PHYS 218 Chapter 13
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Periodic Motion
Motion around a stable point (energy trough)
If there is no work being done by any forces to remove energy, then the motion continues forever.
Terms
- Amplitude
- Maximum displacement from the equilibrium position
- Period
- Time needed to complete a full cycle (measured in seconds)
- Frequency
- Number of cycles that can be done in a given time period (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=\frac{1}{t}} )
- Angular Frequency
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = 2 \pi f} (in radians/second)
Harmonic Oscillation Body
Simple Harmonic Motion (SHM)
Restoring force is directly proportional to the displacement from the equilibrium point: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F = -k \Delta x}
Equation of simple harmonic motion
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d^2 x(t)}{dt^2} = \frac{-k}{m} x(t)}
General Solution for form above
- A is the amplitude
- ω is the angular frequency: square root of "stuff" in front of the x(t) in Equation 1 = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = \sqrt{\tfrac{-k}{m}}}
- φ is the phase
A and φ can be obtained from initial conditions:
Given:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t=0) = x_0} (e.g. 3 m)
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(t=0) = v_0} (e.g. 2 m/s)
We can find:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0 = A cos{\varphi}}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_0 = -A \omega \sin{\varphi}}
Energy of SHM
For a Spring
Proof:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} E &= \frac{1}{2} k(l_0-x)^2 + \frac{1}{2} m\left(\frac{dx}{dt}\right)^2 \\ x(t) &= A \cos(\omega t + \varphi) \\ E &= \frac{1}{2}kA^2\cos^2(\omega t + \varphi) + \frac{1}{2} m A^2 + \omega^2 \sin^2(\omega t + \varphi) \\ &= \frac{1}{2} k A^2 \end{align}}
For a Simple Pendulum
If pendulum has length of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} Only works for very small angles where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(\theta) \approx \theta} ,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s(t) = A \cos\left(\sqrt{\frac{g}{L}} t + \varphi \right)}
For a Physical Pendulum
An extended body that is fixed around an axis (not at center of gravity) Distance from fixed axis to center of gravity is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta(t) = A \cos\left(\sqrt{\frac{Lmg}{I}}t + \varphi\right)}
Damped Oscillation
In SHM, we had Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F = -Cx} . What happens if we add another force... proportional to velocity, per se. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F = -bv}
Now our forces in the system:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum F_x = \underbrace{-kx}_{\mbox{restoring force}} + \underbrace{-bv_x}_{\mbox{friction}} = m \frac{d^2x}{dt^2}}
General Solution:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t) = A\mathrm{e}^{-\frac{b}{2m}t} \cos(\omega t + \varphi)}
Solve for ω by taking derivative and plugging into the first equation:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}}
Determine Damping Case
Note if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{k}{m} - \frac{b^2}{4m^2} = 0} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b = 2\sqrt{km}}
3 Cases:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b < 2\sqrt{km}} — ω is real (underdamping)
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b = 2\sqrt{km}} — ω = 0 (critical damping)
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b > 2\sqrt{km}} — ω is imaginary (overdamping)
Underdamping
System oscillates less and less every time until it completely stops (sine curve tapers as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} increases)
Envelope is approximately e−t
Critical Damping
ω = 0, so system won't even oscillate; asymptotically approaches lowest point of equilibrium Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t) = A\mathrm{e}^{-\frac{b^2}{4m^2}t} \cos{\varphi}}
Overdamping
Same as critical damping, but approaches 0 faster.