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Worksheet: Stress, Strain, and Elastic Modulus

Q1:

Two thin rods, one made of steel and the other of aluminum, are joined end to end. Each rod is 2.0 m long and has cross-sectional area of 9.1 mm2. A tensile force of 10 kN is applied at each end of the combination. Use a value of 2 . 0 × 1 0 1 1 Pa for the Young’s modulus of steel and a use a value of 7 . 0 × 1 0 1 0 Pa for the Young’s modulus of aluminum to determine the effects of the force on the rods.

Find the stress in the rods.

  • A 0 . 9 3 × 1 0 9 N/m2
  • B 0 . 8 8 × 1 0 9 N/m2
  • C 1 . 6 × 1 0 9 N/m2
  • D 1 . 1 × 1 0 9 N/m2
  • E 1 . 9 × 1 0 9 N/m2

Find the strain in the rods.

  • A 5 . 5 × 1 0 3
  • B 6 . 8 × 1 0 3
  • C 6 . 3 × 1 0 3
  • D 5 . 8 × 1 0 3
  • E 7 . 5 × 1 0 3

Find the elongation of the steel rod.

Find the elongation of the aluminum rod.

Q2:

The bulk modulus of a material is 1 0 1 1 N/m2. What is the percent change in volume of a piece of this material when it is subjected to a bulk stress increase of 1 0 7 N/m2? Assume that the force is applied uniformly over the surface.

Q3:

A nylon rope has a Young’s Modulus of 1 . 3 5 × 1 0 9 Pa, a length of 35.0 m when not stretched, and a diameter of 0.800 cm. If a mountain climber of mass 65.0 kg hangs on this rope, how much does it extend?

Q4:

During a walk on a rope, a tightrope walker creates a tension of 3 . 9 4 × 1 0 3 N in a steel wire that is stretched between two supporting poles that are 15.0 m apart. The wire has a diameter of 0.500 cm when it is not stretched. When the walker is on the wire in the middle between the poles the wire makes an angle of 5 0 . 0 0 below the horizontal. Under tension, steel has a Young’s modulus of 2 0 . 0 × 1 0 1 0 Pa. How much does the tension in the wire stretch the wire when the walker is this position?

Q5:

A copper wire is 1 0 0 0 . 0 mm long and its diameter is 1.000 mm. The wire hangs vertically. Determine how much weight must be added to its free end in order to extend its length 3.0 mm. Use a value of 1 . 1 0 × 1 0 1 1 Pa for the Young’s modulus of copper. Ignore any extension of the wire by its own weight.

Q6:

As an oil well is drilled, each new section of drill pipe supports its own weight and the weight of the pipe and the drill bit beneath it. Calculate the stretch in a new 6.00 m-long steel pipe that supports a 100 kg mass drill bit and a 3 . 0 0 × 1 0 3 m length of pipe with a linear mass density of 20.0 kg/m. Treat the pipe as a solid cylinder with a 5.00 cm diameter. The Young's Modulus of Steel is 2 0 9 × 1 0 9 Pa.

Q7:

A 30.0-m-long nylon rope has a diameter of 1.000 cm. A mountain climber of mass 90.0 kg hangs from the rope, extending it by 25.0 cm. What is the Young’s modulus of the nylon in the rope?

  • A 1 . 5 0 × 1 0 9 Pa
  • B 1 . 4 1 × 1 0 9 Pa
  • C 1 . 6 8 × 1 0 9 Pa
  • D 1 . 3 5 × 1 0 9 Pa
  • E 1 . 7 7 × 1 0 9 Pa

Q8:

The “lead” in pencils is a graphite composition with a Young’s modulus of 1 . 0 × 1 0 9 N/m2. Calculate the change in length of the lead in an automatic pencil when it is tapped straight into the pencil with a force of 4.0 N. The lead is 0.50 mm in diameter and 60 mm long.

Q9:

A 20.0-m-tall hollow aluminum flagpole is equivalent in strength to a solid cylinder 4.00 cm in diameter. A strong wind bends the pole equivalently to a horizontal 900.0-N force acting on the the top of the pole. How far to the side does the top of the pole flex? Use a value of 2 . 5 × 1 0 1 0 Pa for the shear modulus of aluminum.

Q10:

Two rods, one made of brass and the other of lead, have the same dimensions. When a particular stress is applied to the brass rod, it stretches by 0.18 mm. How much does the lead rod stretch under the same stress? Use a value of 9 . 0 × 1 0 1 0 Pa for the Young’s modulus of brass and use a value of 1 . 6 × 1 0 1 0 Pa for the Young’s modulus of lead.

