Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 1 Chapter 10: Elasticity and Oscillations Elastic Deformations Hooke’s Law Stress and Strain Shear Deformations Volume Deformations Simple Harmonic Motion The Pendulum Damped Oscillations, Forced Oscillations, and Resonance

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 2 § 10.1 Elastic Deformation of Solids A deformation is the change in size or shape of an object. An elastic object is one that returns to its original size and shape after contact forces have been removed. If the forces acting on the object are too large, the object can be permanently distorted.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 3 § 10.2 Hooke’s Law F F Apply a force to both ends of a long wire. These forces will stretch the wire from length L to L+  L.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 4 Define: The fractional change in length Force per unit cross- sectional area

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 5 Hooke’s Law (F  x) can be written in terms of stress and strain (stress  strain). The spring constant k is now Y is called Young’s modulus and is a measure of an object’s stiffness. Hooke’s Law holds for an object to a point called the proportional limit.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 6 Example (text problem 10.1): A steel beam is placed vertically in the basement of a building to keep the floor above from sagging. The load on the beam is 5.8  10 4 N and the length of the beam is 2.5 m, and the cross-sectional area of the beam is 7.5  m 2. Find the vertical compression of the beam. Force of floor on beam Force of ceiling on beam For steel Y=200  10 9 Pa.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 7 Example (text problem 10.6): A 0.50 m long guitar string, of cross- sectional area 1.0  m 2, has a Young’s modulus of 2.0  10 9 Pa. By how much must you stretch a guitar string to obtain a tension of 20.0 N?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 8 §10.3 Beyond Hooke’s Law If the stress on an object exceeds the elastic limit, then the object will not return to its original length. An object will fracture if the stress exceeds the breaking point. The ratio of maximum load to the original cross- sectional area is called tensile strength.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 9 The ultimate strength of a material is the maximum stress that it can withstand before breaking.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 10 Example (text problem 10.10): An acrobat of mass 55 kg is going to hang by her teeth from a steel wire and she does not want the wire to stretch beyond its elastic limit. The elastic limit for the wire is 2.5  10 8 Pa. What is the minimum diameter the wire should have to support her? Want

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 11 § 10.4 Shear and Volume Deformations A shear deformation occurs when two forces are applied on opposite surfaces of an object.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 12 Hooke’s law (stress  strain) for shear deformations is Define: where S is the shear modulus

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 13 Example (text problem 10.25): The upper surface of a cube of gelatin, 5.0 cm on a side, is displaced by 0.64 cm by a tangential force. If the shear modulus of the gelatin is 940 Pa, what is the magnitude of the tangential force? F F From Hooke’s Law:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 14 An object completely submerged in a fluid will be squeezed on all sides. The result is a volume strain;

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 15 For a volume deformation, Hooke’s Law is (stress  strain): where B is called the bulk modulus. The bulk modulus is a measure of how easy a material is to compress.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 16 Example (text problem 10.24): An anchor, made of cast iron of bulk modulus 60.0  10 9 Pa and a volume of m 3, is lowered over the side of a ship to the bottom of the harbor where the pressure is greater than sea level pressure by 1.75  10 6 Pa. Find the change in the volume of the anchor.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 17 Deformations summary table Tensile or compressiveShear Volume StressForce per unit cross-sectional area Shear force divided by the area of the surface on which it acts Pressure StrainFractional change in length Ratio of the relative displacement to the separation of the two parallel surfaces Fractional change in volume Constant of proportionality Young’s modulus (Y) Shear modulus (S)Bulk Modulus (B)

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 18 § 10.5 Simple Harmonic Motion Simple harmonic motion (SHM) occurs when the restoring force (the force directed toward a stable equilibrium point) is proportional to the displacement from equilibrium.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 19 The motion of a mass on a spring is an example of SHM. The restoring force is F=-kx. x Equilibrium position x y

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 20 Assuming the table is frictionless: Also,

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 21 At the equilibrium point x=0 so a=0 too. When the stretch is a maximum, a will be a maximum too. The velocity at the end points will be zero, and it is a maximum at the equilibrium point.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 22 § Representing Simple Harmonic Motion When a mass-spring system is oriented vertically, it will exhibit SHM with the same period and frequency as a horizontally placed system.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 23 SHM graphically

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 24 A simple harmonic oscillator can be described mathematically by: Or by: where A is the amplitude of the motion, the maximum displacement from equilibrium, A  =v max, and A  2 =a max.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 25 The period of oscillation is where  is the angular frequency of the oscillations, k is the spring constant and m is the mass of the block.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 26 Example (text problem 10.28): The period of oscillation of an object in an ideal mass-spring system is 0.50 sec and the amplitude is 5.0 cm. What is the speed at the equilibrium point? At equilibrium x=0: Since E=constant, at equilibrium (x = 0) the KE must be a maximum. Here v = v max = A .

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 27 The amplitude A is given, but  is not. Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 28 Example (text problem 10.41): The diaphragm of a speaker has a mass of 50.0 g and responds to a signal of 2.0 kHz by moving back and forth with an amplitude of 1.8  m at that frequency. (a) What is the maximum force acting on the diaphragm? The value is F max =1400 N.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 29 (b) What is the mechanical energy of the diaphragm? Since mechanical energy is conserved, E = KE max = U max. The value of k is unknown so use KE max. The value is KE max = 0.13 J. Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 30 Example (text problem 10.47): The displacement of an object in SHM is given by: What is the frequency of the oscillations? Comparing to y(t)= A sin  t gives A = 8.00 cm and  = 1.57 rads/sec. The frequency is:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 31 Other quantities can also be determined: The period of the motion is Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 32 §10.8 The Pendulum A simple pendulum is constructed by attaching a mass to a thin rod or a light string. We will also assume that the amplitude of the oscillations is small.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 33  L m An FBD for the pendulum bob: A simple pendulum: Assume  <<1 radian  T w x y

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 34 Apply Newton’s 2 nd Law to the pendulum bob. If we assume that  <<1 rad, then sin   and cos  1 then the angular frequency of oscillations is found to be: The period of oscillations is

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 35 Example (text problem 10.60): A clock has a pendulum that performs one full swing every 1.0 sec. The object at the end of the string weighs 10.0 N. What is the length of the pendulum? Solving for L:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 36 Example (text problem 10.84): The gravitational potential energy of a pendulum is U=mgy. Taking y=0 at the lowest point of the swing, show that y=L(1-cos  ).  L y=0 L Lcos 

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 37 A physical pendulum is any rigid object that is free to oscillate about some fixed axis. The period of oscillation of a physical pendulum is not necessarily the same as that of a simple pendulum.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 38 §10.9 Damped Oscillations When dissipative forces such as friction are not negligible, the amplitude of oscillations will decrease with time. The oscillations are damped.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 39 Graphical representations of damped oscillations:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 40 §10.10 Forced Oscillations and Resonance A force can be applied periodically to a damped oscillator (a forced oscillation). When the force is applied at the natural frequency of the system, the amplitude of the oscillations will be a maximum. This condition is called resonance.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 41 Summary Stress and Strain Hooke’s Law Simple Harmonic Motion SHM Examples: Mass-Spring System, Simple Pendulum and Physical Pendulum Energy Conservation Applied to SHM