Deck 12: Static Equilibrium and Elasticity

Question 1

A ball of radius r and mass m is hung using a light string of length L from a frictionless vertical wall. The normal force on the ball due to the wall is

A) mgr/L
B)
C)
D) mgL/r
E) None of these is correct.

Answer 1

Question 2

A body is in translational equilibrium when

A) it is acted on by a constant force.
B) no friction forces are involved.
C) it has a constant acceleration.
D) it has a constant velocity.
E) no contact forces are involved.

Answer 2

it has a constant velocity.

Question 3

A body is in mechanical equilibrium when

A) it is moving with constant acceleration.
B) it is being moved by a constant force.
C) the sum of the external forces acting on it is zero.
D) the sum of the external torques acting on it is zero.
E) it is moving with constant velocity.

Answer 3

it is moving with constant velocity.

Question 4

An object has the following two conditions, and . Which of the following statements can be true?

A) The object is at rest.
B) The object is moving at a constant velocity, and rotating with a constant angular velocity.
C) The object is moving at a constant velocity but it not rotating.
D) The object does not have a translation motion but is rotating at a constant angular velocity.
E) all of the above

Question 5

For a rigid body to remain in equilibrium it is necessary and sufficient that

A)
B)
C)
D)
E)

Question 6

Use the figure to the right to answer the next problems.
A ball of mass m and radius r is placed next to a step of height ½ r.

-What is the magnitude of the force the contact point c exerts on the ball when the applied force F just lifts the ball off the floor?

A) mg
B) 1.5 mg
C) 1.73 mg
D) 2 mg
E) 2.14 mg

Question 7

Under which of the following conditions can a ladder standing on a frictionless horizontal floor and leaning against a rough vertical wall be in equilibrium?

A) The normal force exerted by the floor on the ladder equals the normal force the wall exerts on the ladder.
B) The weight of the ladder is equal in magnitude to the frictional force the wall exerts on the ladder.
C) A ladder in this situation cannot be in equilibrium.
D) The normal force exerted by the floor on the ladder equals the weight of the ladder.
E) The normal force exerted by the floor on the ladder equals the frictional force the wall exerts on the ladder.

Question 8

A planet has mass m_P and radius R_P. Suppose a rod of uniform diameter and length R_P extends vertically from the surface of the planet. Where is the center of gravity of the rod as measured from the surface of the planet? The mass of the rod is much less than the mass of the planet.

A) 0
B) R_P/3
C) 2R_P/5
D) R_P/2
E) 3R_P/4

Question 9

A box rests on a horizontal surface with one edge against a small ridge as shown. If the force is 840 N and the bottom right edge of the box is just slightly off the ground, what is the weight of the box? (Hint: With the right edge slightly off the ground, the forces applied by the floor and the ridge must be at the bottom left corner of the box, and the angle can be taken to be negligibly small.)

A) 420 N
B) 590 N
C) 840 N
D) 1.7 $\times$ 10³ N
E) 2.4 $\times$ 10³ N

Question 10

If the weight of the uniform boom shown in the figure is 2.00 kN, the tension in the horizontal guy wire supporting the boom and the 5.60-kN weight is

A) 13.2 kN
B) 9.90 kN
C) 9.20 kN
D) 6.90 kN
E) 3.00 kN

Question 11

A heavy block is supported by three lines as shown. Which of the following statements is true in its entirety?

A) T₁ > T₂ and T₂ > T₃
B) T₂ > T₁ and T₁ > T₃
C) T₂ > T₃ and T₃ > T₁
D) T₃ > T₁ and T₁ > T₂
E) T₁ > T₃ and T₂ > T₃

Question 12

Two kids are trying to pick some apples that are on the other side of the fence which is 1.5m high. Since there is a vicious dog on the other side, they decided to use a ladder and lean it against the fence. The ladder of negligible mass is placed at an angle of 40 ^$\circ$ to the horizontal. One kid of mass 50 kg stands at the bottom of the ladder while another kid of mass 40 kg goes up the ladder. How far up along the ladder can the lighter kid go before it tips over?

