11/11/2015Physics 201, UW-Madison1 Physics 201: Chapter 14 – Oscillations (cont’d)  General Physical Pendulum & Other Applications  Damped Oscillations  Resonances

11/11/2015Physics 201, UW-Madison2 Simple Harmonic Motion: Summary Solution: θ = A cos(  t +  ) Force: k s m 0 k m s 0  L

Question 1 11/11/2015Physics 201, UW-Madison3 The amplitude of a system moving with simple harmonic motion is doubled. The total energy will then be 4 times larger 2 times larger the same as it was half as much quarter as much

Question 2 11/11/2015Physics 201, Fall 2006, UW-Madison4 A glider of mass m is attached to springs on both ends, which are attached to the ends of a frictionless track. The glider moved by 0.2 m to the right, and let go to oscillate. If m = 2 kg, and spring constants are k 1 = 800 N/m and k 2 = 500 N/m, the frequency of oscillation (in Hz) is approximately 6 Hz 2 Hz 4 Hz 8 Hz 10 Hz

11/11/2015Physics 201, UW-Madison5 The harmonic oscillator is often a very good approximation for (non harmonic) oscillations with small amplitude:

General Physical Pendulum 11/11/2015Physics 201, UW-Madison6 Suppose we have some arbitrarily shaped solid of mass M hung on a fixed axis, and that we know where the CM is located and what the moment of inertia I about the axis is. The torque about the rotation (z) axis for small  is (using sin  )  = -Mg d -Mg R   d Mg z-axis R x CM where  

Torsional Oscillator 11/11/2015Physics 201, UW-Madison7 Consider an object suspended by a wire attached at its CM. The wire defines the rotation axis, and the moment of inertia I about this axis is known. l The wire acts like a “rotational spring.” çWhen the object is rotated, the wire is twisted. This produces a torque that opposes the rotation.  In analogy with a spring, the torque produced is proportional to the displacement:  = -k  I wire  

Torsional Oscillator… 11/11/2015Physics 201, UW-Madison8 Since  = -k   = I  becomes where I wire   This is similar to the “mass on spring” Except I has taken the place of m (no surprise).

11/11/2015Physics 201, UW-Madison9 Damped Oscillations l In many real systems, nonconservative forces are present çThe system is no longer ideal çFriction/drag force are common nonconservative forces l In this case, the mechanical energy of the system diminishes in time, the motion is said to be damped l The amplitude decreases with time l The blue dashed lines on the graph represent the envelope of the motion

11/11/2015Physics 201, UW-Madison10 Damped Oscillation, Example l One example of damped motion occurs when an object is attached to a spring and submerged in a viscous liquid l The retarding force can be expressed as F = - b v where b is a constant çb is called the damping coefficient l The restoring force is – kx l From Newton’s Second Law  F x = -k x – bv x = ma x or,

11/11/2015Physics 201, UW-Madison11 Damped Oscillation, Example When b is small enough, the solution to this equation is: with (This solution applies when b<2mω 0.) Time constant:

Energy of weakly damped harmonic oscillator: l The energy of a weakly damped harmonic oscillator decays exponentially in time. 11/11/2015Physics 201, UW-Madison12 The potential energy is ½kx 2 : Time constant is the time for E to drop to 1/e.

11/11/2015Physics 201, UW-Madison13 Types of Damping  0 is also called the natural frequency of the system l If F max = bv max < kA, the system is said to be underdamped When b reaches a critical value b c such that b c / 2 m =  0, the system will not oscillate çThe system is said to be critically damped If F max = bv max > kA and b/2m >  0, the system is said to be overdamped l For critically damped and overdamped there is no angular frequency

11/11/2015Physics 201, UW-Madison14 Forced Oscillations It is possible to compensate for the loss of energy in a damped system by applying an external sinusoidal force (with angular frequency  ) l The amplitude of the motion remains constant if the energy input per cycle exactly equals the decrease in mechanical energy in each cycle that results from resistive forces l After a driving force on an initially stationary object begins to act, the amplitude of the oscillation will increase l After a sufficiently long period of time, E driving = E lost to internal çThen a steady-state condition is reached çThe oscillations will proceed with constant amplitude l The amplitude of a driven oscillation is   0 is the natural frequency of the undamped oscillator

11/11/2015Physics 201, UW-Madison15 Resonance When   an increase in amplitude occurs l This dramatic increase in the amplitude is called resonance The natural frequency   is also called the resonance frequency l At resonance, the applied force is in phase with the velocity and the power transferred to the oscillator is a maximum  The applied force and v are both proportional to sin (  t +  ) çThe power delivered is F. v »This is a maximum when F and v are in phase l Resonance (maximum peak) occurs when driving frequency equals the natural frequency l The amplitude increases with decreased damping l The curve broadens as the damping increases

Resonance Applications: 11/11/2015Physics 201, Fall 2006, UW-Madison16 Extended objects have more than one resonance frequency. When plucked, a guitar string transmits its energy to the body of the guitar. The body’s oscillations, coupled to those of the air mass it encloses, produce the resonance patterns shown.

Question 3 11/11/2015Physics 201, Fall 2006, UW-Madison17 A 810-g block oscillates on the end of a spring whose force constant is k=60 N/m. The mass moves in a fluid which offers a resistive force proportional to its speed – the proportionality constant is b=0.162 N.s/m. çWrite the equation of motion and solution. çWhat is the period of motion? çWrite the displacement as a function of time if at t = 0, x = 0 and at t = 1 s, x = m s