Learning Goal: To understand the meaning of Young's modulus, to perform some real-life calculations related to stretching steel, a common construction material, and to introduce the concept of breaking stress.
You are aready familiar with Hooke's law, which states that for springs and other "elastic" objects
where is the magnitude of the stretching force, is the corresponding elongation of the spring from equilibrium, and is a constant that depends on the geometry and the material of the spring. If the deformations are small enough, most materials, in fact, behave like springs: Their deformation is directly proportional to the external force. Therefore, it may be useful to operate with an expression that is similar to Hooke's law but describes the properties of various materials, as opposed to various objects such as springs. Such an expression does exist. Consider, for instance, a bar of initial length and cross-sectional area stressed by a force of magnitude . As a result, the bar stretches by .
Let us define two new terms:
Tensile stress is the ratio of the stretching force to the cross-sectional area:
Tensile strain is the ratio of the elongation of the rod to the initial length of the bar:
It turns out that the ratio of the tensile stress to the tensile strain is a constant--as long as the tensile stress is not too large. That constant, which is an inherent property of a material, is called Young's modulus and is given by
What is the SI unit of Young's modulus?
Consider a metal bar of initial length and cross-sectional area . The Young's modulus of the material of the bar is . Find the "spring constant" of such a bar for low values of tensile strain.
Express your answer in terms of , , and .
Ten identical steel wires have equal lengths and equal "spring constants" . The wires are connected end to end, so that the resultant wire has the length of . What is the "spring constant" of the resulting wire?
Ten identical steel wires have equal lengths and equal "spring constants" . The wires are slightly twisted together, so that the resultant wire has the length of and is ten times as thick as the individual wire. What is the "spring constant" of the resulting wire?
Ten identical steel wires have equal lengths and equal "spring constants" . The Young's modulus of each wire is . The wires are connected end to end, so that the resultant wire has the length of . What is the Young's modulus of the resulting wire?
Ten identical steel wires have equal lengths and equal "spring constants" . The Young's modulus of each wire is . The wires are slightly twisted together, so that the resultant wire has the length of and is ten times as thick as the individual wire. What is the Young's modulus of the resulting wire?
Consider a steel guitar string of initial length and cross-sectional area . The Young's modulus of steel is . How far () would such a string stretch under a tension of 1500 N?
Use two significant figures in your answer. Express your answer in millimeters.
Although humans have been able to fly hundreds of thousands of miles into outer space, getting inside the earth has proven much more difficult. The deepest mines ever drilled are only about 10 miles deep. To illustrate the difficulties associated with such drilling, consider the following: The density of steel is about and its breaking stress--defined as the maximum stress the material can bear without deteriorating--is about . What is the maximum length of a steel cable that can be lowered into a mine? Assume that the magnitude of the acceleration due to gravity remains constant at .
Use two significant figures in your answer. Express your answer in kilometers.
Learning Goal: To introduce you to the concept of escape velocity for a rocket.
The escape velocity is defined to be the minimum speed with which an object of mass must move to escape from the gravitational attraction of a much larger body, such as a planet of total mass . The escape velocity is a function of the distance of the object from the center of the planet , but unless otherwise specified this distance is taken to be the radius of the planet because it addresses the question "How fast does my rocket have to go to escape from the surface of the planet?"
The key to making a concise mathematical definition of escape velocity is to consider the energy. If an object is launched at its escape velocity, what is the total mechanical energy of the object at a very large (i.e., infinite) distance from the planet? Follow the usual convention and take the gravitational potential energy to be zero at very large distances.
Consider the motion of an object between a point close to the planet and a point very very far from the planet. Indicate whether the following statements are true or false.
Angular momentum about the center of the planet is conserved.
Total mechanical energy is conserved.
Kinetic energy is conserved.
The angular momentum about the center of the planet and the total mechanical energy will be conserved regardless of whether the object moves from small to large (like a rocket being launched) or from large to small (like a comet approaching the earth).
Find the escape velocity for an object of mass that is initially at a distance from the center of a planet of mass . Assume that , the radius of the planet, and ignore air resistance.
Express the escape velocity in terms of , , , and , the universal gravitational constant.
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Properties of Circular Orbits
Learning Goal: To teach you how to find the parameters characterizing an object in a circular orbit around a much heavier body like the earth.
The motivation for Isaac Newton to discover his laws of motion was to explain the properties of planetary orbits that were observed by Tycho Brahe and analyzed by Johannes Kepler. A good starting point for understanding this (as well as the speed of the space shuttle and the height of geostationary satellites) is the simplest orbit--a circular one. This problem concerns the properties of circular orbits for a satellite orbiting a planet of mass .
For all parts of this problem, where appropriate, use for the universal gravitational constant.
Find the orbital speed for a satellite in a circular orbit of radius .