Q11:

A steel suspension rod of a suspension bridge is 23 m long. The suspension rod must not stretch by more than 1.1 cm when a truck with a mass of 3 . 5 × 1 0 4 kg passes over the section of the bridge suspended from the rod. What diameter must the rod have if it supports the entire weight of such a truck? Use a value of 2 . 0 × 1 0 1 1 Pa for the Young’s modulus of steel.

Q12:

A physicist with a mass of 78.0 kg places himself and 5 . 0 × 1 0 2 kg of equipment at the top of a 640 m high TV broadcast antenna to perform experiments on gravity. If the antenna is modeled as a steel cylinder of radius 0.1700 m, by how much is it compressed due to the weight of the physicist and his equipment? Use a value of 2 . 0 × 1 0 1 1 Pa for the Young’s modulus of steel.

Q13:

A piano tuner applies a force to stretch a steel piano wire with a 0.9200 mm diameter by 9.00 mm. The unextended wire length is 1.55 m. Calculate the magnitude of the tension in the wire during stretching. Use a value of 2 . 0 0 × 1 0 1 1 Pa for the Young’s modulus of steel.

Q14:

Normal forces of magnitude 1 . 0 × 1 0 6 N are applied uniformly to a spherical surface enclosing a volume of a liquid. This causes the radius of the surface to decrease from 60.0000 cm to 59.9960 cm. What is the bulk modulus of the liquid?

  • A 1 . 1 × 1 0 1 1 Pa
  • B 1 . 2 × 1 0 1 0 Pa
  • C 9 . 0 × 1 0 1 0 Pa
  • D 1 . 2 × 1 0 9 Pa
  • E 1 . 3 × 1 0 1 1 Pa

Q15:

A copper wire is suspended from the ceiling and hangs vertically. How long must the wire be before the stress at its upper end reaches the proportionality limit of 7 . 0 × 1 0 7 N/m2? Use a value of 8.96 g/cm3 for the density of copper.

Q16:

A uniform rope of cross-sectional area 0.600 cm2 breaks when the tensile stress in it reaches 7 . 2 × 1 0 6 N/m2.

What is the maximum load that can be lifted slowly at a constant speed by the rope?

What is the maximum load that can be lifted by the rope with an acceleration of 4.0 m/s2?

Q17:

A 4 . 0 × 1 0 2 N weight is suspended from a wire, stretching the wire by 3.5 mm. The diameter of the wire is 1.20 mm and the unextended length of the wire is 2.0 m. What is the Young’s modulus of the metal used to manufacture the wire?

  • A 1 . 0 × 1 0 1 1 Pa
  • B 5 . 0 × 1 0 1 0 Pa
  • C 1 . 5 × 1 0 1 1 Pa
  • D 2 . 0 × 1 0 1 1 Pa
  • E 2 . 1 × 1 0 1 1 Pa

Q18:

When bismuth freezes, its volume increases by 3 . 3 2 % . What magnitude of force per unit area is bismuth capable of exerting on a container when it freezes? Use a value of 3 . 4 0 × 1 0 1 0 Pa for the bulk modulus of bismuth.

  • A 4 . 5 3 × 1 0 1 0 N/m2
  • B 1 . 1 3 × 1 0 6 N/m2
  • C 4 . 5 3 × 1 0 8 N/m2
  • D 1 . 1 3 × 1 0 8 N/m2
  • E 3 . 2 9 × 1 0 8 N/m2

Q19:

A boy with a mass of 35 kg falls vertically downward through 3.7 m. The boy lands on one foot and comes to rest in 0.12 s after he hits the ground, decelerating uniformly. The total cross-sectional area of the bones in the boy’s legs just above his ankles is 3.4 cm2. What is the compressive stress in these bones?

  • A 3 . 3 × 1 0 6 Pa
  • B 5 . 8 × 1 0 6 Pa
  • C 1 . 8 × 1 0 6 Pa
  • D 7 . 3 × 1 0 6 Pa
  • E 8 . 0 × 1 0 6 Pa

Q20:

A copper wire of diameter 1.20 cm extends by 1 . 1 % when it is used to lift a load vertically upward with an acceleration of 2.5 m/s2. Find the weight of the load. Use a value of 1 . 1 × 1 0 1 1 Pa for the Young’s modulus of copper.

