PHY 104 Lecture Day 0: Elasticity and Periodic Motion
1) Introduction
a) Personal
i) I am not going to teach you physics. I am going to help you learn physics. How? By Re-interpreting the text material in my lectures, assigning homework that reinforces the concepts, and by administering quizzes and exams to “hold your feet to the fire” and assess just how much you are learning.
b) Review the Syllabus
c) Take Attendance
2) The Context:
a) In the past you have solved problems where the rods were rigid and did not bend or flex
b) This is an idealized model – rigidity.
c) All materials do flex to some extent when under a stress, the flexing or strain is desirous im many applications: airplane wings, earthquake resistance structures
d) We will now study a class of problems that involve materials that flex and some that “flex” with regularity – periodic motion
3) Stress, Strain and Elastic Deformation
a) Define Stress conceptually
i) See Figure 11.1, a measure of the force on an object that causes a deformation
ii) Four types of stress: tensile, shear, compressive, volume
b) Define Strain conceptually- a measure of the stretch, squeeze or twist (i.e. a deformation) in response to a stress
c) In many materials the ratio and the constant is called the “elastic modulus”
i) Relate them to Hooke’s Law is an example of this relationship where the constant in k the spring constant
d) Define some terms
i) See Figure 11.2
ii) Tensile stress is defined at a cross-section of an object under tension
(1) and has units of N/m2 or Pascals (Pa) 1 Pascal = 1 Pa = 1 N/m2
(2) A steel cable is commonly required to withstand a tensile stress of 108 PA
(3) Q: If a 2.54 cm diameter steel cable would fail under a tensile strength of 108 Pa, what is the largest weight it could carry? A = 5.07´104 N or equivalently 5,165 kg (on the Earth’s surface)
(4) Note: Pascals are also units of pressure and other units of pressure (e.g. psi) can also serve as units of tensile stress
(5) If a material is under compression, then the stress is a compressive stretch and is define the same way.
e) Tensile strain is defined as the fractional change in length of an object due to a tensile stress.
i) , strain is a dimensionless number since it is stretch per unit length
ii) Compressive strains are similarly defined and commonly reported as negative numbers.
f) Experiments have shown that under small tensile or compressive stresses and strains, the strain is proportional to the stress, or equivalently the ratio of the stress to the strain is a constant.
i) This constant is known as Young’s Modulus, Y
ii) The units of Young’s Modulus are the same as the units for stress (Pa)
iii) The value of Young’s Modulus depends on the material and not the specific configuration of that material. Young’s modulus measures how easily a material will deform when a stress is applied. Large values of Young’ Modulus denote a material that is fairly rigid and requires a large stress to achieve a measurable strain. Small values of Young’s Modulus denote materials that readily deform under even low to moderate stresses. See Table 11.1
g) Example 11.1
h) Modeling an Elastic Material as a Spring
i) From the definition of Young’s Modulus we can rearrange the terms to so that . This expression has the same form as Hooke’s Law. In fact, this how Hooke originally formulated his not famous law.
(1) What are the units of the parenthetical expression? (N/m)
i) Volume Stress and Strain
i) When an object is submerged in a fluid it feels a pressure force all around it. This will result in the object experiencing a volume strain (i.e. a fractional change in volume).
ii) An object immersed in a fluid experiences a pressure (FPerp/A) that exerts a force that is everywhere perpendicular to the surface of the object.
iii) When we apply pressure to a surface of a fluid in a container, the pressure is transmitted throughout the fluid and also acts on any object within the fluid. If the change in pressure due to depth on the object can be ignored then the pressure will be equal on all sides of the object.
iv) Units of pressure are the same as stress; Pascals, but pressure is also commonly expressed in psi, atmospheres or bars (See the inside cover of the text for conversions between units of pressure) Note: Pressure is a scalar quantity – not a vector. Why?
v) Define the volume strain on an object subject to a volume stress (i.e. pressure) as
When the pressure on an object changes from p to p+Δp and the resulting volume strain is ΔV/V we define the Bulk Modulus B as
The negative sign is included because an increase in pressure always results in a decrease in volume and we wish to report the bulk modulus as a positive number.
vi) Example 11.2
j) Shear Stress and Strain
i) See Figure 11.9
ii) Shear Stress is the ratio of the force tangent to the surface of an object divided by the area of that surface: . Shear stresses can exist only in solid objects; liquids and gases cannot support a shear stress.
iii) Shear strain: See figure 11.10. Define the shear strain as . What are the units of Shear Strain?
iv) Define the Shear Modulus as . As a rule of thumb, S is about 1/3 the value of Y for any given material. See Table 11.3
v) Example 11.3
k) Elasticity and Plasticity for Tensile stresses and strains
i) Examine Figure 11.2
(1) Where is Hooke’s Law Valid? Here the material is in the regime of elastic deformation. Note you can have elastic deformation even if you are outside the limits of Hooke’s Law
(2) For stresses greater than some maximum the object will not return to its original length. It has been permanently deformed. This is the regime of plastic deformation (where a deformed objects does not return to its original length). Some materials can endure a relatively large amount of plastic deformation and are considered to be ductile materials. Other materials that do not deform plastically are described as brittle materials
(3) At some other stress point, the object fractures at the Breaking Stress or in the case of tension, the Tensile Strength. See Table 11.4
l) Homework
i) Problems: 11.1 to 11.22 are fair game for quiz or test questions
ii) I recommend the following problems: 11.2, 11.3, 11.6, 11.8, 11.13, 11.18, 11.62, 11.64, and 11.69. These problems will be posted in Mastering Physics
iii) You are to hand in Problem 11.67 in class on Thursday.
(1) Write out the problem in its entirety
(2) Sketch the problem (before and after)
(3) Solve the parts of the problem clearly showing your work and clearly identifying your result. Express the answer to each question in a complete sentence.
(4) All numbers should have appropriate units – even within calculations.
(5) My Motivation – Mastering Physics really only cares that you get the corrected answer to the problem. On an exam or quiz, I am interested in seeing how you do the problem, that you use good problem solving practices and that you can communicate what you are thinking in your head with the words and equations on paper. I intend on training you to develop the best problem solving practices
4) Review:
a) I recommend that you
i) Review today’s class notes sitting with the text open to use as a source.
ii) Work through the example problems – write them out
iii) Start the Homework Problems on Mastering Physics. It is much more time efficient to review your notes, the text material, and do the example problems BEFORE beginning the Mastering Physics Contact me asap if you have questions or problems. (Contact info in Syllabus)
iv) Rough draft the problem to be turned in and then produce a final copy
v) Skim the remainder of the Chapter that we will cover on Thursday
5) Laboratory Investigation
a) Investigating the frequency of a mass vertically oscillating on a spring.
b) Goal: Determine the specific dependence of the natural frequency f of the oscillating system as a function of mass m.
c) Tasks:
i) Determine the spring constant of the spring
ii) Determine the natural frequency of the oscillating system for approximately 10 different masses
iii) Determine the dependence of the natural frequency on the mass.
iv) Compare to theory.
d) Write up Instructions
i) Title page
ii) Statement of the goal and a brief description of what you did.
iii) Your data: k, f vs. m
iv) Your analysis:
v) Your results and conclusion including a comparison to theory.
vi) Due next Tuesday – when should you start the write up? When will you start the write up?
6) Next Time
a) Periodic Motion
b) Energy in Simple Harmonic Motion Equations of Simple Harmonic Motion
c) The Simple Pendulum