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Torque and Simple Harmonic Motion
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8.01
Week 13D2
Today’s Reading Assignment Young and Freedman: 14.1-14.6
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W013D3 Reading Assignment Young and Freedman: 14.1-14.6
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Simple Pendulum
Table Problem: Simple Pendulum by the Torque Method
(a) Find the equation of motion for θ(t) using the torque method.
(b) Find the equation of motion if θ is always <<1.
Table Problem: Simple Pendulum by the Energy Method
1. Find an expression for the mechanical energy when the pendulum is in motion in terms of θ(t) and its derivatives, m, l, and g as needed.
2. Find an equation of motion for θ(t) using the energy method.
Simple Pendulum: Small Angle Approximation
Equation of motion
Angle of oscillation is small
Simple harmonic oscillator
Analogy to spring equation
Angular frequency of oscillation
Period
sin
d 2dt2
g
l
d 2x
dt2
k
mx
ω0 ≅ g / l
T
0=
2πω0
≅2π l / g
−lmgsin =ml2 d2
dt2
Simple Pendulum: Approximation to Exact Period
Equation of motion:
Approximation to exact period:
Taylor Series approximation:
−lmgsin =ml2 d2
dt2
T ; T0(sin0 / 0 )
−3/8 =T0 + ΔT
ΔT ; T0
116
02
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Concept Question: SHO and the Pendulum
Suppose the point-like object of a simple pendulum is pulled out at by an angle 0 << 1 rad. Is the angular speed of the point-like object equal to the angular frequency of the pendulum?
1.Yes.
2.No.
3.Only at bottom of the swing.
4.Not sure.8
Demonstration Pendulum: Amplitude Effect on
Period
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Table Problem: Torsional Oscillator
A disk with moment of inertia about the center of mass rotates in a horizontal plane. It is suspended by a thin, massless rod. If the disk is rotated away from its equilibrium position by an angle , the rod exerts a restoring torque given by
At t = 0 the disk is released from rest at an angular displacement of . Find the subsequent time dependence of the angular displacement .
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τ cm =−γ
0
(t)
Icm
Worked Example: Physical Pendulum
A general physical pendulum consists of a body of mass m pivoted about a point S. The center of mass is a distance dcm from the pivot point. What is the period of the pendulum.
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Concept Question: Physical Pendulum
A physical pendulum consists of a uniform rod of length l and mass m pivoted at one end. A disk of mass m1 and radius a is fixed to the other end. Suppose the disk is now mounted to the rod by a frictionless bearing so that is perfectly free to spin. Does the period of the pendulum
1. increase?2. stay the same?3. decrease? 12
Physical Pendulum
Rotational dynamical equation
Small angle approximation
Equation of motion
Angular frequency
Period
rτSS =IS
rα
sin
d 2dt2
lcm
mg
IS
ω0
lcm
mg
IS
T 2πω
0
2πI
S
lcm
mg
Demo: Identical Pendulums, Different Periods
Single pivot: body rotates about center of mass.
Double pivot: no rotation about center of mass.14
Small Oscillations
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Small Oscillations
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Potential energy function for object of mass m
Motion is limited to the region
Potential energy has a minimum at
Small displacement from minimum, approximate potential energy by
Angular frequency of small oscillation
U (x)
x1< x< x2
x =x0
U (x) ; U (x0) +
12!
(x−x0 )2 d2U
dx2(x0 )
U (x) ; U (x0 ) +12
keff (x−x0 )2
ω0 = keff / m=
d2Udx2
(x0 ) / m
Concept Question: Energy Diagram 1
A particle with total mechanical energy E has position x > 0 at t = 0
1) escapes to infinity in the – x-direction
2) approximates simple harmonic motion
3) oscillates around a
4) oscillates around b
5) periodically revisits a and b
6) not enough information 17
Concept Question: Energy Diagram 2
A particle with total mechanical energy E has position x > 0 at t = 0
1) escapes to infinity
2) approximates simple harmonic motion
3) oscillates around a
4) oscillates around b
5) periodically revisits a and b
6) not enough information18
Concept Question: Energy Diagram 3
A particle with total mechanical energy E has position x > 0 at t = 0
1) escapes to infinity
2) approximates simple harmonic motion
3) oscillates around a
4) oscillates around b
5) periodically revisits a and b
6) not enough information
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Concept Question: Energy Diagram 4
A particle with total mechanical energy E has position x > 0 at t = 0
1) escapes to infinity
2) approximates simple harmonic motion
3) oscillates around a
4) oscillates around b
5) periodically revisits a and b
6) not enough information20
Concept Question: Energy Diagram 5
A particle with total mechanical energy E has position x > 0 at t = 0
1) escapes to infinity
2) approximates simple harmonic motion
3) oscillates around a
4) oscillates around b
5) periodically revisits a and b
6) not enough information
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Table Problem: Small Oscillations
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A particle of effective mass m is acted on by a potential energy given by
where and are positive constants
a)Sketch as a function of .
b)Find the points where the force on the particle is zero. Classify them as stable or unstable. Calculate the value of at these equilibrium points.
c)If the particle is given a small displacement from an equilibrium point, find the angular frequency of small oscillation.
U (x) =U0 −2xx0
⎛
⎝⎜⎞
⎠⎟
2
+xx0
⎛
⎝⎜⎞
⎠⎟
4⎛
⎝⎜⎜
⎞
⎠⎟⎟
U0 x0
U (x) / U0 x / x
0
U (x) / U0
Appendix
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Simple Pendulum: Mechanical Energy
Velocity
Kinetic energy
Initial energy
Final energy
Conservation of energy
E0=K0 +U0 =mgl(1−cos0 )
v
tan=l
ddt
K
f=
12
mvtan2 =
12
m lddt
⎛
⎝⎜⎞
⎠⎟
2
E
f=K f +U f =
12
m lddt
⎛
⎝⎜⎞
⎠⎟
2
+ mgl(1−cos )
1
2m l
ddt
⎛
⎝⎜⎞
⎠⎟
2
+ mgl(1−cos) =mgl(1−cos0 )
Simple Pendulum: Angular Velocity Equation of Motion
Angular velocity
Integral form
Can we integrate this to get the period?
ddt
=2gl
(cos −cos0 )
d(cos −cos0 )
∫ =2gl
dt∫
Simple Pendulum: Integral Form
Change of variables
“Elliptic Integral”
Power series approximation
Solution
bsin a =sin 2( )
da
1−b2 sin2 a( )1 2∫ =
gl
dt∫
1−b2 sin2 a( )
−1 2=1+
12
b2 sin2 a+38
b4 sin4 a+⋅⋅⋅
2π +
12π sin2 0 2( ) +⋅⋅⋅=
glT
b =sin 0 2( )
Simple Pendulum: First Order Correction
Period
Approximation
First order correction
T =2π
lg
1+14
sin2 0 2( ) +⋅⋅⋅⎛
⎝⎜⎞
⎠⎟=
2πω
1+14
sin2 0 2( ) +⎛
⎝⎜⎞
⎠⎟
sin2 0 2( ) ≅0
2 4
T ≅2πlg
1+116
02⎛
⎝⎜⎞
⎠⎟=T0 1+
116
02⎛
⎝⎜⎞
⎠⎟
=T0 + ΔT1whereΔT1 ≅116
02T0