federica bianco
astro | data science | data for good
PHYS 207.013
Chapter 15
oscillation
Instructor: Dr. Bianco
TAs: Joey Betz; Lily Padlow
University of Delaware - Spring 2021
Simple Harmonic Oscillator
\frac{d^2x}{dt^2} \propto x
T = \frac{2\pi}{\omega}
H&R CH15 oscillations
x = A \cos{(\omega t + \phi)}
Simple Harmonic Oscillator
\frac{d^2x}{dt^2} \propto x
\frac{dx}{dt} = -A\omega \sin{(\omega t + \phi)}
T = \frac{2\pi}{\omega}
H&R CH15 oscillations
x = A \cos{(\omega t + \phi)}
\frac{d^2x}{dt^2} = -A\omega^2 \cos{(\omega t + \phi)}
pendulum
but T changes with θ !
ma_y = m\frac{d^2y}{dt^2} = T\cos{\theta}-mg
ma_x = m\frac{d^2x}{dt^2} = -T\sin{\theta} = -T\frac{x}{l}
H&R CH15 oscillations
H&R CH15 oscillations
pendulum
but T changes with θ !
small angle approximation:
\theta << 1 rad =>
\cos{\theta} \sim 1\\
a_y = 0
ma_y = m\frac{d^2y}{dt^2} = T\cos{\theta}-mg
ma_x = m\frac{d^2x}{dt^2} = -T\sin{\theta} = -T\frac{x}{l}
H&R CH15 oscillations
pendulum
ma_y = m\frac{d^2y}{dt^2} = T\cos{\theta}-mg
ma_y = m\frac{d^2y}{dt^2} = T-mg = 0
ma_x = m\frac{d^2x}{dt^2} = -T\sin{\theta} = -T\frac{x}{l}
ma_y = m\frac{d^2y}{dt^2} = T-mg = 0
T = mg => ma_x = -mg\frac{x}{l}
\frac{d^2x}{dt^2} = -\frac{g}{l}x
H&R CH15 oscillations
spring
H&R CH15 oscillations
\frac{d^2x}{dt^2} = -\frac{k}{m}x
\frac{d^2x}{dt^2} = -\frac{k}{m}x
UNIFORM CIRCULAR MOTION
\vec{v} = -v \sin{\theta} \hat{i} + v \cos{\theta} \hat{j}
H&R CH4 motion in 2D and 3D
r
Uniform cirular motion
SPEED DOES NOT CHANGE - VELOCITY CHANGES!
\alpha
UNIFORM CIRCULAR MOTION
\vec{v} = -v \sin{\theta} \hat{i} + v \cos{\theta} \hat{j}
H&R CH4 motion in 2D and 3D
r
Uniform cirular motion
SPEED DOES NOT CHANGE - VELOCITY CHANGES!
\alpha
x(t) = r \cos{\omega t}
y(t) = r \sin{\omega t}
simple harmonic motion
spring
H&R CH15 oscillations
\frac{d^2x}{dt^2} = -\frac{k}{m}x
pendulum
\frac{d^2x}{dt^2} = -\frac{g}{l}x
spring
H&R CH15 oscillations
\omega^2 = \frac{k}{m
}
pendulum
\omega^2 = \frac{g}{l}
spring
H&R CH15 oscillations
T = \frac{2 \pi}{\omega} = 2\pi\sqrt{\frac{m}{k}}
pendulum
T = \frac{2 \pi}{\omega} = 2\pi\sqrt{\frac{l}{g}}
H&R CH15 oscillations
pendulum
T = \frac{2 \pi}{\omega} = 2\pi\sqrt{\frac{l}{g}}
H&R CH15 oscillations
pendulum
T = \frac{2 \pi}{\omega} = 2\pi\sqrt{\frac{l}{g}}
\tau = -l(mg \sin{\theta}) = I\alpha
\alpha = -\frac{mgl}{I}\sin{\theta} \sim -\frac{mgl}{I}\theta
\frac{d^2\theta}{dt^2} = \sim -\frac{mgl}{I}\theta
\theta(t) = - A \cos(\omega^2 t + \phi)\\
\omega = -\sqrt{\frac{mgl}{I}} = -\sqrt{\frac{mgl}{ml^2}} = -\sqrt{\frac{g}{l}}
T = 2\pi\sqrt{\frac{L}{g}}l
damped oscillations
m\frac{d^2x}{dt^2} = -kx
x(t) = e^{-\frac{bt}{2m}} \cos{\sqrt{\frac{k}{m} - \frac{b^2}{4m}}}
F_d = -bv
F_d = -bv
m\frac{d^2x}{dt^2} = -kx - bv = \\
=-kx -b\frac{dx}{dt}
m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0
not damped:
damped:
forced oscillations and resonances
m\frac{d^2x}{dt^2} = -kx
