Shm

[MedPR]

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Confused about SHM I guess.

A 10 kg mass attached to a frictionless horizontal spring system oscillates with an amplitude of 5cm and a frequency of 1 cycle/second. What is the maximum speed attained by the mass? The answer is 5*2pi m/s

I guess I should get more familiar with the equation that omega=2pi*frequency..

But that equation says that there are 2pi radians per cycle. So why do you multiply by 5cm? If 1 cycle = 2pi radians, then why doesn't it go 2pi m/s? 😕

 

[chiddler]

ω*Amplitude = maximum velocity. You can't use this for any velocity, though. Only max.

-ω^2 * x = acceleration. x = 0 is equilibrium, so when it's position is to the right, acceleration is to the left. This explains the negative sign.

ω^2 * A = max acceleration

so lets see:

A 10 kg mass attached to a frictionless horizontal spring system oscillates with an amplitude of 5cm and a frequency of 1 cycle/second. What is the maximum speed attained by the mass? The answer is 5*2pi m/s

(0.05)(2*π*1) m/s

I think you mean cm/s, right?

To answer the last question you wrote:

it doesn't go 2pi m/s because that is the radian value. Like saying it goes one full cycle per second. Well how much distance is that one cycle? It is 5 cm. That is as much as I can explain. Don't know it in more detail.

 

[MedPR]

ω*Amplitude = maximum velocity. You can't use this for any velocity, though. Only max.

-ω^2 * x = acceleration. x = 0 is equilibrium, so when it's position is to the right, acceleration is to the left. This explains the negative sign.

ω^2 * A = max acceleration

so lets see:

A 10 kg mass attached to a frictionless horizontal spring system oscillates with an amplitude of 5cm and a frequency of 1 cycle/second. What is the maximum speed attained by the mass? The answer is 5*2pi m/s

(0.05)(2*π*1) m/s

I think you mean cm/s, right?

To answer the last question you wrote:

it doesn't go 2pi m/s because that is the radian value. Like saying it goes one full cycle per second. Well how much distance is that one cycle? It is 5 cm. That is as much as I can explain. Don't know it in more detail.

Yea, I meant cm/s.

So ω=2pi*frequency. Frequency = cycles per second. What is the max speed if the oscillation amplitude is 5cm and the frequency is 1.

ω=2pi*1 = 2pi. That's what I'm not understanding :/

 

[chiddler]

Consider what this is implying

2pi m/s

It is going 6.14 m/s with a 5 cm amplitude!

And you can use this calculation with any amplitude. With this logic, anything going 1 Hz will go at 2pi m/s. So we must add in some way to account for the amplitude of oscillation. It is going one cycle per second (2pi) and one cycle is 5 cm (2pi*5 cm/s) so it's going one 5 cm cycle per second.

👍 or 👎?

 

[MedPR]

Consider what this is implying

2pi m/s

It is going 6.14 m/s with a 5 cm amplitude!

And you can use this calculation with any amplitude. With this logic, anything going 1 Hz will go at 2pi m/s. So we must add in some way to account for the amplitude of oscillation. It is going one cycle per second (2pi) and one cycle is 5 cm (2pi*5 cm/s) so it's going one 5 cm cycle per second.

👍 or 👎?

Nope, still not getting it. I'm sorry.

I'm pretty sure I'm missing something in regard to the difference between angular frequency (&#969😉 and angular velocity, which is what we are really trying to calculate here...

Edit: The problem tells you the frequency is 1. So that means the angular frequency ω is 2pi. However, this doesn't really tell you much. You know that every 1 second the thing travels 5cm. So the linear velocity is 5cm/s.

Now I'm stuck. Wikipremed says to use an analogy of a circle, and I'm not really understanding that either. I understand that 2pi radians = 360 degrees = a circle, so it kind of makes sense. I think I'm missing one little piece of the puzzle and I just can't wrap my head around it until I get that piece.

I mean, I understand it from the point of view that the oscillating spring is analogous to a circle.
However, I don't understand why or how you know that the oscillating spring is analogous to a circle.

 

[chiddler]

Nope, still not getting it. I'm sorry.

I'm pretty sure I'm missing something in regard to the difference between angular frequency (&#969😉 and angular velocity, which is what we are really trying to calculate here...

Edit: The problem tells you the frequency is 1. So that means the angular frequency ω is 2pi. However, this doesn't really tell you much. You know that every 1 second the thing travels 5cm. So the linear velocity is 5cm/s.