Q21:

A disk between vertebrae in the spine is subjected to a shearing force of 7 . 0 0 × 1 0 2 N. The disk is equivalent to a solid cylinder 0.650 cm high and 3.800 cm in diameter. Find its shear deformation. Use a value of 1 . 0 0 × 1 0 9 N/m2 for the shear modulus of the disk.

  • A 1 . 5 3 × 1 0 4 cm
  • B 1 . 0 0 × 1 0 4 cm
  • C 1 . 5 0 × 1 0 4 cm
  • D 4 . 0 1 × 1 0 4 cm
  • E 4 . 0 1 × 1 0 3 cm

Q22:

A cube of steel with a volume of 1.0 m3 is placed in a fluid that is compressed, applying a force of 1 . 0 × 1 0 7 N to each of the cube’s faces. Find the reduction in the volume of the cube. Use a value of 1 . 6 × 1 0 1 1 for the bulk modulus of steel.

  • A 1 . 6 × 1 0 3 m−3
  • B 5 . 4 × 1 0 4 m3
  • C 1 . 9 × 1 0 5 m3
  • D 6 . 3 × 1 0 5 m3
  • E 3 . 8 × 1 0 4 m3

Q23:

A wire that is 2.0 m long has a load suspended from it and the wire extends by 1.0 mm. What is the wire’s tensile strain?

  • A 2 . 0 × 1 0 4
  • B 1 . 0 × 1 0 4
  • C 2 . 5 × 1 0 4
  • D 5 . 0 × 1 0 4
  • E 5 . 0 × 1 0 3

Q24:

A sculpture that has a weight of 10.0 kN rests on the level top of a 6.00 m high vertical column that has a cross-sectional area of 0.200 m2. The column has a density of 2 7 0 0 kg/m3.

What is the compressive stress applied to a cross-sectional layer of the column 3.0 m below the base of the sculpture?

Find the compressive strain of the column section extending from the base of the sculpture to 3.0 m vertically below that point. Use a value of 4 . 5 0 × 1 0 7 kPa for the Young’s modulus of granite.

  • A 2 . 8 5 × 1 0 6
  • B 2 . 9 8 × 1 0 6
  • C 3 . 4 8 × 1 0 6
  • D 2 . 3 0 × 1 0 6
  • E 3 . 0 2 × 1 0 6

Q25:

The figure shows a set of traffic lights suspended by two wires attached to posts at the sides of the road. The left wire makes an angle of 3 0 . 0 above the horizontal and carries a tension of 125 N. The left post is a 14.0 m tall hollow aluminum pole and is equivalent in strength to a 6.50 cm diameter solid cylinder. The Young’s modulus of aluminum is 7 0 . 0 × 1 0 9 N/m2 and the shear modulus is 2 5 . 0 × 1 0 9 N/m2.

How far is the left post bent to the side?

  • A 2 . 9 7 × 1 0 7 m
  • B 1 . 4 3 × 1 0 5 m
  • C 5 . 2 7 × 1 0 6 m
  • D 1 . 8 3 × 1 0 5 m
  • E 1 . 6 6 × 1 0 5 m

By how much is the left post compressed?

  • A 3 . 7 7 × 1 0 6 m
  • B 1 . 7 1 × 1 0 7 m
  • C 4 . 5 7 × 1 0 6 m
  • D 1 . 9 2 × 1 0 7 m
  • E 1 . 2 2 × 1 0 7 m

Q26:

During a wrestling match, a 120 kg wrestler briefly stands on one hand during a maneuver designed to perplex his already moribund adversary. By how much does the upper arm bone shorten in length? The bone can be represented by a uniform rod 42.0 cm in length and 2.30 cm in radius. The Young’s modulus of the bone under compression is 9 . 0 0 × 1 0 9 N/m2.

  • A 7 . 6 0 × 1 0 7 m
  • B 1 . 0 4 × 1 0 4 m
  • C 3 . 3 0 × 1 0 6 m
  • D 3 . 3 0 × 1 0 5 m
  • E 3 . 3 7 × 1 0 6 m

Q27:

During a circus act, one performer swings upside down hanging from a trapeze holding another performer, also upside down, by the legs. If the upward force on the lower performer is three times her weight, how much do the bones (the femurs) in her upper legs stretch? Assume that each is equivalent to a uniform rod 35.0 cm long and 1.80 cm in radius. The Young’s modulus of the bone under tension is 1 6 . 0 × 1 0 9 N/m2. The performer’s mass is 60.0 kg.

