Deck 10: Equilibrium and Elasticity

Question 1

A 3.00-kg ball rests in a frictionless groove as shown in the figure.
(a) What is the magnitude of the force that the left side of the groove exerts on the ball?
(b) What is the magnitude of the force that the right side of the groove exerts on the ball?

Answer 1

(a) 26.4 N
(b) 21.5 N

Question 2

The graph in the figure shows the force on an object as a function of the elongation caused by that force. Which statement about this object is true?

A) The object obeys Hooke's law at all points from A to C.
B) The object obeys Hooke's law at all points from B to C.
C) The object obeys Hooke's law at all points from A to B.
D) The elastic limit occurs at point C.
E) The region of elastic behavior occurs from B to C.

Answer 2

The object obeys Hooke's law at all points from A to B.

Question 3

Two identical ladders are 3.0 m long and weigh 600 N each. They are connected by a hinge at the top and are held together by a horizontal rope, 1.0 m above the smooth floor forming a symmetric "A" arrangement. The angle between the ladders is 60° and both ladders have their center of gravity at their midpoint. What is the tension in the rope?

A) 240 N
B) 300 N
C) 220 N
D) 260 N
E) 280 N

Answer 3

280 N

Question 4

A 120-kg refrigerator, 2.00 m tall and 85.0 cm wide, has its center of mass at its geometrical center. You are attempting to slide it along the floor by pushing horizontally on the side of the refrigerator. The coefficient of static friction between the floor and the refrigerator is 0.300. Depending on where you push, the refrigerator may start to tip over before it starts to slide along the floor. What is the highest distance above the floor that you can push the refrigerator so that it won't tip before it begins to slide?

A) 0.710 m
B) 1.00 m
C) 1.21 m
D) 1.42 m
E) 1.63 m

Question 5

A 30.0-kg child sits on one end of a long uniform beam having a mass of 20.0 kg, and a 40.0-kg child sits on the other end. The beam balances when a fulcrum is placed below the beam a distance 1.10 m from the 30.0-kg child. How long is the beam?

A) 2.12 m
B) 1.98 m
C) 1.93 m
D) 2.07 m
E) 2.20 m

Question 6

Two compressible solids are formed into spheres of the same size. The bulk modulus of sphere two is twice as large as the bulk modulus of sphere one. You now increase the pressure on both spheres by the same amount. As a result of the increased pressure, how is the change in volume of sphere two (△V₂) related to the change in volume of sphere one (ΔV₁)?

A) ΔV₂ = 2ΔV₁
B) ΔV₂ = 4ΔV₁
C) vV₂ = ΔV₁
D) ΔV₂ = 1/2ΔV₁
E) ΔV₂ = 1/4ΔV₁

Question 7

In the figure, the horizontal lower arm has a mass of 2.8 kg and its center of gravity is 12 cm from the elbow joint pivot. How much force FM must the vertical extensor muscle in the upper arm exert on the lower arm to hold a 7.5 kg shot put?

A) 100 N
B) 500 N
C) 750 N
D) 1000 N
E) 1500 N

Question 8

A stepladder consists of two halves, hinged at the top, and connected by a tie rod that keeps the two halves from spreading apart. In this particular instance, the two halves are 2.50 m long, the tie rod is connected to the center of each half and is 70.0 cm long. An 800-N person stands 3/5 of the way up the stepladder, as shown in the figure. Neglecting the weight of the ladder, and assuming that the ladder is resting on a smooth floor, what is the tension in the tie rod? Note: To solve this problem you must "cut" the ladder in half and consider the equilibrium of forces and torques acting on each half of the ladder.

A) 140 N
B) 240 N
C) 280 N
D) 360 N
E) 560 N

Question 9

An 82.0 kg-diver stands at the edge of a light 5.00-m diving board, which is supported by two narrow pillars 1.60 m apart, as shown in the figure. Find the magnitude and direction of the force exerted on the diving board
(a) by pillar A.
(b) by pillar B.

