And Oscillations. Objectives Oscillations Typical example - a simple pendulum (a mass attached to a vertical string). When the mass is displaced to one.
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Slide 1
And Oscillations
Slide 2
Objectives
Slide 3
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.
Slide 4
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.
Slide 5
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 Hookes law).
Slide 6
Kinematics of SHM A = amplitude (maximum displacement) = phase
angle (determines the initial displacement)
Slide 7
Example
Slide 8
SHM behavior of displacement, velocity & acceleration vs.
time
Slide 9
Example A particle undergoes SHM with an amplitude of 8.00 cm
and an angular frequency of 0.250 s -1. At t=0, the velocity is
1.24 cm/s. Write the equations giving the displacement and velocity
for this motion.
Slide 10
Example A particle undergoes SHM with an amplitude of 8.00 cm
and an angular frequency of 0.250 s -1. At t=0, the velocity is
1.24 cm/s. Calculate the initial displacement.
Slide 11
Example A particle undergoes SHM with an amplitude of 8.00 cm
and an angular frequency of 0.250 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
Slide 12
Kinematics of SHM: Frequency
Slide 13
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 particles
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
Slide 14
Kinematics of SHM The displacement of the particle is the
length of the arc joining points E and P. x = r; = 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.
Slide 15
Kinematics of SHM
Slide 16
Slide 17
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.
Slide 18
Kinematics of SHM
Slide 19
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?
Slide 20
Assignment Complete problems 1-4 on the back page
Slide 21
+ Damping & Resonance
Slide 22
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.
Slide 23
Energy in SHM
Slide 24
Slide 25
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.
Slide 26
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.
Slide 27
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
Slide 28
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).
Slide 29
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.
Slide 30
Forced oscillations & resonance
Slide 31
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.
Slide 32
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]