# 15.6 Forced oscillations  (Page 4/7)

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## Key equations

 Relationship between frequency and period $f=\frac{1}{T}$ $\text{Position in SHM with}\phantom{\rule{0.2em}{0ex}}\varphi =0.00$ $x\left(t\right)=A\phantom{\rule{0.1em}{0ex}}\text{cos}\left(\omega t\right)$ General position in SHM $x\left(t\right)=A\text{cos}\left(\omega t+\varphi \right)$ General velocity in SHM $v\left(t\right)=\text{−}A\omega \text{sin}\left(\omega t+\varphi \right)$ General acceleration in SHM $a\left(t\right)=\text{−}A{\omega }^{2}\text{cos}\left(\omega t+\varphi \right)$ Maximum displacement (amplitude) of SHM ${x}_{\text{max}}=A$ Maximum velocity of SHM $|{v}_{\text{max}}|=A\omega$ Maximum acceleration of SHM $|{a}_{\text{max}}|=A{\omega }^{2}$ Angular frequency of a mass-spring system in SHM $\omega =\sqrt{\frac{k}{m}}$ Period of a mass-spring system in SHM $T=2\pi \sqrt{\frac{m}{k}}$ Frequency of a mass-spring system in SHM $f=\frac{1}{2\pi }\sqrt{\frac{k}{m}}$ Energy in a mass-spring system in SHM ${E}_{\text{Total}}=\frac{1}{2}k{x}^{2}+\frac{1}{2}m{v}^{2}=\frac{1}{2}k{A}^{2}$ The velocity of the mass in a spring-mass system in SHM $v=±\sqrt{\frac{k}{m}\left({A}^{2}-{x}^{2}\right)}$ The x -component of the radius of a rotating disk $x\left(t\right)=A\text{cos}\left(\omega \phantom{\rule{0.1em}{0ex}}t+\varphi \right)$ The x -component of the velocity of the edge of a rotating disk $v\left(t\right)=\text{−}{v}_{\text{max}}\text{sin}\left(\omega \phantom{\rule{0.1em}{0ex}}t+\varphi \right)$ The x -component of the acceleration of the edge of a rotating disk $a\left(t\right)=\text{−}{a}_{\text{max}}\text{cos}\left(\omega \phantom{\rule{0.1em}{0ex}}t+\varphi \right)$ Force equation for a simple pendulum $\frac{{d}^{2}\theta }{d{t}^{2}}=-\frac{g}{L}\theta$ Angular frequency for a simple pendulum $\omega =\sqrt{\frac{g}{L}}$ Period of a simple pendulum $T=2\pi \sqrt{\frac{L}{g}}$ Angular frequency of a physical pendulum $\omega =\sqrt{\frac{mgL}{I}}$ Period of a physical pendulum $T=2\pi \sqrt{\frac{I}{mgL}}$ Period of a torsional pendulum $T=2\pi \sqrt{\frac{I}{\kappa }}$ Newton’s second law for harmonic motion $m\frac{{d}^{2}x}{d{t}^{2}}+b\frac{dx}{dt}+kx=0$ Solution for underdamped harmonic motion $x\left(t\right)={A}_{0}{e}^{-\frac{b}{2m}t}\text{cos}\left(\omega t+\varphi \right)$ Natural angular frequency of a mass-spring system ${\omega }_{0}=\sqrt{\frac{k}{m}}$ Angular frequency of underdamped harmonic motion $\omega =\sqrt{{\omega }_{0}^{2}-{\left(\frac{b}{2m}\right)}^{2}}$ Newton’s second law for forced, damped oscillation $\text{−}kx-b\frac{dx}{dt}+{F}_{o}\text{sin}\left(\omega t\right)=m\frac{{d}^{2}x}{d{t}^{2}}$ Solution to Newton’s second law for forced, damped oscillations $x\left(t\right)=A\text{cos}\left(\omega t+\varphi \right)$ Amplitude of system undergoing forced, damped oscillations $A=\frac{{F}_{o}}{\sqrt{m{\left({\omega }^{2}-{\omega }_{o}^{2}\right)}^{2}+{b}^{2}{\omega }^{2}}}$

## Conceptual questions

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

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.

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?

Can any one give me the definition for Bending moment plz...
I need a question for moment
what is charge
An attribution of particle that we have thought about to explain certain things like Electomagnetism
Nikunj
please what is the formula instantaneous velocity in projectile motion
A computer is reading from a CD-ROM that rotates at 780 revolutions per minute.What is the centripetal acceleration at a point that is 0.030m from the center of the disc?
change revolution per minute by multiplying from 2pie and devide by 60.and take r=.030 and use formula centripital acceleration =omega sqare r.
Kumar
OK thank you
Rapqueen
observation of body boulded
a gas is compressed to 1/10 0f its original volume.calculate the rise temperature if the original volume is 400k. gamma =1.4
the specific heat of hydrogen at constant pressure and temperature is 14.16kj|k.if 0.8kg of hydrogen is heated from 55 degree Celsius to 80 degree Celsius of a constant pressure. find the external work done .
Celine
hi
shaik
hy
Prasanna
g
what is imaginary mass and how we express is
what is imaginary mass how we express it
Yash
centre of mass is also called as imaginary mass
Lokmani
l'm from Algeria and fell these can help me
Many amusement parks have rides that make vertical loops like the one shown below. For safety, the cars are attached to the rails in such a way that they cannot fall off. If the car goes over the top at just the right speed, gravity alone will supply the centripetal force. What other force acts and what is its direction if: (a) The car goes over the top at faster than this speed? (b) The car goes over the top at slower than this speed?
how can I convert mile to meter per hour
1 mile * 1609m
Boon
hey can someone show me how to solve the - "Hanging from the ceiling over a baby bed ...." question
i wanted to know the steps
Shrushti
sorry shrushti..
Rashid
which question please write it briefly
Asutosh
Olympus Mons on Mars is the largest volcano in the solar system, at a height of 25 km and with a radius of 312 km. If you are standing on the summit, with what initial velocity would you have to fire a projectile from a cannon horizontally to clear the volcano and land on the surface of Mars? Note that Mars has an acceleration of gravity of 3.7 m/s2 .
what is summit
Asutosh
highest point on earth
Ngeh
पृथवी को इसके अक्ष पर कितने कोणीय चाल से घूमाऐ कि भूमधय पे आदमी का भार इसके वासतविक भार से 3/5अधिक हो
best
Murari
At a post office, a parcel that is a 20.0-kg box slides down a ramp inclined at 30.0° 30.0° with the horizontal. The coefficient of kinetic friction between the box and plane is 0.0300. (a) Find the acceleration of the box. (b) Find the velocity of the box as it reaches the end of the plane, if the length of the plane is 2 m and the box starts at rest.
As an IT student must I take physics seriously?
yh
Bernice
hii
Raja
IT came from physics and maths so I don't see why you wouldn't
conditions for pure rolling
Md