Question 10

A 5.0-m long, 12-kg uniform ladder rests against a smooth vertical wall with the bottom of the ladder 3.0 m from the wall. The coefficient of static friction between the floor and the ladder is 0.28. What distance, measured along the ladder from the bottom, can a 60-kg person climb before the ladder starts to slip?

A) 4.0 m
B) 3.7 m
C) 1.7 m
D) 1.3 m
E) 3.3 m

Question 11

If the torque on an object adds up to zero

A) the forces on it also add up to zero.
B) the object is at rest.
C) the object cannot be turning.
D) the object could be accelerating linearly but it could not be turning.
E) the object could be both turning and accelerating linearly.

Question 12

A 20.0-kg uniform plank is supported by the floor at one end and by a vertical rope at the other as shown in the figure. A 50.0-kg mass person stands on the plank a distance three-fourths of the length plank from the end on the floor.
(a) What is the tension in the rope?
(b) What is the magnitude of the force that the floor exerts on the plank?

Question 13

A child is trying to stack two uniform wooden blocks, 12 cm in length, so they will protrude as much as possible over the edge of a table, without tipping over, as shown in the figure. What is the maximum possible overhang distance d?

A) 5 cm
B) 6 cm
C) 7 cm
D) 8 cm
E) 9 cm

Question 14

A uniform sign is supported against a wall at point P as shown in the figure. If the sign is a square 0.40 m on a side and its mass is 4.0 kg, what is the magnitude of the horizontal force that the wall at P experiences?

A) 20 N
B) 0.00 N
C) 7.8 N
D) 98 N

Question 15

A nonuniform, 80.0-g, meterstick balances when the support is placed at the 51.0-cm mark. At what location on the meterstick should a 5.00-g tack be placed so that the stick will balance at the 50.0 cm mark?

A) 16.0 cm
B) 67.0 cm
C) 66.0 cm
D) 35.0 cm
E) 34.0 cm

Question 16

A light board, 10 m long, is supported by two sawhorses, one at one edge of the board and a second at the midpoint. A small 40-N weight is placed between the two sawhorses, 3.0 m from the edge and 2.0 m from the center. What forces are exerted by the sawhorses on the board?

Question 17

Tensile stress is

A) the strain per unit length.
B) the same as force.
C) the ratio of the change in length to the original length.
D) the applied force per cross-sectional area.
E) the ratio of elastic modulus to strain.

Question 18

Tensile train is

A) the ratio of the change in length to the original length.
B) the stress per unit area.
C) the same as force.
D) the applied force per cross-sectional area.
E) the ratio of stress to elastic modulus.

Question 19

A heavy boy and a lightweight girl are balanced on a massless seesaw. If they both move forward so that they are one-half their original distance from the pivot point, what will happen to the seesaw? Assume that both people are small enough compared to the length of the seesaw to be thought of as point masses.

A) It is impossible to say without knowing the masses.
B) It is impossible to say without knowing the distances.
C) The side the boy is sitting on will tilt downward.
D) Nothing will happen; the seesaw will still be balanced.
E) The side the girl is sitting on will tilt downward.

Question 20

Which one of the following stresses would be most likely to cause a bone to fracture?

A) tensile stress
B) compressive stress
C) sheer stress
D) bulk stress

Question 21

In the figure, a 10.0-m long bar is attached by a frictionless hinge to a wall and held horizontal by a rope that makes an angle θ of 53° with the bar. The bar is uniform and weighs 39.9 N. How far from the hinge should a 10.0-kg mass be suspended for the tension T in the rope to be 125 N?

Question 22

What is the maximum length of a metal cable that can hang vertically supported from one end of the cable? The Young's modulus of this metal is 2.10 × 10¹¹ N/m², its tensile strength (the maximum tensile stress it can support without breaking) is 7.40 × 10⁸ N/m², and its density is 7.60 × 10³ kg/m³.