A) 1.4 m
B) 2.28 m
C) 2.72 m
D) 3.50 m
E) 5.25 m

Question 13

If you neglect the weight of the boom, what is the maximum weight W that can be suspended from the boom if the horizontal guy wire will break under a tension of 5.60 kN?

A) 4.00 kN
B) 5.60 kN
C) 6.70 kN
D) 4.23 kN
E) 3.00 kN

Question 14

When a certain truck stands on a level roadway, its front wheels support 1.11 kN, and its rear wheels support 18.9 kN. If the front and rear axles are 3.66 m apart, the horizontal distance between the rear axle and the center of gravity of the truck is approximately

A) 0.305 m
B) 0.610 m
C) 0.244 m
D) 0.229 m
E) 0.203 m

Question 15

The horizontal bar in the figure will remain horizontal if

A) L₁ = L₂ and R₁ = R₂
B) L₁ = L₂ and M₁ = M₂
C) R₁ = R₂ and M₁ = M₂
D) L₁M₁ = L₂M₂
E) R₁L₁ = R₂L₂

Question 16

After a shell explodes at the top of its trajectory, the center of gravity of the fragments has an acceleration, in the absence of air resistance,

A) of less than g downward.
B) equal to g and downward.
C) greater than g and downward.
D) of g downward plus a forward component.
E) of g downward plus a backward component.

Question 17

Use the figure to the right to answer the next problems.
A ball of mass m and radius r is placed next to a step of height ½ r.

-What is the force F needed to just lift the ball off the floor?

A) 0.5 mg
B) 1.41 mg
C) 1.73 mg
D) mg
E) None of these is correct.

Question 18

A 10-m long plank (of negligible mass) is supported at each end by vertical cables, also of negligible mass. A person of unknown weight sits on the plank between the cables. The tension in the left cable is 300 N, and in the right cable it is 200 N. The person's weight is

A) 100 N
B) 300 N
C) 400 N
D) 500 N
E) 700 N

Question 19

A flat object resting on a horizontal frictionless surface is acted on by forces of equal magnitude as shown. The center of gravity of the object is indicated in each case. The object is in translational equilibrium in figure(s)

A) 3 and 4
B) 1 and 2
C) 2 and 4
D) 1 only
E) 4 only

Question 20

The center of gravity and the center of mass coincide if

A) the body is in rotational equilibrium.
B) the body is in translational equilibrium.
C) the acceleration due to gravity is constant over the body.
D) the body is in both translational and rotational equilibrium.
E) any forces accelerating the body are constant.

Question 21

Use the following to answer question :
The horizontal boom supporting the sign is of uniform construction and weighs 50 N. If the sign weighs 150 N, the angle between the force exerted by the hinge on the boom and the boom is

A) ~85º
B) ~67º
C) ~47º
D) ~37º
E) None of these is correct.

Question 22

A 10-m long plank (of negligible mass) is supported at each end by vertical cables. A person of unknown weight sits on the plank between the cables. The tension in the left cable is 300 N, and in the right cable it is 200 N. How far is the person sitting from the left cable?

A) 2.0 m
B) 3.0 m
C) 4.0 m
D) 6.0 m
E) 7.0 m

Question 23

The force combination that forms a simple couple is

A) and
B) and
C) and
D) and
E) and

Question 24

Large cranes have stabilizers that extend beyond the wheels of the machines. If a crane has stabilizers that extend out by the width of the machine, by what factor can the load be increased before it tips over? Assume that the center of gravity of the crane without the load is in the middle of the crane.

A) 1
B) 2
C) 3
D) 4
E) 5

Question 25

The object shown in the diagram is a cube of uniform density resting on a rough surface. The applied force is balanced by the frictional force _fr. When the block is on the verge of tipping, the point of application of the normal force acting on the cube will be

A) 1
B) 2
C) 3
D) 4
E) 5

Question 26

A ladder is extended to a length of 8 m. It rests against a smooth frictionless vertical wall at an angle of 55 degrees to the horizontal. The ladder has a mass of 30 kg, and its center of mass is three-eighths of the way up the ladder from the floor. Calculate the magnitude of the resultant of the horizontal and vertical forces at the bottom of the ladder and the floor.

