15.6 Forced oscillations (Page 4/7)

University physics volume 1 Page 4 / 7

Key equations

Relationship between frequency and period	$f = \frac{1}{T}$
$Position in SHM with ϕ = 0.00$	$x (t) = A cos (ω t)$
General position in SHM	$x (t) = A cos (ω t + ϕ)$
General velocity in SHM	$v (t) = − A ω sin (ω t + ϕ)$
General acceleration in SHM	$a (t) = − A ω^{2} cos (ω t + ϕ)$
Maximum displacement (amplitude) of SHM	$x_{max} = A$
Maximum velocity of SHM	$\| v_{max} \| = A ω$
Maximum acceleration of SHM	$\| a_{max} \| = A ω^{2}$
Angular frequency of a mass-spring system in SHM	$ω = \sqrt{\frac{k}{m}}$
Period of a mass-spring system in SHM	$T = 2 π \sqrt{\frac{m}{k}}$
Frequency of a mass-spring system in SHM	$f = \frac{1}{2 π} \sqrt{\frac{k}{m}}$
Energy in a mass-spring system in SHM	$E_{Total} = \frac{1}{2} k x^{2} + \frac{1}{2} m v^{2} = \frac{1}{2} k A^{2}$
The velocity of the mass in a spring-mass system in SHM	$v = \pm \sqrt{\frac{k}{m} (A^{2} - x^{2})}$
The x -component of the radius of a rotating disk	$x (t) = A cos (ω t + ϕ)$
The x -component of the velocity of the edge of a rotating disk	$v (t) = − v_{max} sin (ω t + ϕ)$
The x -component of the acceleration of the edge of a rotating disk	$a (t) = − a_{max} cos (ω t + ϕ)$
Force equation for a simple pendulum	$\frac{d^{2} θ}{d t^{2}} = - \frac{g}{L} θ$
Angular frequency for a simple pendulum	$ω = \sqrt{\frac{g}{L}}$
Period of a simple pendulum	$T = 2 π \sqrt{\frac{L}{g}}$
Angular frequency of a physical pendulum	$ω = \sqrt{\frac{m g L}{I}}$
Period of a physical pendulum	$T = 2 π \sqrt{\frac{I}{m g L}}$
Period of a torsional pendulum	$T = 2 π \sqrt{\frac{I}{κ}}$
Newton’s second law for harmonic motion	$m \frac{d^{2} x}{d t^{2}} + b \frac{d x}{d t} + k x = 0$
Solution for underdamped harmonic motion	$x (t) = A_{0} e^{- \frac{b}{2 m} t} cos (ω t + ϕ)$
Natural angular frequency of a mass-spring system	$ω_{0} = \sqrt{\frac{k}{m}}$
Angular frequency of underdamped harmonic motion	$ω = \sqrt{ω_{0}^{2} - {(\frac{b}{2 m})}^{2}}$
Newton’s second law for forced, damped oscillation	$− k x - b \frac{d x}{d t} + F_{o} sin (ω t) = m \frac{d^{2} x}{d t^{2}}$
Solution to Newton’s second law for forced, damped oscillations	$x (t) = A cos (ω t + ϕ)$
Amplitude of system undergoing forced, damped oscillations	$A = \frac{F_{o}}{\sqrt{m {(ω^{2} - ω_{o}^{2})}^{2} + b^{2} ω^{2}}}$

Conceptual questions

Why are soldiers in general ordered to “route step” (walk out of step) across a bridge?

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Do you think there is any harmonic motion in the physical world that is not damped harmonic motion? Try to make a list of five examples of undamped harmonic motion and damped harmonic motion. Which list was easier to make?

All harmonic motion is damped harmonic motion, but the damping may be negligible. This is due to friction and drag forces. It is easy to come up with five examples of damped motion: (1) A mass oscillating on a hanging on a spring (it eventually comes to rest). (2) Shock absorbers in a car (thankfully they also come to rest). (3) A pendulum is a grandfather clock (weights are added to add energy to the oscillations). (4) A child on a swing (eventually comes to rest unless energy is added by pushing the child). (5) A marble rolling in a bowl (eventually comes to rest). As for the undamped motion, even a mass on a spring in a vacuum will eventually come to rest due to internal forces in the spring. Damping may be negligible, but cannot be eliminated.

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Some engineers use sound to diagnose performance problems with car engines. Occasionally, a part of the engine is designed that resonates at the frequency of the engine. The unwanted oscillations can cause noise that irritates the driver or could lead to the part failing prematurely. In one case, a part was located that had a length L made of a material with a mass M . What can be done to correct this problem?

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Source: OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5

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