A) 9.94 km
B) 4.22 km
C) 456 km
D) 1.75 km
E) 24.8 km

Question 23

A 1000-kg object hangs from the lower end of a steel rod 5.0 m long that is suspended vertically. The diameter of the rod is 0.80 cm and Young's modulus for the rod is 210,000 MN/m². What is the elongation of the rod due to this object?

A) 1.2 cm
B) 0.46 cm
C) 0.12 cm
D) 1.8 cm
E) 0.047 cm

Question 24

In the figure, a uniform ladder 12 meters long rests against a vertical frictionless wall. The ladder weighs 400 N and makes an angle θ of 79° with the floor. A man weighing 790 N climbs slowly up the ladder. When he has climbed to a point that is 7.8 m from the base of the ladder, the ladder starts to slip. What is the coefficient of static friction between the floor and the ladder?

Question 25

A uniform 300-kg beam, 6.00 m long, is freely pivoted at P, as shown in the figure. The beam is supported in a horizontal position by a light strut, 5.00 m long, which is freely pivoted at Q and is loosely pinned to the beam at R. A load of mass is suspended from the end of the beam at S. A maximum compression of 23,000 N in the strut is permitted, due to safety. The maximum mass M of the load is closest to

A) 789 kg.
B) 554 kg.
C) 1020 kg.
D) 1090 kg.
E) 1320 kg.

Question 26

A uniform 300-kg beam, 6.00 m long, is freely pivoted at P, as shown in the figure. The beam is supported in a horizontal position by a light strut, 5.00 m long, which is freely pivoted at Q and is loosely pinned to the beam at R. A load of mass is suspended from the end of the beam at S. A maximum compression of 23,000 N in the strut is permitted, due to safety. Under maximum load, find the magnitude of the x component of the force exerted on the beam by the pivot at P.

A) 13,800 N
B) 12,800 N
C) 11,200 N
D) 14,400 N
E) 16,000 N

Question 27

A solid uniform brick is placed on a sheet of wood. When one end of the sheet is raised (see figure), you observe that the maximum that the angle θ can be without tipping over the brick is 49.6°. There is enough friction to prevent the brick from sliding. What is the width w of the brick?

A) 5.18 cm
B) 6.09 cm
C) 6.81 cm
D) 9.40 cm
E) 10.5 cm

Question 28

A very light 1.00-m wire consists of two segments of equal length, one of steel (Young's modulus is 2.00 × 10¹¹ N/m²) and one of brass (Young's modulus is 9.0 × 1010 N/m²). The steel segment is 1.50 mm in diameter, and the brass segment has twice this diameter. When a weight w is hung from the ceiling by this wire, the steel segment stretches by 1.10 mm. By what distance does the brass segment stretch?

A) 0.50 mm
B) 0.61 mm
C) 1.2 mm
D) 2.4 mm
E) 9.8 mm

Question 29

A steel lift column in a service station is a solid cylinder 4.0 m long and 0.20 m in diameter. Young's modulus for this steel is 20 × 10¹⁰ N/m². By what distance does the column compress when a 5000-kg truck is on it?

A) 4.7 × 10^-7 m
B) 8.0 × 10^-7 m
C) 3.2 × 10^-6 m
D) 7.8 × 10^-6 m
E) 3.1 × 10^-5 m

Question 30

The tensile strength (the maximum tensile stress it can support without breaking) for a certain steel wire is 3000 MN/m². What is the maximum load that can be applied to a wire with a diameter of 3.0 mm made of this steel without breaking the wire?

A) 64 kN
B) 9.0 kN
C) 42 kN
D) 85 kN
E) 21 kN

Question 31

A dump truck has a large cubical concrete block in its bed. The coefficients of friction between this block and the floor of the bed are µk = 0.450 and µs = 0.650. As the bed is slowly tilted above the horizontal, will the brick first begin to slide or will it first tip over?

