And Oscillations

Objectives

Oscillations Typical example - a simple pendulum (a mass attached to a vertical string). When the mass is displaced to one side and released, the mass begins to oscillate. An oscillation involves repetitive (periodic) motion; the body moves back and forth around an equilibrium position A characteristic of oscillatory motion is the period, T = time taken to complete one full oscillation Examples of oscillations include: the motion of a mass at the end of a horizontal/vertical spring after the mass is displaced from equilibrium; the motion of a ball inside a bowl after it has been displaced from the bottom of the bowl; the simple pendulum – a mass attached to a vertical string, like a tire swing the motion of a diving board as a diver prepares to dive; the motion of a tree branch or skyscraper under the action of the wind.

Kinematics of SHM A mass at the end of a horizontal spring: Consider a mass, m, attached to a horizontal spring with spring constant, k. If the particle is moved a distance A to the right and then released, oscillations will take place because the particle will experience a restoring force (the tension in the spring). The particle will oscillate around its equilibrium position.

Kinematics of SHM Consider the particle when it is an arbitrary position (like b in the diagram below). At that position, the extension of the spring is x. The magnitude of the force from the spring is F=kx (by Hooke’s law).

Kinematics of SHM A = amplitude (maximum displacement) Φ = phase angle (determines the initial displacement)

Example

SHM behavior of displacement, velocity & acceleration vs. time

Example A particle undergoes SHM with an amplitude of 8.00 cm and an angular frequency of s -1. At t=0, the velocity is 1.24 cm/s. Write the equations giving the displacement and velocity for this motion.

Example A particle undergoes SHM with an amplitude of 8.00 cm and an angular frequency of s -1. At t=0, the velocity is 1.24 cm/s. Calculate the initial displacement.

Example A particle undergoes SHM with an amplitude of 8.00 cm and an angular frequency of s -1. At t=0, the velocity is 1.24 cm/s. Calculate the first time at which the particle is at x = 2.00 cm and x = -2.00cm

Kinematics of SHM: Frequency

Kinematics of SHM A particle in a bowl: We consider now a particle of mass m that is placed inside a spherical bowl of radius r, as shown. In the first diagram, the letter E marks the particle’s equilibrium position at the bottom of the bowl. In the second diagram, the particle is shown displaced away from the equilibrium. The particle will be let go from the position P. In the absence of friction, the particle will perform oscillations about the equilibrium position. Will these oscillations be simple harmonic? To answer this question we must relate the acceleration to the displacement

Kinematics of SHM The displacement of the particle is the length of the arc joining points E and P. x = rθ; θ = the angle between the normal force and a line drawn up from the equilibrium position, r. The force trying to bring the particle back towards equilibrium is found by taking the components of the weight along the dashed set of axes shown. The forces on the particle are its weight and the normal force from the bowl, as shown in the second figure.

Kinematics of SHM

A simple pendulum: Consider a mass m that is attached to a vertical string of length L that hangs from the ceiling. The first figure shows the equilibrium position.

Kinematics of SHM

Example (a) Calculate the length of a pendulum that has a period equal to 1.00 s. (b) Calculate the percentage increase in the period of a pendulum when the length is increased by 4.00% What is the new period?

Assignment Complete problems 1-4 on the back page

+ Damping & Resonance

Objectives solve problems with kinetic energy and elastic potential energy in SHM; understand that in SHM there is a continuous transformation of energy, from kinetic energy into elastic potential energy and vice versa; describe the effect of damping on an oscillating system; understand the meaning of resonance and give examples of its occurrence; Discuss qualitatively the effect of a periodic external force on an oscillating system.

Energy in SHM

Example The graph below shows the variation with the square of the displacement (x 2 ) of the potential energy of a particle of mass 40 g that is executing SHM. Using the graph, determine (a) the period of the oscillation, and (b) the maximum speed of the particle during an oscillation.

Damping The SHM described earlier is unrealistic in that we have completely ignored frictional and other resistance forces. The effect of these forces on an oscillating system is that the oscillations will eventually stop and the energy of the system will be dissipated mainly as thermal energy to the environment and the system itself. Oscillations taking place in the presence of resistive forces are called damped oscillations. The behavior of the system depends on the degree of damping. We may distinguish 3 distinct cases: under-damping, critical damping, and over-damping.

Under-damping Whenever the resistance forces are small, the system will continue to oscillate but with a frequency that is somewhat smaller than that in the absence of damping. The amplitude gradually decreases until it approaches zero and the oscillations stop. The amplitude decreases exponentially. A typical example of under- damped SHM is shown at left. Heavier damping would make the oscillations die off faster

Critical damping In this case, the amount of damping is large enough that the system returns to its equilibrium state as fast as possible without performing oscillations. A typical case of critical damping is shown below (the shaded region).

Over-damping In this case, the degree of damping is so great that the system returns to equilibrium without oscillations (as in the case of critical damping) but much slower than in the case of critical damping. Belowis an image that combines all three types of damping.

Forced oscillations & resonance

In general, sometime after the external force is applied, the system will switch to oscillations with a frequency equal to the driving frequency, f D. However, the amplitude of the oscillations will depend on the relation between f D and f 0, and the amount of damping.

Forced oscillations & resonance The state in which the frequency of the externally applied force equals the natural frequency of the system is called resonance. This results in oscillations with large amplitude, as seen here.as seen here Resonance can be disastrous or good:disastrousgood Bad resonanceGood resonance Airplane wing in flightMicrowave ovens use it to warm food Building in an earthquakeRadios use it to tune into stations Car by bumps on the road or poorly tuned engine. Quartz oscillator (used in electronic watches) [Google piezoelectricity for more information]

Assignment Complete the follow-up questions