Post on 16-Mar-2020
transcript
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08 m
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Today:• SHO• Electric force and field
SHM (the SHO)
Simple harmonic motion approximates many, many physical systems!
Terms associated with SHM: Hooke’s Law = F = -kx, where k is the ... frequency = f
period = T = 1/f amplitude = xm
angular frequency = T = 2B/T = 2Bf phase constant = n displacement = x = xm cos(T t + n)
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SHO
The solution to the SHO differential equation,
is
2
2
d xm k x
dt
( ) cos( )x t A t
SHO - Oscillation Relate to potential energy “wells” and
turning points.
With no friction (“damping”), mechanical energy is conserved: K + Us= constant
Linear oscillations (motion in regions where Hooke’s Law is valid) depend on k and m:
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2sU k x
k
m 2
mT
k
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Visualizing E conservation via potential energy curves
E
U(-2)
K(-2)
Turning pointsU(J)
Animation of Circular/SHM
SHO – relation to circular motion
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The Equation of Motion
Using Hooke’s Law for the net force on a horizontal oscillator (mass on a spring):
Fnet = m a
- k x = m d2x/dt2
d2x/dt2 = -(k/m) x
This is the “differential equation” of SHM.It’s “solutions” are sines (or cosines).
They’re really the same; just differ by phase!
Phase Shifts
If Φ is positive, the shift is to the left.
In the plot, the increments are in 0.1 radians
x = xm cos(T t + n)
ωt
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Phase Shifts
If Φ is negative, the shift is to the right.
In the plot, the increments are in 0.1 radians
x = xm cos(T t + n)
ωt
For
sol
ns
of f
orm
x =
xm
cos(T
t + n
)
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Which of the following describe φfor the SHM of the figure?
A. 0 < φ < π/2
B. π/2 < φ < π
C. π < φ < 3π/2
D. - π/2 < φ < 0
E. - π < φ < - π/2
F. - 3π/2 < φ < - π
Assume x = xm cos(T t + n)
ωt
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Assume x = xm cos(T t + n)
Which of the following describe φ for the SHM of the figure?
ωt
A. 0 < φ < π/2
B. π/2 < φ < π
C. π < φ < 3π/2
D. - π/2 < φ < 0
E. - π < φ < - π/2
F. - 3π/2 < φ < - π
Now, a little tangent …
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Newton’s Law of Gravity
A uniform spherical shell of mass exerts no net gravitational force on a particle inside.
A uniform spherical shell of mass attracts a particle outside the shell as if all the shell’s mass were concentrated at its center.
rr
mmGF ˆ
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The Shell Theorems
whereG = 6.67 x 10-11 Nm2/kg2
The superposition principle holds for the gravitational force!
j
i
m1
m3m2
x
y
12F 13F
1 22
ˆgrav
m mF G r
r
θ
j
i
q1
q3q2
x
y
12F 13F
1 22
1ˆ
4Coulo
q qF r
r
θ
[for q1 opposite in sign to q2 & q3]
Gravitation! Electrostatics!
In both cases, the free-body diagrams and the general net forceExpressions are identical:
12 13 13 13 12ˆ ˆsin ( cos )netF F F i F j F F
we just have to calculate the magnitudes of the forces from the respective force laws.
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Happy weekend!