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When a weight is
added to a spring
and stretched, the
released spring will
follow a back and
forth motion.
At the equilibrium
position, velocity
is at its maximum.
At maximum
displacement, spring
force and acceleration
reach a maximum.
It is found that the
force applied is
directly proportional
to the distance the
spring stretches
(or is compressed).
As long as force remains
proportional to distance, a
plot of force vs. distance is a
straight line. Actually there is
a point at which force vs.
distance are no longer
proportional, this is called
the “proportionality limit”.
Once the force is removed,
the object will return to its
original shape. If the
“elastic limit” is exceeded,
the object will not return to
its original shape and will
be permanently deformed.
For most springs: F = kx
x is how much the spring is
displaced from its original
length,
k is the spring constant (N/m),
F is the force.
This is known as
Hooke’s Law.
A spring that
behaves according
to Hooke’s Law is
called an ideal
spring.
If a mass of 0.55 kg
attached to a vertical
spring stretches the
spring 2.0 cm from its
original equilibrium
position, what is the
spring constant?
F= kx
0.55 kg x 10 =
k(0.02)
5.5 = k(0.02)
k = 275 N/m
The spring
constant k is
often referred to
as the “stiffness”
of the spring.
A force must be applied
to a spring to stretch or
compress it. By Newton’s
third law, the spring must
apply an equal force to
whatever is applying the
force to the spring.
This reaction force
is often called the
“restoring force” and
is represented by
the equation
F = -kx.
This force varies with
the displacement.
Therefore the
acceleration
varies with the
displacement.
When the restoring force
has the mathematical
form given by F = -kx, the
type of motion resulting is
called “simple harmonic
motion”.
A graph of this motion is
sinusoidal. When an object
is hung from a spring, the
equilibrium position is
determined by how far the
weight stretches the spring
initially.
When a spring
is stretched or
compressed it has
elastic potential
energy.
PEelastic = 1/2
2
kx
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.
E f = E0
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=
2
2
1/2 mv + mgh + 1/2 kx
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 from
rest. How far does the ball fall
before being momentarily
stopped by the spring?
1/2 mv2 + mgh + 1/2 kx2 =
2
2
1/2 mv + mgh + 1/2 kx
1/2 mv2 + mgh + 1/2 kx2 =
2
2
1/2 mv + mgh + 1/2 kx
2
kx
mgh = 1/2
2
(0.2)10h = 1/2 (28)x
2
2h = 14x
1 = 7x
1/7 = x
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.
When a pendulum
swings through 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.
In some instances it is
more useful to use the
period T of vibration rather
than frequency f.
T = 2π √L/g
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 attached
to the ceiling almost
touches the floor and
its period is 12 s.
How high is the
ceiling?
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.
Formulas for frequency
and period of an
oscillating spring:
2πf = √ k/m
T = 2π√ m/k
The body of a 1275 kg car is
supported on a frame by four
springs. Two people riding in the
car have a combined mass of
153 kg. When driven over a
pothole, the frame approximates
simple harmonic motion with a
period of 0.840 s. Find the spring
constant of a single spring.