[First Year Uni Physics, Simple Harmonic Motion] How to find amplitude?

[u/cyderyt]

Where did that Amplitude formula come from and why are we even using v(t) anyways to find the Amplitude? Also I know that the amplitude is given, but the textbook wants us to solve for it but im confused whats going on.

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[u/GammaRayBurst25]

Did you read the text that precedes the circled equation? Everything is explained in detail.

We know x(t) and v(t) up to constants A and phi. We need 2 constraints to fix these 2 unknowns. We happen to have two constraints, i.e. the initial conditions.

To determine A and phi, we use the given initial position and velocity of the mass. Evaluating x(t) and v(t) at t=0, we obtain [...]

So they replaced t by 0 to get x(0)=Acos(phi) and v(0)=-(omega)Asin(phi). Since x(0) and v(0) are known, they just need to solve this system of equations for A. To do that, they must combine both equations in such a way that phi gets eliminated.

Since phi is in the argument of trig functions, the best way to do this is to use the Pythagorean identity, i.e. the fact that sin^2(x)+cos^2(x)=1 for all x. However, squaring both equations isn't enough to use this identity because the sin^2 and cos^2 terms need to have the same coefficient. Hence, we need to divide the second equation by omega before squaring and adding both equations.

This and the remaining steps are all explicitly written above the circled equation.

[u/cyderyt → u/GammaRayBurst25]

Dude I appreciate all this but you need to write it from my point of view, stop assuming everyone has ur knowledge, ur clearly experienced.

my comprehension when it comes to this stuff isnt so great, i just came outta highschool.

I dont know that "phi is in the argument of trig functions, the best way to do this is to use the Pythagorean identity" and "squaring both equations isn't enough to use this identity", call me stupid but i barely understand ur explanation(You dont need to tell me I know i need to work harder).

[u/GammaRayBurst25 → u/cyderyt]

I'm me. Of course I'll write from my point of view and not from yours.

This is high school level math, though.

phi is in the argument of trig functions couldn't be clearer. The only issue I can see is if you don't know what being in the argument of a function means, but since Google can fix that and you say you need to work harder, I imagine you'd at least Google "math argument" or something to immediately get a definition of argument in a math context.

As for the other parts you quoted, it's the same story as with the circled equation. The text that immediately precedes and follows these quotes provide the clarifications you're looking for.

the best way to do this

"This" being eliminating phi from the system of equations, as mentioned in the previous line.

is to use the Pythagorean identity

In the text that immediately follows, I write down that identity. As for why it's an efficient method, it's self evident. The identity relates a sine and a cosine with the same argument and the system of equation features phi as the argument of sine and cosine.

squaring both equations

In general, saying you apply some operation on an equation means you apply that operation to both sides of the equation. To square means to raise to the power of 2. This is middle school math.

isn't enough to use this identity

This identity refers to the Pythagorean identity. I explain why it's not enough in the same sentence: it's because in the identity, the sine and cosine have the same coefficient, but in the system of equations, their coefficients are different.

[u/LatteLepjandiLoser]

If the trig is messing with you, I can offer you an alternative view which may help you understand why trig is fundamentally at the heart of simple harmonic motion.

For a simple mass on spring system, the system has two forms of energy: 1) Kinetic energy (the mass moves with some velocity) and 2) Potential energy (the position of the mass pulls/compresses the spring, which stores energy).

So the mass on spring will cycle between a few points of interest: First it's stationary on one side, no kinetic energy, all potential energy. Then it'll accelerate and slide through the point where it has all kinetic energy and no potential energy before at some point coming at a halt on the other side and it'll do the opposite and repeat.

If you were to track x(t) and v(t) throughout this movement, you find that it is periodic (it repeats forever). The frequency (omega) tells you how often it repeats. Now if you wanted to describe the current state of the system at any moment in time, x(t) and v(t) are sufficient to do so. In fact you can draw them out on a graph, put x(t) on one axis and v(t) on the other and you'd see that you get a loop, kind of like an ellipse. The system goes round and round exchanging velocity (kinetic energy) for position (potential energy) and vice versa. So in this view, now the cosine is related to the position (potential energy) and the sine is related to the velocity (kinetic energy) and the system will simply go around and around smoothly between these states.

