Deck 9: Rotation of Rigid Bodies

Question 1

Volume stress and strain: 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) ΔV₂ = ΔV₁
D) ΔV₂ = 1/2ΔV₁
E) ΔV₂ = 1/4ΔV₁

Answer 1

ΔV₂ = 1/2ΔV₁

Question 2

Tensile stress and strain: 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

Answer 2

128 N

Question 3

Tensile stress and strain: 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

Answer 3

1.0 m

Question 4

Shear stress and strain: 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 5

Tensile stress and strain: 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 6

Volume stress and strain: 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 7

Tensile stress and strain: 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 8

Tensile stress and strain: 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 9

Tensile stress and strain: 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 10

Volume stress and strain: 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 11

Tensile stress and strain: 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 12

Tensile stress and strain: 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 13

Tensile stress and strain: 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 14

Volume stress and strain: 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 15

Tensile stress and strain: 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 16

Tensile stress and strain: 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 17

Tensile stress and strain: 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 stress and strain: 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.

Question 19

Tensile stress and strain: Tensile strain 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 20

Tensile stress and strain: 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 21

Shear stress and strain: 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 22

Shear stress and strain: 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 23

Volume stress and strain: 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 24

Volume stress and strain: 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 25

Volume stress and strain: 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 26

Shear stress and strain: 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