A) 371 N
B) 294 N
C) 77.0 N
D) 304 N
E) 284 N

Question 27

The sign in the figure weighs 200 N. The boom is of uniform construction. If the force exerted by the hinge on the boom is 300 N and acts at an angle of 20º above the horizontal, the weight of the boom is

A) ~250 N
B) ~540 N
C) ~460 N
D) ~413 N
E) None of these is correct.

Question 28

The sign in the figure weighs 200 N. The boom is of uniform construction. If the force exerted by the hinge on the boom is 300 N and acts at an angle of 20º above the horizontal, the angle the tie cable makes with the positive x axis is

A) ~47º
B) ~137º
C) ~43º
D) ~163º
E) None of these is correct.

Question 29

The magnitude of the resultant couple acting on the rectangular plate is

A) 160 N·m
B) 45 N·m
C) 115 N·m
D) 205 N·m
E) None of these is correct.

Question 30

Use the following to answer question :
The horizontal boom supporting the sign is of uniform construction and weighs 50 N. If the sign weighs 150 N, the force exerted by the hinge on the boom is

A) ~350 N
B) ~304 N
C) ~25 N
D) ~550 N
E) None of these is correct.

Question 31

Use the following to answer question :

-The horizontal boom supporting the sign is of uniform construction and weighs 50 N. If the sign weighs 150 N, the tension in the supporting cable is

A) ~350 N
B) ~303 N
C) ~25 N
D) ~550 N
E) None of these is correct.

Question 32

Daniel and Jonathan are playing on the seesaw with their mother in the park. The 6-m long uniform board is hinged at its midpoint. If Daniel and Jonathan weigh 30kg and 20kg and sit at distances of 2m and 3m respectively from the fulcrum, calculate where their mother (mass = 60 kg) has to sit on the other side of the fulcrum for static equilibrium.

A) 2.5 m
B) 1.5 m
C) 1.0 m
D) 2.0 m
E) none of the above

Question 33

Two unequal, antiparallel forces

A) can be replaced by a single force.
B) can be replaced by a couple whose torque equals the torque about the center of mass of the original forces.
C) can be replaced by equal, parallel forces.
D) can be replaced by a single force plus a couple whose torque equals the torque about the center of mass of the original forces.
E) can be replaced by none of these.

Question 34

Which of the objects is nearest to being in unstable equilibrium?

A) 1
B) 2
C) 3
D) 4
E) 5

Question 35

The boom in the figure is uniform and weighs 2.20 kN. The magnitude of the tension in the guy wire is

A) ~0.635 kN
B) ~1.83 kN
C) ~1.94 kN
D) ~1.47 kN
E) None of these is correct.

Question 36

Four happy kids are lining up and jumping off a diving board one after another. The kids, each with a mass of 40 kg, are situated as in the above figure, and the uniform board has a total mass of 60 kg. Calculate the force on the support closest to the water.

A) 3140 N, up
B) 4320 N, down
C) 4320 N, up
D) 3140 N, down
E) 3920 N, down

Question 37

The boom in the figure is uniform and weighs 2.20 kN. The magnitude of the force exerted on the boom by the hinge is

A) ~0.635 kN
B) ~1.83 kN
C) ~1.94 kN
D) ~1.47 kN
E) None of these is correct.

Question 38

A 6.0-kg ball is held in the hand with the lower arm making a 90-degree angle with the upper arm. The lower arm and hand have a combined mass of 2.5 kg and a center of mass 18 cm from the elbow joint. The ball is 40 cm from the elbow joint. If the biceps muscle is 12 cm from the center-of-mass point of the lower arm and hand, calculate the force that the biceps muscle must exert in order to keep the lower arm and hand horizontal.

