Download When spring is stretched or compressed it has elastic potential energy.

When spring is
stretched or
compressed it has
elastic potential
energy.
2
Welastic = 1/2 kx0 - 1/2
2
kxf
or
Initial elastic potential
energy minus Final
elastic potential
energy.
2
kx
PEelastic = 1/2
where k is the spring
constant, and x is the
distance the spring is
compressed or stretched
beyond its unstrained
length.
The unit is the joule (J).
When external
nonconservative forces
do no net work on a
system then total
mechanical energy
must be conserved.
Ef = E 0
Total mechanical energy =
translational kinetic energy +
rotational kinetic energy +
gravitational potential energy +
elastic potential energy.
If there is no rotation,
this becomes this
equation:
E = 1/2
2
mv
+ mgh + 1/2
2
kx
Ex. 10 - A m = 0.200 kg object
is vibrating on a horizontal,
frictionless table. The spring
(k = 545 N/m) is stretched initially
to x0 = 0.0450 m and then
released from rest. Determine the
final translational speed vf
of the object when the final
displacement of the spring is
(a) xf = 0.0255 m and (b) xf = 0.
Ex. 12 - A 0.20-kg ball is attached
to a vertical spring. The spring
constant is 28 N/m. The ball,
supported initially so that the spring
is neither stretched nor
compressed, is released form rest.
How far does the ball fall before
being momentarily stopped by the
spring?
A simple pendulum is a mass
m suspended by a pivot P.
When the object is pulled to
one side and released, it will
swing back and forth in a
motion approximating simple
harmonic motion.
A series of substitutions
finds that, for small
angles,
2πf = √g/L
f is frequency, g is 9.80,
and L is length.
2πf = √g/L
Mass is algebraically
eliminated, and it has
no bearing on the
frequency of a
pendulum.
Ex. 13 - Determine
the length of a simple
pendulum that will
swing back and forth
in simple harmonic
motion with a period
of 1.00 s.
A pendulum can be
a real object, in
which case it is
called a physical
pendulum.
In reality, an object in simple
harmonic motion will not
vibrate forever. Friction, or
some such force, will
decrease the velocity and
amplitude of the motion. This
is called damped harmonic
motion.
A shock absorber
introduces damping forces
to reduce the vibrations of
the ride. The smallest
degree of damping that
completely eliminates the
oscillations is called
critical damping.
When damping exceeds
the critical value, this is
called overdamping.
Underdamping is when
the damping forces are
less than what would
eliminate the vibration.
Automobile shocks
are generally
designed to produce
underdamped
motion.
In driven harmonic
motion, a driving
force is applied to
assist the object in its
vibrational motion.
If the frequency of this
driving force is the
same as the natural
frequency of the
system, the total
mechanical energy of
the system increases.
Resonance is when a
periodic force can
transmit large amounts
of energy to an
oscillating object,
leading to a large
amplitude motion.
Resonance usually
occurs when the
frequency of the
driving force is equal
to the natural frequency
of the object.
Formulas for
frequency and
period of an
oscillating spring:
2•π•f = √ k/m
2•π/ T = √ k/m