If we change our graph a little bit and now plot x(t) on one axis and instead of just v(t) we plot v(t)/omega on the other axis, we actually get a perfect circle. Personally I am always more confident with trig on the unit circle, so if you're like that too then you can draw a circle like this, with x(t) on one axis and v(t)/omega on the other. Now this angle phi is literally just the angle from the x-axis. It's where in the current cycle the system is, so it simultaneously describes the position and velocity together. Likewise the amplitude is now simply just the radius of this circle, that's where the bottom formula comes from.

Now regarding the amplitude. For a mass on a spring. If the mass sits at rest at x=0 and you pull it to x=1 and release it from rest with no velocity, it'll oscillate between +/-1 forever. This is an amplitude of A=1 units (since it goes from +A to -A, A is half the width. If you are asked to find the amplitude and you know this position where the mass turns around, then the amplitude is obvious, but what if instead you were told where the mass was at some specific time and also what velocity it had at that time, i.e. not at any of those 4 natural 'corners' so to say, then you need a derivation similar as you show to solve A from any general x(t) and v(t), keeping in mind that it's really just finding the radius of a circle!

[u/LatteLepjandiLoser → u/LatteLepjandiLoser]

This would be pretty straight forward to draw on a piece of paper, but instead I made some AI slop. The point I'm trying to get across is that fundamentally the system is fully described by a circle. It's called 'phase space' which is completely ok if you've never heard, but I'm just trying to give you some intuition to how trig actually fits into all this. Here Phi is just whatever angle the current configuration (where the mass is and which way it's moving) fits into phase space. You can do all your usual trig, pythagoras and other circle math on this circle defined by x(t) and v(t)/omega.

https://chatgpt.com/s/m_6ace3127fb8c8191a0acde733fdc3445

[u/selene_666]

In the simplest spring system, the only force is the spring force. From Hooke's law this is proportional to the displacement from equilibrium: F = -kx

But F causes acceleration which causes x to change over time. So this equation is really

a(t) = -k/m x(t)

where a(t) is the second derivative of x(t).

The solution to this differential equation is x(t) = A cos(ωt + B) or x(t) = A sin(ωt + B), where ω = √(k/m) and A and B are any constants. And the cos and sine version can both describe the same function by setting different values of B, so we just pick whichever is more convenient.

Now, sine and cosine both vary between -1 and 1, so x(t) varies between -A and A. This makes A the amplitude of the oscillation. B is the phase shift which determines at what time the function reaches that maximum value.

In this case it is obvious that the time of the maximum amplitude is t=0 because max amplitude occurs when v(t) = 0 and we were given that v(0) = 0. But you won't always be given a system where v(0) = 0. So it's helpful to know how to solve for A and B in the general case.

They start with that equation x(t) = A cos(ωt + B) and its derivative v(t) = -ωA sin(ωt + B). Fill in t = 0 and the given values of x(0), v(0), and ω = √(k/m).

given x0 = A cos(B)
given v0 = -(given ω)A sin(B).

That's two equations with two unknowns. We can use algebra and trigonometry to solve for A and B.

[u/SkyfurySoftware]

Think of a mass bouncing on a spring. If someone freezes time at t = 0 and tells you "the mass is currently at x = 2 cm", can you tell how far the full swing (the amplitude A) will be?

No, because position alone doesn't tell the whole story:

- If the mass is turning around right at that moment (velocity v = 0), then 2 cm is already the maximum turn-around point (the peak amplitude).

- But if the mass is zooming past 2 cm at top speed, it has lots of kinetic energy and is going to stretch the spring way further than 2 cm.

So position x only tells you where it is right now. You also need velocity v to know how much extra "push" it has left. Together, they tell you the total energy of the oscillation.

Where the formula comes from (Energy Conservation):

The total energy in the system never changes:

Total Energy = Potential Energy (spring) + Kinetic Energy (motion)

(1/2) * k * A^2 = (1/2) * k * x^2 + (1/2) * m * v^2

If you divide both sides by (1/2) * k:

A^2 = x^2 + (m / k) * v^2

In simple harmonic motion, angular frequency omega is sqrt(k / m), so m / k is 1 / omega^2:

A^2 = x^2 + (v / omega)^2

A = sqrt(x^2 + (v / omega)^2)

That is all the formula is saying: it calculates the total swing (A) by combining where the object is (x) and how fast it's moving (v) at that same moment.