A) 466 N
B) 48.0 N
C) 392 N
D) 233 N
E) none of the above

Question 39

The sign shown in the figure weighs 200 N. The boom is of uniform construction. If the force exerted by the hinge on the boom is 300 N and acts at an angle of 20º above the horizontal, the tension in the tie cable is

A) ~205 N
B) ~145 N
C) ~413 N
D) ~625 N
E) None of these is correct.

Question 40

A ladder is extended to a length of 8 m. It rests against a smooth frictionless vertical wall at an angle of 55 degrees to the horizontal. The ladder has a mass of 30 kg, and its center of mass is three-eighths of the way up the ladder from the floor. If the coefficient of static friction between the ladder and the floor is 0.2, calculate how far a man of 80 kg can stand along the ladder without it slipping away at the floor.

A) 3.7 m
B) 6.3 m
C) all the way to the top
D) 5.4 m
E) none of the above

Question 41

An 800-kg mass is hung from a 1.5-m copper wire with a cross-sectional area of 0.35 cm². Young's modulus for copper is 1.1 $\times$ 10¹¹ N/m². How much does the wire stretch?

A) 2.56 mm
B) 2.97 mm
C) 3.14 mm
D) 3.86 mm
E) 4.13 mm

Question 42

A wire of circular cross section of diameter d and length L is stretched an amount $\Delta$ L by a steady force F. An equal force would produce how much stretch in a similar wire of diameter 2d and length 2L?

A) $\Delta$ L/8
B) $\Delta$ L/4
C) $\Delta$ L/2
D) 2 $\Delta$ L
E) 4 $\Delta$ L

Question 43

Which of the following could not have units of N/m²?

A) Young's modulus
B) shear modulus
C) bulk modulus
D) stress
E) strain

Question 44

The bulk modulus of water is 2.0 $\times$ 10⁹ N/m². By how much must the pressure be increased to reduce the volume of 1.0 kg of water from 1.00 to 0.998 L?

A) 2.0 $\times$ 10⁶ N/m²
B) 4.0 $\times$ 10⁶ N/m²
C) 20 $\times$ 10⁶ N/m²
D) 40 $\times$ 10⁶ N/m²
E) 8.0 $\times$ 10⁶ N/m²

Question 45

Use the following to answer question :
The point that identifies the fracture point of the material whose graph of stress as a function of strain is shown below is

A) 1
B) 2
C) 3
D) 4
E) 5

Question 46

The form of Hooke's law relating shear stress to shear strain is
F/A = E $\Delta$ s/h
The figure shows a block subjected to shear stress. Which number corresponds to the surface area A in the equation just given?

A) 1
B) 2
C) 3
D) 4
E) 5

Question 47

An 800-kg mass is hung from a 2-m steel wire with a cross-sectional area of 0.25 cm². Young's modulus for steel is 2.0 $\times$ 10¹¹ N/m². How much does the wire stretch?

A) 2.56 mm
B) 2.97 mm
C) 3.14 mm
D) 3.86 mm
E) 4.13 mm

Question 48

A wire 2.5-m long has a cross-sectional area of 2.5 mm². It is hung vertically, and a 4.5-kg mass is hung from it. By how much does the wire stretch if Young's modulus for that material is 2.0 $\times$ 10¹¹ N/m²?

A) 0.22 mm
B) 2.2 mm
C) 0.022 mm
D) 0.72 mm
E) 0.072 mm

Question 49

Young's modulus Y has units of

A) force.
B) mass density.
C) energy.
D) pressure.
E) mass per unit time.

Question 50

Use the picture on the right to answer the next question.
A steel wire and an aluminum wire of the same length L and diameter d are joined to form a wire of length 2L. The wire is fastened to the ceiling and a mass m is attached to the other end.
The stress in the aluminum portion is ____ the steel portion.

A) less than
B) equal to
C) greater than
D) depends on how tight m is attached to the steel
E) unable to tell

Question 51

If you measure the force required to stretch a typical laboratory spring as a function of the amount of stretch, which of the graphs shown below would best represent your data?

