Various problems & questions - help,suggestion?

[R A V E N]

Question 1.

From kinematics we have $$a=\frac{\mathrm{d}^2s}{\mathrm{d}t^2}$$ which means,as I understood it:to get the acceleration,we first derive distance with respect to time,then again derive the result of that operation with respect to time.

If so,wouldn`t be more appropriate to write $$a=\frac{\mathrm{d}^2s}{\mathrm{d}t}$$ - I mean,what $$2$$ represents in $$\mathrm{d}t^2$$?

 

[LowlyPion]

d²x/dt² is the correct notation for the second derivative.

Think of it more as Δ(Δx/Δt)/Δt, where the limit is applied as Δ tends toward 0.

 

[R A V E N]

Question 2:

The second question is about equation which describes a wave in air which originates from vibrating string attached to both of its ends and strained by some force.I know that people in United States and UK represent this equation in different form,so I`ll describe every parameter.The equation is:

$$y=A\sin(\omega t-kx)$$.

However,in one book it is given like:

$$y=A\cos(\omega t-kx)$$.

Why this second form is used and where it comes from?If I understood it correctly,it is the first equation where phase of the wave is shifted by $$+\frac{\pi}{2}$$,so the origin point of the Cartesian coordinate system used to analyze wave is moved for the corresponding length to the right,but for what purpose?

______________________________________________________________________________

$$y$$ - displacement of the particle of air caused by the wave in the moment of $$t$$ and at the distance of $$x$$ from the origin point of the Cartesian coordinate system used to analyze the wave

$$A$$ - amplitude of the wave

$$\omega=\frac{2\pi}{T}$$ and $$k=\frac{2\pi}{\lambda}$$ where $$T$$ is the period of the wave and $$\lambda$$ is the wavelenght of the wave.

 

[R A V E N]

Question 3:

We have a damped harmonic mechanical oscillator - a moving body on the spring.If force of damping is proportional to the speed of that body then:

a)none of the statements below is true - CORRECT
b)frequency decreases with time - INCORRECT,SINCE FREQUENCY IS NOT FUNCTION OF TIME:
$$\omega=\sqrt{\omega_0^2-\frac{c^2}{4m^2}}$$
c)displacement of body is sinusoidal function of time - INCORRECT,INSTEAD IT IS A SINUSOIDAL FUNCTION OF ANGULAR DISPLACEMENT
d)velocity of body is sinusoidal function of time - INCORRECT,INSTEAD IT IS A SINUSOIDAL FUNCTION OF ANGULAR DISPLACEMENT
e)mechanical energy is constant - INCORRECT,MECHANICAL ENERGY APPROACHES TO ZERO AS TIME PASSES

Have I answered and explained all this right?Actually,this is a trick question,since force of damping is ALWAYS quantitatively proportional to the speed of damping $$F=cv$$.

 

[R A V E N]

Have I wrote something non-understandable since English is not my first language?

 

[Hootenanny]

Question 2:

The second question is about equation which describes a wave in air which originates from vibrating string attached to both of its ends and strained by some force.I know that people in United States and UK represent this equation in different form,so I`ll describe every parameter.The equation is:

$$y=A\sin(\omega t-kx)$$.

However,in one book it is given like:

$$y=A\cos(\omega t-kx)$$.

Why this second form is used and where it comes from?If I understood it correctly,it is the first equation where phase of the wave is shifted by $$+\frac{\pi}{2}$$,so the origin point of the Cartesian coordinate system used to analyze wave is moved for the corresponding length to the right,but for what purpose?

The two questions described two different scenarios.

For example, the first could describe a pendulum that starts from its equilibrium position. The second equation could represent a pendulum that starts from it's maximal displacement: imagine someone pulling a pendulum to the side and then releasing it.

Does that make sense?