

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