Solving Oscillation & SHM Difficulties

[Amr Elsayed]

I have some difficulties regarding oscillation and SHM, i hope someone makes it clearer to me.
Firstly, I don't have a good intuition of the formulas for velocity and acceleration as functions of time. I have no idea why the negative sign is present in the formula and what it's supposed to mean & adjust. I tried to derive at on my own but there was no negative sign

Secondly, the period of a string doesn't depend on the amplitude, right ? In case I have a vertical string with some mass attached to it, isn't this mass supposed to change the equilibrium position of the spring, and then I don't account for it in my calculations as i use the new equilibrium position ? How can being in an accelerating elevator affect the period however the formula for the period seems not to relate ??

 

[beamie564]

For the first part (negative sign of velocity), could you be more specific?
For the second part, when you attach a mass to the spring its equilibrium position changes. However, for any subsequent oscillations this new equilibrium position is what matters.
Now, for the spring in an elevator the time period changes because the spring is now in a non inertial frame, so the apparent value of g ( acc. due to gravity) changes.

 

[Amr Elsayed]

could you be more specific

I'm talking about the formula: v= -A* omega * sin( 2*pi*f*t) where did this negative sign come from ? and for what values is x positive and for what values is it negative ?

for the third part, i don't get how changing g relates to the period. I know that the period equals 2*pi / omega

 

[beamie564]

For the negative sign: I'll assume we're talking about a horizontal spring on a frictionless surface ( to eliminate gravity).
You have x=Acos( omega*t)
We'll assume x +ve towards the right.
At t=0 the spring is at the extreme position ( note that x=+A). When you release the spring, the velocity is in the negative direction, while the displacement is still positive.. This changes as the mass passes the equilibrium point. Both the displacement and velocity are now in the negative direction ( for more understanding- in the second quadrant cos@ is -ve and sin@is +ve. You can see that this gives -ve x and v).

 

[beamie564]

I'm sorry if I couldn't properly explain it to you.. It's so much easier if I could scribble some diagrams ☹

 

[Amr Elsayed]

I'm sorry if I couldn't properly explain it to you.. It's so much easier if I could scribble some diagrams ☹

would you please explain more how 2*pi*f*t could give values ranging from 0 to 180 or to pi and so on...
what about the third part ?? how acceleration due to gravity relates to the period of a vertical spring ?? isn't it supposed to merely change the equilibrium position ?

 

[sophiecentaur]

Firstly, I don't have a good intuition of the formulas for velocity and acceleration as functions of time.

I hate to be a drag about this but I just don't believe it's possible to derive how a mass on a spring (or a pendulum on a string) produces Simple Harmonic Motion, "intuitively". It involves solving a second order differential equation of motion.
If you don't like the Maths then you can only believe that it works and accept the results. This wiki link explains the whole business quite well. You may be trying to jump in too deep too quickly in this subject. I think it would be best to start with a more basic approach.

 

[beamie564]

how acceleration due to gravity relates to the period of a vertical spring ?? isn't it supposed to merely change the equilibrium position ?

I misspoke. The time period for a vertical spring doesn't depend upon the value of g, the time period for a string ( pendulum) does..
I had string in my mind and I wrote spring instead..
( in my defence you mentioned both string and spring in your doubt). Sorry for any confusion caused.

 

[Amr Elsayed]

It involves solving a second order differential equation of motion.

Actually I don't want to have the intuition how it's derived. I want to have the intuition how it works. Specifically, I want to know how the equation and the negative sign in it relate to the fact that velocity can be either in the same or opposite direction of the displacement. How can all the stuff in front of Sin give an angle ranging from 0 to 180 and from 180 to 360, and the same about cosine.
BTW, I believe it can be derived using substitution, since we have a formula for position as a function of time and a formula for velocity as a function of position.

I had string in my mind and I wrote spring instead..

Does this mean that non inertial frames have no effect on the period of a vertical spring ??

 

[beamie564]

Actually I don't want to have the intuition how it's derived. I want to have the intuition how it works. Specifically, I want to know how the equation and the negative sign in it relate to the fact that velocity can be either in the same or opposite direction of the displacement. How can all the stuff in front of Sin give an angle ranging from 0 to 180 and from 180 to 360, and the same about cosine.
BTW, I believe it can be derived using substitution, since we have a formula for position as a function of time and a formula for velocity as a function of position.
Does this mean that non inertial frames have no effect on the period of a vertical spring ??

Looks that way, provided that you don't accelerate in between an observation.

 

[sophiecentaur]

I want to have the intuition how it works.

Doesn't the maths give the best description about how it works? The Maths includes the 'sign' in the equation of motion, which tells you it's a restoring force and solving the equation tells you the way the position / velocity / acceleration etc all very in time. Arm waving and intuition will only give you a qualitative idea and not show why the resulting motion wwillould be regular or what waverorm it would take. It works like the Maths tells you. Can one get better than that?

 

[Amr Elsayed]

Doesn't the maths give the best description about how it works?

Yes, I got it... I had some mathematical little problem that I could solve, I now have the intuition how it works.
Would you please help me with this ? does instantaneous conversion from an inertial frame to a non-inertial frame like an accelerating elevator have something to do with the period of oscillation of vertical and horizontal springs ?? isn't the mass we're concerned with the inertial mass not the invariant one ?

 

[jbriggs444]

Yes, I got it... I had some mathematical little problem that I could solve, I now have the intuition how it works.
Would you please help me with this ? does instantaneous conversion from an inertial frame to a non-inertial frame like an accelerating elevator have something to do with the period of oscillation of vertical and horizontal springs ??

Suppose you hang an ideal massless spring from the ceiling of an elevator and attach a mass to the bottom. You know the spring constant and you know the mass of the object. Can you calculate the period of its up and down oscillation? Does that period depend on the value of g?

 

[Amr Elsayed]

Suppose you hang an ideal massless spring from the ceiling of an elevator and attach a mass to the bottom. You know the spring constant and you know the mass of the object. Can you calculate the period of its up and down oscillation? Does that period depend on the value of g?

Actually it's not, but it depends on the inertial mass, right ? if the elevator accelerates the inertial mass would increase or decrease.. but in case the mass we're concerned with is the invariant mass, no change will occur

 

[Amr Elsayed]

Is the change in weight due to being in a non inertial frame adjusted with a change in g to keep the m constant or what ?

 

[jbriggs444]

Actually it's not, but it depends on the inertial mass, right ? if the elevator accelerates the inertial mass would increase or decrease.. but in case the mass we're concerned with is the invariant mass, no change will occur

Inertial mass and invariant mass are one and the same thing.

 

[Amr Elsayed]

Inertial mass and invariant mass are one and the same thing.

Oh.. so it's g that changes to match the change in the weight when an elevator accelerates, and I don't need to be concerned with the inertial mass in classical mechanics. okay, thanks