A) It will first tip over.
B) It will first begin to slide.
C) It will tip over just as it begins to slide.
D) It is impossible to answer without knowing the mass of the block.
E) It is impossible to answer without knowing the dimensions of the block.

Question 32

A steel rod 55 cm long has a diameter of 30 cm. The compressive strength (the maximum stress it can support without breaking) of this steel is 500 × 10⁶ N/m². What is the compression force that would break the rod?

A) 3.5 × 10⁷ N
B) 2.4 × 10⁷ N
C) 1.4 × 10⁸ N
D) 4.7 × 10⁸ N
E) 8.9 × 10⁸ N

Question 33

A cable is 100 m long, has a cross-sectional area of 1.0 mm², and is made of a material having a Young's modulus of 1.0 × 10¹¹ N/m². If a 1000-N force is applied to stretch the cable, how far does it stretch?

A) 0.0010 m
B) 0.010 m
C) 0.10 m
D) 1.0 m
E) 10 m

Question 34

A steel guitar string with a diameter of 0.300 mm and a length of 70.0 cm is stretched by 0.500 mm while being tuned. How much force is needed to stretch the string by this amount? Young's modulus for steel is 2.0 × 1011 N/m².

Question 35

An aluminum wire and a steel wire, each of length 2.0 m, are hung from the ceiling. A 5.0-kg mass is suspended from the lower end of each wire. The aluminum wire has a diameter of 2.2 mm. What must be the diameter of the steel wire if it is to stretch the same distance as the aluminum wire, so that the two wires maintain equal lengths after the masses are attached? Young's modulus for aluminum is 0.70 × 10¹¹ N/m² and for steel it is 2.0 × 10¹¹ N/m².

Question 36

A 10.0-kg uniform ladder that is 2.50 m long is placed against a smooth vertical wall and reaches to a height of 2.10 m, as shown in the figure. The base of the ladder rests on a rough horizontal floor whose coefficient of static friction with the ladder is 0.800. An 80.0-kg bucket of concrete is suspended from the top rung of the ladder, right next to the wall, as shown in the figure. What is the magnitude of the friction force that the floor exerts on the ladder?

A) 538 N
B) 706 N
C) 1290 N
D) 833 N
E) 601 N

Question 37

A very light 1.00-m wire consists of two segments of equal length, one of steel (Young's modulus is 2.00 × 1011 N/m²) and one of brass (Young's modulus is 9.0 × 1010 N/m²). The steel segment is 1.50 mm in diameter, and the brass segment has twice this diameter. When a weight w is hung from the ceiling by this wire, the steel segment stretches by 1.10 mm. Find the weight w.

A) 190 N
B) 390 N
C) 780 N
D) 1000 N
E) 3100 N

Question 38

In the figure, a uniform rectangular crate 0.40 m wide and 1.0 m tall rests on a horizontal surface. The crate weighs 930 N, and its center of gravity is at its geometric center. A horizontal force F is applied at a distance h above the floor. If h = 0.61 m, what minimum value of F is required to make the crate start to tip over? Static friction is large enough that the crate does not start to slide.

Question 39

A 0.600-mm diameter wire stretches 0.500% of its length when it is stretched with a tension of 20.0 N. What is the Young's modulus of this wire?

A) 5.66 × 1010 N/m²
B) 3.54 × 109 N/m²
C) 1.41 × 1010 N/m²
D) 6.43 × 109 N/m²
E) 2.78 × 109 N/m²

Question 40

A 20.0-kg uniform door has a width of 1.20 m and a height of 2.50 m. The door is mounted on a post by a pair of hinges, marked 1 and 2 in the figure, at the top and bottom of the door. An external force of 60.0 N, at an angle of 30.0° above the horizontal, is applied to the small doorknob, as shown in the figure. The doorknob is 1.00 m above the bottom of the door.
(a) Find the x component of the force that hinge 1 exerts on the door at the top.
(b) Find the SUM of the y components of the forces that hinges 1 and 2 together exert on the door.