m\frac{d^2x}{dt^2} = -kx + F \cos{\omega t}
A = \frac{F}{m}\frac{1}{\frac{k}{m} - \omega^2}
not forced:
forced:
natural frequency
SHO
forcing frequency
\omega << \frac{k}{m} => A = \frac{F}{k}
\omega ->\infty => A = 0
\omega =\frac{k}{m}~~~ => A -> \infty
forced oscillations and resonances
m\frac{d^2x}{dt^2} = -kx
m\frac{d^2x}{dt^2} = -kx + F \cos{\omega t}
A = \frac{F}{m}\frac{1}{\frac{k}{m} - \omega^2}
not forced:
forced:
natural frequency
SHO
forcing frequency
\omega << \frac{k}{m} => A = \frac{F}{k}
\omega ->\infty => A = 0
\omega =\frac{k}{m}~~~ => A -> \infty
\omega << \frac{k}{m} => A = \frac{F}{k}
\omega ->\infty => A = 0
\omega =\frac{k}{m}~~~ => A -> \infty
forced oscillations and resonances
m\frac{d^2x}{dt^2} = -kx
m\frac{d^2x}{dt^2} = -kx + F \cos{\omega t}
A = \frac{F}{m}\frac{1}{\frac{k}{m} - \omega^2}
not forced:
forced:
natural frequency
SHO
forcing frequency
\omega << \frac{k}{m} => A = \frac{F}{k}
\omega ->\infty => A = 0
\omega =\frac{k}{m}~~~ => A -> \infty
forced oscillations and resonances
\omega = \frac{2\pi}{f}\\ f_n = n f_1
n=1
n=2
n=3
n=4
n=5
n=6
f \propto
f \propto l T m
A classic example.... that is actually wrong!
A classic example.... that is actually wrong!
vortex shedding resonance
SHO energy
U =
Springs
H&R CH7 kinetic energy and work
\vec{F}_s = -k\vec{d}
Hooke's law
Conservation of Elastic Energy
U_{spring} = \frac{1}{2}kx^2
SHO energy
U =
U = \frac{1}{2}{kx^2}
K = \frac{1}{2}{mv^2} =
K = \frac{1}{2}{mv^2} = \frac{1}{2}{m\frac{d^2x}{dt^2}}
SHO energy
U =
U = \frac{1}{2}{kx^2}
K =\frac{1}{2}{m\frac{d^2x}{dt^2}}
K = \frac{1}{2}{m\frac{d^2x}{dt^2}} = \frac{1}{2}mA^2 \omega^2 \sin^2(\omega t + \phi)
\frac{dx}{dt} = -A\omega \sin{(\omega t + \phi)}
x = A \cos{(\omega t + \phi)}
\frac{d^2x}{dt^2} = -A\omega \cos{(\omega t + \phi)}
U = \frac{1}{2}{kx^2} = \frac{1}{2}kA^2 \cos^2(\omega t + \phi)
\omega = \sqrt{\frac{k}{m}}
K = \frac{1}{2}{m\frac{d^2x}{dt^2}} = \frac{1}{2}kA^2 \sin^2(\omega t + \phi)
SHO energy
U =
U = \frac{1}{2}{kx^2}
\frac{dx}{dt} = -A\omega \sin{(\omega t + \phi)}
x = A \cos{(\omega t + \phi)}
\frac{d^2x}{dt^2} = -A\omega \cos{(\omega t + \phi)}
U = \frac{1}{2}{kx^2} = \frac{1}{2}kA^2 \cos^2(\omega t + \phi)
\omega = \sqrt{\frac{k}{m}}
K = \frac{1}{2}{m\frac{dx}{dt}^2} = \frac{1}{2}kA^2 \sin^2(\omega t + \phi)
M.E. =
M.E. = U + K =
=\frac{1}{2} k A^2 (\cos^2({\omega t + \phi}) + \sin^2({\omega t + \phi})^2)=
=\frac{1}{2} k A^2
SHO energy
U =
U = \frac{1}{2}{kx^2}
\frac{dx}{dt} = -A\omega \sin{(\omega t + \phi)}
x = A \cos{(\omega t + \phi)}
\frac{d^2x}{dt^2} = -A\omega \cos{(\omega t + \phi)}
U = \frac{1}{2}{kx^2} = \frac{1}{2}kA^2 \cos^2(\omega t + \phi)
\omega = \sqrt{\frac{k}{m}}
K = \frac{1}{2}{m\frac{dx}{dt}^2} = \frac{1}{2}kA^2 \sin^2(\omega t + \phi)
M.E. =
M.E. = U + K =
=\frac{1}{2} k A^2
A = x_\mathrm{max}
phys207 oscillations
By federica bianco
phys207 oscillations
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