A) 1
B) 2
C) 3
D) 4
E) 5

Question 52

The bulk modulus B has units of

A) force.
B) energy.
C) pressure.
D) mass per unit time.
E) mass density.

Question 53

A wire of length L and cross-sectional area A is stretched an amount $\Delta$ L by a force F. The strain is

A) $\Delta$ L
B) $\Delta$ L/L
C) F
D) F/A
E) FL/(A $\Delta$ L)

Question 54

Use the picture on the right to answer the next question.
A steel wire and an aluminum wire of the same length L and diameter d are joined to form a wire of length 2L. The wire is fastened to the ceiling and a mass m is attached to the other end.
The amount of expansion in the aluminum portion is ____ the steel portion.

A) less than
B) equal to
C) greater than
D) depends on how tight m is attached to the steel
E) unable to tell

Question 55

The configuration(s) that is/are in unstable equilibrium is/are

A) 3 and 5
B) 1 and 4
C) 1
D) 2 and 4
E) 2

Question 56

Use the following to answer question :
The point that identifies the elastic limit of the material whose graph of stress as a function of strain is shown below is

A) 1
B) 2
C) 3
D) 4
E) 5

Question 57

Use the picture on the right to answer the next question.
A steel wire and an aluminum wire of the same length L and diameter d are joined to form a wire of length 2L. The wire is fastened to the ceiling and a mass m is attached to the other end.
The strain in the aluminum portion is ____ the steel portion.

A) less than
B) equal to
C) greater than
D) depends on how tight m is attached to the steel
E) unable to tell

Question 58

Use the following to answer question :
The point that identifies the proportional limit of the material whose graph of stress as a function of strain is shown below is

A) 1
B) 2
C) 3
D) 4
E) 5

Question 59

An upright refrigerator tips over if its

A) height is greater than its width.
B) center of mass is below its middle.
C) center of mass is above its middle.
D) center of mass is at its middle.
E) center of mass projects onto the floor at a point outside the outline of its base.

Question 60

The configuration that is in neutral equilibrium is

A) 1
B) 2
C) 3
D) 4
E) 5

Question 61

Two wires, A and B, have the same initial length, but the radius of A is four times that of B, and the Young's modulus of A is a third that of B. If the same weight is attached to both wires, what is the ratio of the extension of A divided by the extension of B?

A) 0.19
B) 48
C) 5.3
D) 1.3
E) 0.75

Question 62

A wire has a diameter of 0.75 mm. It is stretched 0.2% of its original length when a mass of 5 kg is hung from it. What is the Young's modulus of the wire?

A) 1.4 $\times$ 10¹⁰ N/m²
B) 5.6 $\times$ 10¹⁰ N/m²
C) 1.7 $\times$ 10¹¹ N/m²
D) 5.6 $\times$ 10⁸ N/m²
E) 1.4 $\times$ 10⁸ N/m²

Question 63

A 3-m long wire with a diameter of 1.5 mm has a Young's modulus of 9 $\times$ 10¹⁰ N/m². What mass hung from the wire would extend it by 3 mm?

A) 159 kg
B) 5.20 kg
C) 16.2 kg
D) 64.8 kg
E) None of the above

Question 64

A design project requires you to construct a crane to lift a maximum load of 5000 kg and with a maximum upward acceleration of 2 m/s². What size diameter steel cable should you use if the cable is to have a safety factor of 5?

A) 1.64 cm
B) 1.10 cm
C) 2.68 cm
D) 3.00 cm
E) 1.20 cm

Question 65

A design project requires you to construct a crane to lift a maximum of 5000 kg and with a maximum upward acceleration of 2 m/s², and with a safety factor of 5. If the cable is
20m long, by how much does the cable stretch when the crane is operating at the design limits?

A) 2.80 cm
B) 1.05 cm
C) 6.21 cm
D) 0.84 cm
E) 5.22 cm