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Katzenstein Lecture: Nobel Laureate Gerhard t’HooftFriday at 4:00 in P-36 …
TopicsSimple Harmonic Motion – masses on springs Pendulum Energy of the SHO
Physics 1501: Lecture 26, Pg 2
New topic (Ch. 13) New topic (Ch. 13) Simple Harmonic Motion (SHM)Simple Harmonic Motion (SHM)
We know that if we stretch a spring with a mass on the end and let it go the mass will oscillate back and forth (if there is no friction).
This oscillation is called Simple Harmonic Motion,and is actually very easy to understand...
km
km
km
Physics 1501: Lecture 26, Pg 3
SHM So FarSHM So Far
We showed that (which came from F=ma)
has the most general solution x = Acos(t + )
where A = amplitude
= frequency
= phase constant
For a mass on a spring
The frequency does not depend on the amplitude !!!We will see that this is true of all simple harmonic
motion ! The oscillation occurs around the equilibrium point where
the force is zero!
Physics 1501: Lecture 26, Pg 4
The Simple PendulumThe Simple Pendulum
A pendulum is made by suspending a mass m at the end of a string of length L. Find the frequency of oscillation for small
displacements.
L
m
mg
z
Physics 1501: Lecture 26, Pg 5
The Simple Pendulum...The Simple Pendulum...
Recall that the torque due to gravity about the rotation (z) axis is = -mgd.
d = Lsin Lfor small
so = -mg L
L
dm
mg
z
where
Differential equation for simple harmonic motion !
= 0 cos(t + )
But = II=mL2
Physics 1501: Lecture 26, Pg 6
The Rod PendulumThe Rod Pendulum
A pendulum is made by suspending a thin rod of length L and mass M at one end. Find the frequency of oscillation for small displacements.
Lmg
z
xCM
Physics 1501: Lecture 26, Pg 7
The Rod Pendulum...The Rod Pendulum...
The torque about the rotation (z) axis is
= -mgd = -mg{L/2}sin-mg{L/2}for small
In this case
Ldmg
z
L/2
xCM
where
d I
So = Ibecomes
Physics 1501: Lecture 26, Pg 8
Lecture 26, Lecture 26, Act 1Act 1PeriodPeriod
(a) (b) (c)
What length do we make the simple pendulum so that it has the same period as the rod pendulum?
LR
LS
Physics 1501: Lecture 26, Pg 9
Suppose we have some arbitrarily shaped solid of mass M hung on a fixed axis, that we know where the CM is located and what the moment of inertia I about the axis is.
The torque about the rotation (z) axis for small is (sin ≈ )
= - Mgd ≈ - MgR
General Physical PendulumGeneral Physical Pendulum
Energy of the Spring-Mass SystemEnergy of the Spring-Mass System
Add to get E = K + U
1/2 m (A)2sin2(t + ) + 1/2 k (Acos(t + ))2
Remember that
U~cos2K~sin2
E = 1/2 kA2
so, E = 1/2 kA2 sin2(t + ) + 1/2 kA2 cos2(t + )
= 1/2 kA2 [ sin2(t + ) + cos2(t + )]
= 1/2 kA2
ActiveFigure
Physics 1501: Lecture 26, Pg 16
Energy in SHMEnergy in SHM
For both the spring and the pendulum, we can derive the SHM solution using energy conservation.
The total energy (K + U) of a system undergoing SMH will always be constant!
This is not surprising since there are only conservative forces present, hence energy is conserved.
-A A0 s
U
U
KE
Physics 1501: Lecture 26, Pg 17
SHM and quadratic potentialsSHM and quadratic potentials SHM will occur whenever the potential is quadratic. Generally, this will not be the case: For example, the potential between
H atoms in an H2 molecule lookssomething like this:
-A A0 x
U
U
KEU
x
Physics 1501: Lecture 26, Pg 18
SHM and quadratic potentials...SHM and quadratic potentials...However, if we do a Taylor expansion of this function about the minimum, we find that for smalldisplacements, the potential IS quadratic:
U
x
U(x) = U(x0 ) + U(x0 ) (x- x0 )
+ U (x0 ) (x- x0 )2+....
U(x) = 0 (since x0 is minimum of potential)
x0
U
x Define x = x - x0 and U(x0 ) = 0
Then U(x) = U (x0 ) x 2
Physics 1501: Lecture 26, Pg 19
SHM and quadratic potentials...SHM and quadratic potentials...
U
x
x0
U
x
U(x) = U (x0) x 2
Let k = U (x0)
Then:
U(x) = k x 2
SHM potential !!
Physics 1501: Lecture 26, Pg 20
What about Friction?What about Friction? Friction causes the oscillations to get
smaller over time This is known as DAMPING. As a model, we assume that the force due
to friction is proportional to the velocity.
Physics 1501: Lecture 26, Pg 21
What about Friction?What about Friction?
We can guess at a new solution.
With,
Physics 1501: Lecture 26, Pg 22
What about Friction?What about Friction?
What does this function look like?(You saw it in lab, it really works)
Physics 1501: Lecture 26, Pg 23
What about Friction?What about Friction?
There is a cuter way to write this function if you remember that exp(ix) = cos x + i sin x .