Question 41

A sample of tendon 3.00 cm long and 4.00 mm in diameter is found to break under a minimum force of 128 N. If instead the sample had been 1.50 cm long and of uniform composition and cross-sectional area, what minimum force would have been required to break it?

A) 32 N
B) 64 N
C) 128 N
D) 256 N
E) 512 N

Question 42

A copper sphere has a radius of 2.50 m under normal room pressure of 1.0 × 10⁵ N/m². If we increase the pressure on this sphere to 10 times the normal room pressure, what is the change in its volume? The bulk modulus for copper is 1.4 × 10¹¹ Pa.

Question 43

The base of an aluminum block, which is fixed in place, measures 90 cm by 90 cm, and the height of the block is 60 cm. A force, applied to the upper face and parallel to it, produces a shear strain of 0.0060. The shear modulus of aluminum is 3.0 × 10¹⁰ Pa. What is the sheer stress on the block?

A) 180 × 106 N/m²
B) 360 × 106 N/m²
C) 600 × 106 N/m²
D) 720 × 106 N/m²
E) 900 × 106 N/m²

Question 44

A 12-L volume of oil is subjected to a pressure change, which produces a volume strain on the oil of -3.0 × 10^-4. The bulk modulus of the oil is 6.0 × 10⁹ N/m² and is independent of the pressure. By how many milliliters does this pressure reduce the volume of the oil?

A) 2.0 mL
B) 2.4 mL
C) 2.8 mL
D) 3.2 mL
E) 3.6 mL

Question 45

A solid steel sphere with a radius of 2.0 m falls off a ship and sinks to a depth where the pressure is 15 MN/m². The bulk modulus for this steel is 1.6 × 10¹¹ N/m². What is the change in the radius of the sphere?

A) -0.021 mm
B) -4.2 mm
C) -0.42 mm
D) -0.19 mm
E) -0.062 mm

Question 46

A shear force of 400 N is applied to one face of an aluminum cube with sides of 30 cm while the opposite face is held fixed in place. What is the resulting displacement of the face? (The shear modulus for aluminum is 2.5 × 10¹⁰ N/m²)

A) 1.9 × 10^-8 m
B) 2.6 × 10^-8 m
C) 4.4 × 10^-8 m
D) 5.3 × 10^-8 m
E) 8.2 × 10^-8 m

Question 47

A 12-L volume of oil is subjected to a pressure change, which produces a volume strain on the oil of -3.0 × 10^-4. The bulk modulus of the oil is 6.0 × 10⁹ N/m² and is independent of the pressure. What is the pressure change that produced the volume strain in the oil?

A) 1.2 MN/m²
B) 1.4 MN/m²
C) 1.6 MN/m²
D) 1.8 MN/m²
E) 2.0 MN/m²

Question 48

At a depth of about 1030 m in the sea, the pressure has increased by 100 atmospheres (to 1.0 × 10⁷ N/m²). By how much has 1.0 m³ of water been compressed by this pressure? The bulk modulus of water is 2.3 × 10⁹ N/m².

A) 2.3 × 10^-3 m³
B) 3.3 × 10^-3 m³
C) 4.3 × 10^-3 m³
D) 5.3 × 10^-3 m³
E) 6.3 × 10^-3 m³

Question 49

When the pressure applied to an unknown liquid is increased from 1.0 × 10⁷ N/m² to 5.5 × 10⁷ N/m², the volume of the liquid decreases by 0.70%. Calculate the bulk modulus of the liquid.

Question 50

The base of an aluminum block, which is fixed in place, measures 90 cm by 90 cm, and the height of the block is 60 cm. A force, applied to the upper face and parallel to it, produces a shear strain of 0.0060. The shear modulus of aluminum is 3.0 × 10¹⁰ N/m². What is the displacement of the upper face in the direction of the applied force?

A) 3.0 mm
B) 3.6 mm
C) 4.2 mm
D) 4.8 mm
E) 5.4 mm