Elasticity and Oscillations

Intrinsic Strength of MaterialsHanging by a hairCan a single hair support your weight? Steel wire?A hair, cable etc. can hold more weight if it has greater cross section.

Intrinsic Strength of MaterialsWould we build a suspension bridge from Rapunzals hair?Steel cable is intrinsically stronger than hair, rope

Intrinsic Strength of MaterialsCompare strength of materials: consider not just how much weight they can support but how much weight for a given cross section F/A. Stress = F/A

* Hookes LawF = - k xFFApply a force to both ends of a long wire. These forces will stretch the wire from length L to L+L.F = kL

StrainIf we pull on a spring it increases in lengthIf we pull on a wire increases in length.Longer wires (and springs) will stretch more than short ones

*Define:The fractional change in lengthForce per unit cross-sectional area

*Hookes Law (Fx) can be written in terms of stress and strain (stress strain).The spring constant k is now (F=kx)Y is called Youngs modulus and is a measure of an objects stiffness. Hookes Law holds for an object to a point called the proportional limit.

*Example (text problem 10.7): A 0.50 m long guitar string, of cross-sectional area 1.0106 m2, has a Youngs modulus of 2.0109 Pa. By how much must you stretch a guitar string to obtain a tension of 20 N?

*Beyond Hookes LawIf 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.The ultimate strength of a material is the maximum stress that it can withstand before breaking.

*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 2.5108 Pa. What is the minimum diameter the wire should have to support her?

* Simple Harmonic MotionWhat is motion of the cart if we move it slightly?Stable repeats, oscillatesGone! How do we describe a)?

OscillationsTime it takes for the ball to roll down, up opposite side, down and back is the period TT is the time to complete one full cycle.Equivalent information is number of cycles in one second ,frequency f.e.g. T= 1/10 f = 10; T = 5 f= 1/5f=1/T T = 1/fFrequency is measured in Hertz = 1 cycle /second

*We can plot the position of the ball as a function of time. A is the distance from the bottom of the cup.

* Simple Harmonic MotionSimple harmonic motion (SHM) occurs when the restoring force (the force directed toward a stable equilibrium point) is proportional to the displacement from equilibrium.

*The motion of a mass on a spring is an example of SHM.The restoring force is F = kx.

*Assuming the table is frictionless:Also,

*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.

* 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.

*SHM graphically

*t=0: m starts with x=A, v=0 and a negative.As soon as m released v becomes negative.t=1/4 T: equilibrium x=0, v max(-), a=0t= T: x=-A, v=0 a positive, begins + motiont= T: x=0 equilibrium v max (+), a=0t= T; back at the beginning. Here we go again.

Simple Harmonic MotionThe x graph looks like a cosine x= A cos(2t/T)A is maximum displacement, =2/TThe v graph also is SHOv = vmax sintvmax should get larger if A is large ( fixed)vmax should get larger if is large (A fixed)Vmax ~A

*At endpoints v=0, x maxAt equilibrium x=0 v maxv2max =k/m A2 v= k/m Aa =-2 x

*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 = vmax, and A2 = amax.

*The period of oscillation iswhere is the angular frequency of the oscillations, k is the spring constant and m is the mass of the block.

*Example (text problem 10.30): 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 = vmax = A.

*The amplitude A is given, but is not.Example continued:

Question A mass on a spring has amplitude A. If A is doubled, the total energy of the system is:A)doubled.B)quadrupled.C)the same.D)halved.E)1/4 as much.

*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 1.8104 m at that frequency.(a) What is the maximum force acting on the diaphragm?The value is Fmax=1400 N.

*(b) What is the mechanical energy of the diaphragm?Since mechanical energy is conserved, E = Kmax = Umax.The value of k is unknown so use Kmax.The value is Kmax= 0.13 J.

*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 sint gives A = 8.00 cm and = 1.57 rads/sec. The frequency is:

*Other quantities can also be determined:The period of the motion isExample continued:

*Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site (www.mhhe.com/grr), Instructor Resources: CPS by eInstruction, Chapter 10, Questions 1, 2, 3, 4, 5, and 6.*Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site (www.mhhe.com/grr), Instructor Resources: CPS by eInstruction, Chapter 10, Question 7.*Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site (www.mhhe.com/grr), Instructor Resources: CPS by eInstruction, Chapter 10, Questions 8, 9, 11, and 12.*Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site (www.mhhe.com/grr), Instructor Resources: CPS by eInstruction, Chapter 10, Questions 8, 9, 11, and 12.*Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site (www.mhhe.com/grr), Instructor Resources: CPS by eInstruction, Chapter 10, Questions 10, 13, 14, and 15.