Solving Oscillation Problems with Steel Ruler Measurements

[Noiro]

Homework Statement

Steel ruler tip's oscillation amplitude is 1cm, frequency 5Hz. What is the path the ruler's tip goes within 2 seconds?

Homework Equations

I think I need to find a period so the equations would be T=1/5 s but I'm not sure.

The Attempt at a Solution

I don't know how to solve it so I would be grateful if someone could help me.

 

[Hootenanny]

Homework Statement

Steel ruler tip's oscillation amplitude is 1cm, frequency 5Hz. What is the path the ruler's tip goes within 2 seconds?

Homework Equations

I think I need to find a period so the equations would be T=1/5 s but I'm not sure.

The Attempt at a Solution

I don't know how to solve it so I would be grateful if someone could help me.

What does the question mean by "path"? Does it mean the arc length, or the normal deviation, or the tangential deviation?

 

[Noiro]

What does the question mean by "path"? Does it mean the arc length, or the normal deviation, or the tangential deviation?

It is the arc length

 

[Hootenanny]

It is the arc length

Oke doke.

As you correctly say, the period of oscillations is 1/5 seconds. What else do you think we need to work out in order to solve this problem?

 

[Noiro]

Oke doke.

As you correctly say, the period of oscillations is 1/5 seconds. What else do you think we need to work out in order to solve this problem?

Maybe a phase of an oscillation? I'm not sure though. What do we need?

 

[Hootenanny]

Maybe a phase of an oscillation? I'm not sure though. What do we need?

What do you usually need to work out the arc length (two things)?

 

[Noiro]

What do you usually need to work out the arc length (two things)?

Measure of a central angle and radius?

 

[RoyalCat]

Measure of a central angle and radius?

Scratch what I said for now.

Spoiler

Not to interlope, but the arc length is given, and you can't calculate it using the central angle and radius as those are unknowns for this problem.

I think what Hootenanny was trying to steer you towards was the distance traveled per period, try finding that and then try and see what distance the tip travels in 2 seconds.

 

[Hootenanny]

Measure of a central angle and radius?

Correct. So, what is the radius?

 

[Hootenanny]

but the arc length is given

No it isn't, the amplitude is given, but that isn't the arc length.

and you can't calculate it using the central angle and radius as those are unknowns for this problem.

Erm... yes you can.

 

[Noiro]

Correct. So, what is the radius?

I don't know. What is it?

 

[Hootenanny]

I don't know. What is it?

You have to work it out...

HINT: If you weren't given the frequency, but were instead given the radius of the pendulum, what equation would you use to determine the period of the oscillations?

 

[Noiro]

You have to work it out...

HINT: If you weren't given the frequency, but were instead given the radius of the pendulum, what equation would you use to determine the period of the oscillations?

I think it's T=2pi sqrt(L/g)

 

[Hootenanny]

I think it's T=2pi sqrt(L/g)

That is correct.

Just one minor point, do you have a diagram associated with this question? I'm assuming that this steel rule is hanging vertically and is oscillating in plane like a pendulum. Would that be correct?

 

[Noiro]

That is correct.

Just one minor point, do you have a diagram associated with this question? I'm assuming that this steel rule is hanging vertically and is oscillating in plane like a pendulum. Would that be correct?

No it's embedded in a clamp. I hope you can imagine what it looks like.

 

[ideasrule]

C'mon, get on MS Paint and draw a diagram. A lot of us here are lazy. :shy:

 

[Hootenanny]

No it's embedded in a clamp. I hope you can imagine what it looks like.

Ahh I see. I had originally envisaged a pendulum-like set up. However, this case actually turns out to be a lot simpler than the case that I had originally envisaged. The steel rule is acting like a mass-spring system and in this case to a fairly high degree of accuracy the tip of the rule doesn't trace an arc. Instead, it traces [approximately] a straight line.

So, let's forget about determining the radius. Instead, can you tell me how far the tip of the rule travels in one complete oscillation?

 

[Noiro]

Ahh I see. I had originally envisaged a pendulum-like set up. However, this case actually turns out to be a lot simpler than the case that I had originally envisaged. The steel rule is acting like a mass-spring system and in this case to a fairly high degree of accuracy the tip of the rule doesn't trace an arc. Instead, it traces [approximately] a straight line.

So, let's forget about determining the radius. Instead, can you tell me how far the tip of the rule travels in one complete oscillation?

Well if amplitude is 1cm then I guess it's 2cm. Am I right?

 

[RoyalCat]

Amplitude is the maximal displacement from the point of equilibrium.
The period is the length of time it takes the particle to return to its initial position, the length of how many amplitudes would a particle have to cover in distance over the course of 1 period?

Another hint:

Spoiler

This little diagram should hopefully make the point a bit more clear:
|---eq---|
--- is the amplitude, not --- ---.

 

[Hootenanny]

Well if amplitude is 1cm then I guess it's 2cm. Am I right?

As RoyalCat has said, 1cm is the maximal displacement (i.e. the distance between its equilibrium position and one of the turning points). What is a full oscillation in terms of turning points and the equilibrium position?

 

[Noiro]

As RoyalCat has said, 1cm is the maximal displacement (i.e. the distance between its equilibrium position and one of the turning points). What is a full oscillation in terms of turning points and the equilibrium position?

So it's 4cm right?

 

[Hootenanny]

So it's 4cm right?

Correct. So, if the ruler has a period of 1/5 seconds, how many oscillations occur in 2 seconds? Hence how far does the tip of the ruler travel in 2 seconds?

 

[Noiro]

Correct. So, if the ruler has a period of 1/5 seconds, how many oscillations occur in 2 seconds? Hence how far does the tip of the ruler travel in 2 seconds?

So in 2 seconds occur 10 oscillations and tip of the ruler travels 40cm. Am I right?

 

[Hootenanny]

So in 2 seconds occur 10 oscillations and tip of the ruler travels 40cm. Am I right?

Sounds good to me!

Apologies for the earlier confusion regarding the orientation of the ruler. Thanks to RoyalCat for pointing out the correct set up.

 

[Noiro]

Sounds good to me!

Apologies for the earlier confusion regarding the orientation of the ruler. Thanks to RoyalCat for pointing out the correct set up.

No problem. Thanks Hootenanny and RoyalCat you really helped me!