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Following points are the summary of these Lecture Slides : Vibrations and Oscillations, Spring, Oscillate, Equilibrium Position, Mass, Pulled, Pushed, Equilibrium, Spring Pushes, Spring Constant
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A mass on a spring will oscillate if the mass is
pushed or pulled from its equilibrium position. Why? We saw from the Hooke’s Law experiment that the
force of a spring is related to how far the spring ispulled or pushed from equilibrium:
F
spring
= -k(x-x
)o^
where the minus simply indicates that if you pushin on the spring, the spring pushes out.
The spring constant, k, describes how “stiff”
the spring is. A large k indicates that alarge force is needed to stretch the spring. But why does the mass oscillate?From Newton’s Second Law, if we ignore
other forces like friction or air resistance:
F = ma
and
F = -k(y-y
) - mgo
leads to
-k(y-y
) - mg = mao
Now as it passes the equilibrium position, the mass
has velocity so it will pass through the equilibriumposition and end up with a positive (compressed)position. Here the force becomes negative givinga negative acceleration. This will make thepositive velocity less positive, but still positive.Hence the mass will go to an even more positiveposition, with an even bigger negative force andacceleration that will continue to slow it downuntil it reaches zero speed.
At this point where the speed is zero, we have
a positive (compressed) position whichgives a negative force and hence negativeacceleration. This acceleration will thencause the speed to decrease to a negativevalue which will cause the mass to moveback towards equilibrium.
y = y
e^
t +
)o
The sine function is operating on the quantity:
(
t +
)o
.^
This expression must be an angle. But
what angle? There is
no “geometric” angle
in the problem
because the problem is only in one dimension. Instead, we call this kind of angle a
phase
angle. A
phase angle simply describes where in theoscillation the wave is!
The crest of the
sine wave islocated at 90
o^ ,
the trough at 270
o
and it crosses zeroat 0
o^ , 180
o^
and
starts repeatingat 360
o
(or 2
radians).
Sine function
(^10) -
Phase angle in degrees
o f V a l u e
s i n e
Series
y = y
e^
t +
)o
The amplitude, A, describes how far up and
how far down x goes. Since sine has amaximum value of 1 and a minimum valueof -1, A is used to put in units and to givethe amplitude of the oscillation.
y = A sin(
t +
)o
We have seen that
describes how fast the mass
oscillates
. But what does this oscillation speed
(ω
=d
phase
/dt
) depend on?
By putting in our solution for
y
into Newton’s
Second Law (the differential equation), we can geta prediction:
=
(k/m)
.
For stiffer springs and lighter masses, the frequency
of the oscillation increases. Note: the Amplitude does NOT affect the frequency!
total
(1/2)m
2 A
2 cos
2 (
t+
) + (1/2)kAo
2 sin
2
(
t+
)o
From the above, it looks like E
total
depends on
time. Does this violate Conservation ofEnergy? We saw that
(k/m)
. Substituting this into
the first term gives:
total
(1/2) kA
2 cos
2 (
t+
) + (1/2)kAo
2 sin
2
(
t+
)o
= (1/2)kA
2
m
2 A
2 , which does NOT
depend on time!
Since
Energy =
m
2 A
2
, as the frequency
goes up (
ω
), to keep the same energy the
amplitude (A) needs to go down.
Can you make
sense of that relationship? Since kinetic energy depends on velocity (squared),
and since v = dx/dt , a higher frequency meansthat for the same distance (amplitude) we have asmaller dt. To keep the same v, we need a smallerdistance (amplitude) to go with the smaller dt(higher frequency).
What happens when we continue to apply a
force on the mass? Specifically, whathappens when we apply an oscillating force: F
applied
o^
sin(
applied
t)
How does the mass on the spring respond?
Does it’s response depend on F
o^
Does it’s response depend on
applied
We’ll see a demonstration in class.
As we saw in the demonstration, the mass on
the spring does not receive much energyfrom the oscillating force,
unless
applied
is
very close to
free
. This is because the.
force is sometimes in the same direction asthe motion (giving energy) and sometimesin the opposite direction (receiving energy),so on average very little energy is actuallytransmitted.
In the Oscillations Lab, we will
experimentally develop equations todescribe the oscillations of a pendulum. In this case, we will have a real geometric
angle AND we will have a phase angle.The geometric angle
(the angle the pendulum
makes with the vertical)
oscillates, so we will
have:
pendulum
max
sin(
t+
)o
In the lab we will investigate what physical
parameters the period (T) depends on. Again, we will have the relations we had for
the spring and for circular motion ingeneral:
f = 1/T,
f^