Now I'm stuck. Wikipremed says to use an analogy of a circle, and I'm not really understanding that either. I understand that 2pi radians = 360 degrees = a circle, so it kind of makes sense. I think I'm missing one little piece of the puzzle and I just can't wrap my head around it until I get that piece.

I mean, I understand it from the point of view that the oscillating spring is analogous to a circle.
However, I don't understand why or how you know that the oscillating spring is analogous to a circle.

No a full cycle is two amplitudes. So it goes amplitude to the left, and then amplitude to the right. Bam, full cycle. That is 10 cm. The average linear velocity is therefore 10 cm/s, but this is not the max velocity. Remember it is always changing as it oscillates, with its maximum at equilibrium position. This is also the reason you can't say: "if, in SHM, spring goes 0.1 cm in the first two seconds, how far does it go in the next two seconds?" The answer is not 0.1 cm again.

A circle or sin graph. Both are repeating and are comparable to oscillation. Here

edit: my graph should be cos, not sin, because the spring starts cocked.

 

[MedPR]

No a full cycle is two amplitudes. So it goes amplitude to the left, and then amplitude to the right. Bam, full cycle. That is 10 cm. The average linear velocity is therefore 10 cm/s, but this is not the max velocity. Remember it is always changing as it oscillates, with its maximum at equilibrium position. This is also the reason you can't say: "if, in SHM, spring goes 0.1 cm in the first two seconds, how far does it go in the next two seconds?" The answer is not 0.1 cm again.

A circle or sin graph. Both are repeating and are comparable to oscillation. Here

edit: my graph should be cos, not sin, because the spring starts cocked.

Ok so it starts from x=10cm. Then since its amplitude is 5cm, it extends out to 15cm. This 15cm point represents pi, right? Then it comes back to 10cm, the starting point, representing 2pi?

Ok, I guess I see how it mimics a circle. So one period = one circumference = 2pi*r, where r is the amplitude! So the maximum speed is 10pi or ~31.4. Thank you!

Now my next question. When does it reach this speed? I know that KE is maximal at pi/2 and 3pi/2... Isn't the velocity decreasing as it approaches 2pi from 3pi/2? Since the acceleration is in the opposite direction?

 

[chiddler]

yes, but it's a lot easier to think of it as equilibrium is x=0 as opposed to what you wrote x=10. it doesn't make a difference, but...you know...easier.

 

[MedPR]

yes, but it's a lot easier to think of it as equilibrium is x=0 as opposed to what you wrote x=10. it doesn't make a difference, but...you know...easier.

I didn't want to use x=0 because it doesn't leave any room for the overshoot displacement...

 

[chiddler]

yes it should be decreasing because it is returning to its loaded 100% potential energy position from its 100% kinetic energy position moments prior. So it slows down to convert KE to PE as spring is loaded.

welcomes :-3

 

[MedPR]

yes it should be decreasing because it is returning to its loaded 100% potential energy position from its 100% kinetic energy position moments prior. So it slows down to convert KE to PE as spring is loaded.

welcomes :-3

Ok so the max speed is at intervals of 90deg or pi/2?

 

[chiddler]

I didn't want to use x=0 because it doesn't leave any room for the overshoot displacement...

on the contrary, xo=0, then x = 10, x = -10, x=0 again. nice and symmetric. but lets not argue this 😛. just preference anyway.

Ok so the max speed is at intervals of 90deg or pi/2?

The oscillating system reaches max speed every pi or 180 degrees. look at the graph I posted: max speed (or maximum and minimum velocity) occur at 1/4 period and 3/4 period. So every 1/2 period, max speed is reached. Half of one cycle is pi, or 180 degrees.

another way to think about this is: how often in one cycle does the oscillation go through equilibrium position?

 

[thais]

I worked the problem out a little different, this might help (I hope it is correct).
Due to the conservation of energy theorem, when the spring is at maximum speed PE energy is 0.
PE for spring = kx^2/2

Then: kx^2/2 = mv^2/2 (PE = KE)

then solve for v. v^2 = kx^2/m

v = sqrt (k/m) * x then, from memory we know that ω = sort(k/m)

v = ω * x because ω = 2pif

v = 2pi*5

A mnemonic for ω and f is f=1/2pi sqrt(k/m ) → Mnemonic: "I frequently drop my 2 pies" then alphabetical (k/m)

I hope it helps.