  • A 8 . 2 9 × 1 0 3 cm
  • B 3 . 4 5 × 1 0 3 cm
  • C 1 . 8 5 × 1 0 4 cm
  • D 2 . 5 3 × 1 0 3 cm
  • E 1 . 2 2 × 1 0 3 cm

Q28:

A large uniform cylindrical nickel rod is 2.0 m long and has a diameter of 5.00 cm. The rod is fastened to a concrete floor with its long axis vertical. The rod is compressed by its own weight. Use a value of 8.9 g/cm3 for the density of nickel.

What is the normal stress in the rod at the cross section located 1.0 m vertically above its lower end?

What is the normal stress in the rod at the cross section located 1.5 m vertically above its lower end?

Q29:

A strain gauge increases resistance by 0.0800 Ω/mm of extension. What increase in strain is indicated by an increase of 0.0230 Ω in resistance?

Q30:

A strain gauge increases resistance by 0.043 Ω/cm of strain. What increase in strain is indicated by an increase in resistance of 0.019 Ω?

Q31:

A strain gauge increases resistance by 0.093 Ω/cm of strain. What increase in strain is indicated by an increase in resistance of 0.012 Ω?

Q32:

What deformation characteristics do brittle materials exhibit?

  • Aplastic deformation followed by elastic deformation
  • Bplastic deformation only
  • Celastic deformation followed by plastic deformation
  • Delastic deformation only
  • Eplastic deformation followed by fracture

Q33:

What are the units of mechanical stress?

  • AN
  • Bm2
  • CIt is a dimensionless quantity.
  • DN/m2
  • EN/m

Q34:

For ductile metals that strain harden, what is the relationship between the tensile strength and the yield stress?

  • AThe tensile strength is always less than the yield stress.
  • BThe tensile strength may be either less than or greater than the yield stress.
  • CThe tensile strength is always equal to the yield stress.
  • DThe tensile strength is always greater than the yield stress.
  • EThe tensile strength is inversely proportional to the yield stress.

Q35:

What property should a strong material have a high value of?

  • Apercent elongation
  • Bshear modulus
  • Ccreep rate
  • Dyield stress
  • Etensile strength

Q36:

What property should a stiff material have a high value of?

  • Ayield stress
  • BPoisson’s ratio
  • Ctensile strength
  • Delastic modulus
  • Ehardness

Q37:

What property should a ductile material have a high value of?

  • Aelastic modulus
  • Bhardness
  • Cultimate tensile strength
  • Dpercent elongation
  • Emelting temperature

Q38:

What would be an appropriate unit for stiffness?

  • Akg/m
  • Blbf
  • Cpsi/ft
  • Dlbf/ft
  • Elbm/s

Q39:

What would be an appropriate unit for Young’s modulus?

  • APa⋅s
  • Btorr/ft
  • CN⋅m
  • Dkpsi
  • EN/yard

Q40:

How does plastic flow differ from elastic deformation?

  • APlastic flow releases more heat than elastic deformation.
  • BThe phenomena are very similar.
  • CPlastic flow is reversible upon removal of the load; elastic deformation is not.
  • DElastic deformation is instantaneous upon loading and reversible upon removal of load; plastic deformation may be slow and persists after removal of the load.
  • EPlastic flow is reversible upon removal of the load; elastic deformation is not.

Q41:

What electrical property varies in electrical strain gauges during their operation?

  • ARadio-frequency penetration
  • BInductance
  • CReactance
  • DResistance
  • EDielectric constant

Q42:

What property is exploited when using photoelasticity to monitor strains?

  • AAbsorbance
  • BRefractive index
  • CTemperature-dependent refraction
  • DBirefringence
  • EReflection

Q43:

For a material with a Young’s modulus of 112 MPa, estimate the stored strain energy per unit volume for a linear elastic strain of 0 . 0 5 0 % .

Q44:

For a material with a Young’s modulus of 57 GPa, calculate the stored strain energy per unit volume for a linear elastic strain of 1 . 0 % .

  • A 57 GPa
  • B 28 GPa
  • C 2 . 8 × 1 0 6 GPa
  • D 2 . 8 × 1 0 3 GPa
  • E 5 . 7 × 1 0 3 GPa

Q45:

What is the most basic difference between elastic and plastic deformation?

  • AElastic deformation occurs in metals; plastic deformation occurs in polymers.
  • BPlastic deformation stores energy; elastic deformation dissipates energy.
  • CElastic deformation occurs in metals; plastic deformation occurs in polymers.
  • DElastic deformation automatically reverses upon removal of the stress; plastic deformation does not.
  • EElastic deformation is not as reproducible as plastic deformation.

