Is the Velocity Equation for SHM Correctly Derived Without Calculus?

[ProPM]

Hi

I am a bit lost with the equations for velocity:

I don't know Calculus yet, so my teacher just gave me the equation:

-wx₀cos(wt) (w being omega)

He then said: v₀ = wx₀

and therefore, concluded: -v₀cos(wt)

and then for when the displacement is maximum at time = 0: v₀cos(wt)

Is this correct? I mean, I am obviously not doubting him but I am a bit confused plus my notes were not very organized on this day...

Thanks in advance

 

[rl.bhat]

when t =0, ωt = 0 and cos(ωt) = 1

So vo = -xo*ω

 

[ProPM]

So the equation would be:

v₀cos(wt)

and for sin, how would it work?

 

[mukundpa]

The sign and the function sin or cos depends on the instant you are taking t=0. If t is taken zero when the particle is at the equilibrium position (x=0) than the equation for displacement will be x= A sin wt and that for velocity will be v = Aw cos wt

thus at extreme position wt = 90 deg, gives x = A and v = 0.
(A is amplitude and Aw = Vo, the maximum velocity)

 

[rwisz]

Hi there,

First of all, don't worry if you are not familiar with Calculus yet. The equations for Simple Harmonic Motion (SHM) can be quite complex, and it takes time to fully understand them. Let's break down the equations step by step to help you understand them better.

The equation you were given, -wx0cos(wt), is the equation for the displacement of an object undergoing SHM. The variable w represents the angular frequency, and x0 represents the amplitude (maximum displacement) of the motion. This equation can be derived using Calculus, but for now, it's important to understand that it represents the back and forth motion of an object in SHM.

Next, your teacher mentioned that v0 = wx0. This is the equation for the initial velocity of the object. In SHM, the initial velocity is equal to the product of the angular frequency and the amplitude. This means that the initial velocity is directly proportional to the amplitude of the motion.

Using this equation, your teacher concluded that the velocity equation for SHM is -v0cos(wt). This is because the velocity of an object in SHM is always changing, and it can be represented by the derivative of the displacement equation. Again, don't worry if you don't fully understand this yet, as Calculus is needed to fully grasp this concept.

Finally, when the displacement is maximum at time = 0, the velocity equation becomes v0cos(wt). This is because at this point, the object is moving at its maximum speed in the positive direction, and the cosine function represents this motion.

I hope this helps clarify the equations for you. Keep practicing and asking questions, and you will soon have a better understanding of SHM and its equations. Good luck!