Understanding Simple Harmonic Motion: FAQs and Examples

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Not all periodic motions are classified as simple harmonic motion, as many periodic motions, like a swinging pendulum or the Earth's orbit around the sun, do not fit this definition. Simple harmonic motion is characterized by a single frequency, represented mathematically as x1(t)=Asin(ωt). When combining two harmonics, such as x(t)=x1(t)+x2(t)=Asin(ωt)+Asin(2ωt), the resulting motion's period and graph can be analyzed for further understanding. A periodic function is defined as one that repeats itself after a constant interval T, satisfying the condition f(t)=f(t+T) for all t. Understanding these distinctions is crucial for grasping the principles of periodic motion.
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Are all periodic motions simple harmonic? Why or why not? What is an example of a periodic motion that is not simple harmonic?
 
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1. No; there's a lot of periodic motion which is not simple harmonics.
For example:
A swinging pendulum is not a simple harmonic oscillator.
The Earth's motion around the sun is periodic, but not that of a simple harmonic..
 
A simple harmonic motion function has only a single frequency, as follows:

x1(t)=Asin(ωt)

Question: What happens when you add the following 2 harmonics together?

x(t)=x1(t)+x2(t)=Asin(ωt)+Asin(2ωt)

What is the period of this motion? What does the graph of x vs t look like?

The answers to those questions should get you started.
 
Just to state what is ordinarily meant by a PERIODIC function:
We say that a function f is periodic in t with period T, if it exists a constant T so that
f(t)=f(t+T) (for all choices of t)
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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