Chapter 10: Oscillations Simple Harmonic Motion The Pendulum Damped Oscillations, Forced Oscillations, and Resonance

§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. 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.

The motion of a mass on a spring is an example of SHM. Equilibrium position y 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.

§10.6-7 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. 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.

SHM graphically

A simple harmonic oscillator can be described mathematically by: where A is the amplitude of the motion, the maximum displacement from equilibrium, A=vmax, and A2 =amax. Or by:

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.

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 = vmax = A.

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

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10-4 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? Example continued: (b) What is the mechanical energy of the diaphragm? Since mechanical energy is conserved, E = KEmax = Umax. The value of k is unknown so use KEmax. The value is KEmax= 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 sint gives A = 8.00 cm and  = 1.57 rads/sec. The frequency is:

Other quantities can also be determined: Example continued: Other quantities can also be determined: The period of the motion is

§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. 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 16, 17, 18, and 19.

An FBD for the pendulum bob: A simple pendulum:  L m An FBD for the pendulum bob:  T w x y Assume <<1 radian

Apply Newton’s 2nd 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

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:

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 Lcos

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.

§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.

Graphical representations of damped oscillations:

§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. 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 20.

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