Transcript where 14-3 Energy in the Simple Harmonic Oscillator This graph

Simple Harmonic Motion
Waves
14.2 Simple Harmonic motion (SHM
)
14-3 Energy in the Simple Harmonic Oscillator
14-5 The Simple Pendulum
14-6 The Physical Pendulum and the Torsional
Pendulum
15-1 Characteristics of Wave Motion
15-2 Types of Waves: Transverse and Longitudinal
15-3 Energy Transported by Waves
FINAL STUDY GUIDE IS
POSTED IN HW section
14-3 Energy in the Simple Harmonic
Oscillator
Example 14-7: Energy calculations.
For the simple harmonic oscillation of Example
14–5 (where m=0.300kg, k = 19.6 N/m, A =
0.100 m, x = -(0.100 m) cos 8.08t, and v =
(0.808 m/s) sin 8.08t), determine (a) the total
energy, (b) the kinetic and potential energies as
a function of time, (c) the velocity at half
amplitude (x = ± A/2), and (d) the kinetic and
potential energies when the mass is 0.050 m
from equilibrium
14-3 Energy in the Simple Harmonic
Oscillator
We already know that the potential energy of a
spring is given by:
The total mechanical energy is then:
The total mechanical energy will be conserved,
as we are assuming the system is frictionless.
14-3 Energy in the Simple
Harmonic Oscillator
If the mass is at the limits
of its motion, the energy is
all potential.
If the mass is at the
equilibrium point, the energy
is all kinetic.
We know what the potential
energy is at the turning
points:
14-3 Energy in the Simple Harmonic Oscillator
The total energy is, therefore,
And we can write:
This can be solved for the velocity as a function
of position:
where
14-3 Energy in the Simple Harmonic
Oscillator
This graph shows the potential energy function
of a spring. The total energy is constant.
14-5 The Simple Pendulum
Example 14-9: Measuring g.
A geologist uses a simple pendulum that
has a length of 37.10 cm and a frequency
of 0.8190 Hz at a particular location on
the Earth. What is the acceleration of
gravity at this location?
14-5 The Simple Pendulum
Problem 41
41. (I) A pendulum has a period of 1.35 s on
Earth. What is its period on Mars, where
the acceleration of gravity is about 0.37
that on Earth?
14-5 The Simple Pendulum
A simple pendulum
consists of a mass at
the end of a
lightweight cord. We
assume that the cord
does not stretch,
and that its mass is
negligible.
14-5 The Simple Pendulum
In order to be in SHM, the
restoring force must be
proportional to the negative
of the displacement. Here we
have:
which is proportional to sin θ
and not to θ itself.
However, if the angle is small,
sin θ ≈ θ.
14-5 The Simple Pendulum
Therefore, for small angles, we have:
where
The period and frequency are:
14-5 The Simple Pendulum
So, as long as the cord
can be considered
massless and the
amplitude is small, the
period does not depend on
the mass.