Q46:

The strain produced in a rod for different applied tensile loads is shown in the accompanying diagram. The equilibrium length of the rod is 5.00 cm and its equilbirum diameter is 25.00 mm. What is the modulus of elasticity of the rod’s material?

Q47:

The strain produced in a rod for different applied tensile loads is shown in the accompanying diagram. The equilibrium length of the rod is 0.10 m and its equilbirum diameter is 50.0 mm. What is the modulus of elasticity of the rod’s material?

Q48:

A uniform square prism with cross-sectional side length of 15.0 mm and an unloaded lateral length of 20.0 cm is subject to a tensile load of magnitude 1.0 kN. The Young’s modulus of the material composing the prism is 250 GPa. What is the magnitude of the extension of the prism?

Q49:

A uniform square prism with cross-sectional side length of 10.0 mm and an unloaded lateral length of 20.0 cm is subject to a tensile load of magnitude 1.0 kN. The Young’s modulus of the material composing the prism is 250 GPa. What is the magnitude of the extension of the prism?

Q50:

What is the definition of stress as used in mechanics?

  • AIt is the sum of all torques acting on a system.
  • BIt is the vector sum of all forces acting on a system.
  • CIt is the square of the magnitude of the total force acting on a system.
  • DIt is a loading force per unit area.
  • EIt is a force per unit volume.

Q51:

How is the axial tensile strength of a cylindrical member related to the diameter of the member?

  • AIt increases in proportion to the cube of the diameter.
  • BIt increases linearly with diameter.
  • CIt is independent of diameter.
  • DIt increases approximately as the square of the diameter.
  • EIt increases approximately as the square root of the diameter.

Q52:

Which of the following statements best describes an elastic material?

  • AAn elastic material deforms instantaneously under a load and gradually rebounds when the load is removed.
  • BAn elastic material gradually deforms under load and rebounds instantaneously when the load is removed.
  • CAn elastic material absorbs heat when stretched.
  • DAn elastic material deforms instantaneously and reversibly to changes in load.
  • EAn elastic material stretches under load but fractures at some critical load.

Q53:

A hydraulic press uses a small piston to apply a pressure of 156.5 atm to a 250 L volume of oil. The oil compresses by 2 . 0 × 1 0 5 m3 per atm of pressure applied to it.

What is the bulk compressive strain of the oil?

  • A 3 . 2 × 1 0 5
  • B 5 . 0 × 1 0 3
  • C 5 . 6 × 1 0 4
  • D 3 . 1 × 1 0 3
  • E 6 . 3 × 1 0 3

What is the reduction in volume of the oil?

Q54:

A steel rod of length 2.0 m has a cross-sectional area of 0.30 cm2. The rod supports a platform of mass 550 kg that hangs from the rod’s lower end.

Find the tensile stress of the rod, taking the weight of the rod to be negligible.

  • A 0 . 8 6 × 1 0 8 kPa
  • B 1 . 3 × 1 0 8 kPa
  • C 2 . 2 × 1 0 8 kPa
  • D 1 . 8 × 1 0 8 kPa
  • E 1 . 6 × 1 0 8 kPa

Find the rod’s elongation. Use a value of 200 GPa for the Young’s modulus of steel and consider the weight of the rod to be negligible.

Q55:

A vertically suspended cylindrical rod is subjected to a gravitational acceleration of 9.80 m/s2. The ultimate tensile stress of the rod material is 1 8 0 0 . 0 MPa. The density of the rod material is 1 0 0 0 . 0 kg/m3. What is the maximum length of the rod that will not break due to its own weight?

Q56:

A vertically suspended cylindrical rod is subjected to a gravitational acceleration of 9.80 m/s2. The ultimate tensile stress of the rod material is 1 8 0 0 . 0 MPa. The density of the rod material is 4 0 0 0 . 0 kg/m3. What is the maximum length of the rod that will not break due to its own weight?

Q57:

The strain produced in a rod for different applied tensile loads is shown in the accompanying diagram. The equilibrium length of the rod is 5 cm and its equilbirum diameter is 5.0 mm. What is the 0 . 1 % -strain-offset yield stress of the rod’s material?

Q58:

A rod is being used to carry a tensile load of 5 0 0 0 N. The ultimate tensile stress of the material used in the rod is 1.0 GPa. What should be the absolute minimum diameter of the rod to safely carry the load?