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Chapter 4: Q37P (page 167)

Suppose that we hang a heavy ball with a mass of 10 kg (about 22 lb ) from a steel wire 3m long that is 3 mm in diameter. Steel is very stiff, and Young’s modulus for steel is unusually large, 2×1011N/m2.Calculate the stretchΔL of the steel wire. This calculation shows why in many cases it is a very good approximation to pretend that the wire doesn’t stretch at all (“ideal non extensible wire”).

Short Answer

Expert verified

ΔLof steel wire is=2.079×10−4m≈0.2mm, Stretch is very negligible.

Step by step solution

01

Given data

Mass = 10kg

Diameter = 3mm

Length = 3m

02

Step 2:Solution

Young’s modulus,

Y=F/AΔL/L⇒ΔLL=FAY⇒ΔL=FLAY

From equation,

F=mg=10×9.8=98NL=3m

A =cross-sectional area =πr2

=π×(1.5×10−3)2=7.068×10−6m2

Y=2×1011N/m2ΔL=98×37.068×10−6×2×1011=2.079×10−4m≈0.2mm

Stretch is very negligible.

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Most popular questions from this chapter

An object of mass m is attached by two stretched springs (stiffnessKs and relaxed lengthL0 ) to rigid walls, as shown in figure 4.60. The springs are initially stretched by an amount (L-L0). When the object is displaced to the right and released, it oscillates horizontally. Starting from the momentum principle, find a function of the displacement xof the object and the timet describes the oscillatory motion.

(a) What is the period of the motion?

(b) If L0 were shorter (so the springs are initially stretched more), would the period be larger, smaller, or the same?

Two wires are made of the same kind of metal. Wire A has a diameter of 2.4 mm and is initially 2.8m long. You have a 8kg mass from wire A, measure the amount of stretch, and determine Young’s modulus to be Wire B, which is made of the same kind of metal as wire A, has the same length as wire A but twice the diameter. You hang the same 8kg mass from wire B, measure the amount of stretch, and determine Young’s modulus YB,

Which one of the following is true?

Two metal rods are made of different elements. The interatomic spring stiffness of element A is three times larger than the interatomic spring stiffness for element B. The mass of an atom of element A is three times greater than the mass of an atom of element B. The atomic diameters are approximately the same for A and B. What is the ratio of the speed of sound in rod A to the speed of sound in rod B?

For a vertical spring-mass oscillator that is moving up and down, which of the following statements are true? (More than one statement may be true.) (a) At the lowest point in the oscillation, the momentum is zero. (b) At the lowest point in the oscillation, the rate of change of the momentum is zero. (c) At the lowest point in the oscillation, mg=kss (d) At the lowest point in the oscillation, mg>k5s (e) At the lowest point in the oscillation, mg<k5s.

(a)A climber whose mass is 55 kg hangs motionless from a rope. What is the tension in the rope?

(b)Later, a different climber whose mass is 88 kg hangs from the same rope. Now what is the tension in the rope?

(c)Compare the physical state of the rope when it supports the heavier climber to the state of the rope when it supports the lighter climber. Which statements about the physical state of the rope are true? Check all that apply. (1) Because the same rope is used, the tension in the rope must be the same in both cases. (2) The interatomic bonds in the rope are stretched more when the rope supports the heavier climber than when the rope supports the lighter climber. (3) The rope is slightly longer when it supports the heavier climber than when it supports the lighter climber.
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