A Trolley oscillates horizonally between two springs....

[Mr Smailes]

Homework Statement

Help needed with part 4c.ii of the attached paper.

Relevant Equations

K.E. = 1/2mv^2 P.E = 1/2kx^2

How do I know the position where the K.E is equal to the P.E?

 

[haruspex]

When they are equal, how does the PE compare with the maximum PE?
How does the PE vary with displacement?

 

[BvU]

Homework Statement:: Help needed with part 4c.ii of the attached paper.
Relevant Equations:: K.E. = 1/2mv^2 P.E = 1/2kx^2

How do I know the position where the K.E is equal to the P.E?

Hello mr @Mr Smailes ,

$\qquad$!​

Our guidelines require a bit more effort from you in tackling this exercise...
[edit]Ah, I see @haruspex already asked the relevant question

 

[Mr Smailes]

When they are equal, how does the PE compare with the maximum PE?
How does the PE vary with displacement?

I'm confused due to there being two springs so would the PE for one not be maximum when it is minimum for the other?

 

[TSny]

I'm confused due to there being two springs so would the PE for one not be maximum when it is minimum for the other?

I think you are supposed to assume that, when the trolley is in the middle at the equilibrium position, then the springs are neither stretched nor compressed. So, at that position, neither spring has any potential energy.

As you stretch a spring, the potential energy of the spring increases. What happens to the potential energy of the spring as you compress it?

 

[haruspex]

I think you are supposed to assume that, when the trolley is in the middle at the equilibrium position, then the springs are neither stretched nor compressed.

Probably, but you could also consider that the PE of the spring system means only the available energy. At the equilibrium position, none is available.

 

[jbriggs444]

I'm confused due to there being two springs so would the PE for one not be maximum when it is minimum for the other?

Indeed, that is likely so. But in spite of this, is the sum of the two potentials constant?

Also, remember that only differences in potential energy are physically meaningful. If you find yourself trying to equate potential energy with kinetic energy, you are likely doing something wrong. If you find yourself trying to relate the change in potential energy with the change in kinetic energy then you are more likely to be correct.

 

[TSny]

Probably, but you could also consider that the PE of the spring system means only the available energy. At the equilibrium position, none is available.

Yes. Maybe I am not correctly interpreting @Mr Smailes remark:

I'm confused due to there being two springs so would the PE for one not be maximum when it is minimum for the other?

He seems to be thinking that as one spring increases its PE, the other spring decreases its PE. Now that would actually be possible if, say, both springs are highly compressed when the trolley is at the equilibrium position. But, if we assume the springs have their natural length at the equilibrium position, then both springs increase their PE as the trolley moves away from equilibrium. This is what I hoped he would realize.

 

[haruspex]

Yes. Maybe I am not correctly interpreting @Mr Smailes remark:

He seems to be thinking that as one spring increases its PE, the other spring decreases its PE. Now that would actually be possible if, say, both springs are highly compressed when the trolley is at the equilibrium position. But, if we assume the springs have their natural length at the equilibrium position, then both springs increase their PE as the trolley moves away from equilibrium. This is what I hoped he would realize.

Yes, there are two issues here:
- the possible misconception that, even if they are relaxed at equilibrium, adding PE to one reduces it for the other
- if they are not relaxed at equilibrium and we interpret their overall PE as the sum of their individual PEs then a) that might always exceed the KE; and b) there would be phases in which one loses PE as the other gains it.

 

[TSny]

Yes, there are two issues here:
- the possible misconception that, even if they are relaxed at equilibrium, adding PE to one reduces it for the other
- if they are not relaxed at equilibrium and we interpret their overall PE as the sum of their individual PEs then a) that might always exceed the KE; and b) there would be phases in which one loses PE as the other gains it.

Yes.

 

[Mr Smailes]

Thanks for all your help I now understand the potential energy is increasing from the equilibrium position and the kinetic energy is decreasing from the equilibrium position where it is max. So they will be equal at some point between the equilibrium position and the max amplitude but how do I know where?

 

[jbriggs444]

By itself, potential energy is meaningless. Only differences are meaningful.

If you change the reference zero point for your measurement of potential energy, potential energy changes. But nothing physical is any different. If you are trying to compare potential energy with kinetic energy, you are doing something wrong.

 

[haruspex]

By itself, potential energy is meaningless. Only differences are meaningful.

If you change the reference zero point for your measurement of potential energy, potential energy changes. But nothing physical is any different. If you are trying to compare potential energy with kinetic energy, you are doing something wrong.

As I wrote in post #6, I think it is reasonable to consider the PE here as only that which is available (for conversion to KE).

 

[haruspex]

So they will be equal at some point between the equilibrium position and the max amplitude but how do I know where?

Have you tried answering my questions in post #2?

 

[Mr Smailes]

The PE increases with displacement but I am still unsure of your first question in how does PE compare with max PE?

Have you tried answering my questions in post #2?

 

[haruspex]

The PE increases with displacement but I am still unsure of your first question in how does PE compare with max PE?

No, I asked

When they are equal, how does the PE compare with the maximum PE?

I.e. when PE and KE are equal.
Perhaps I should start with a more basic question: what is the relationship between max PE and max KE (assuming we are defining PE as that which is available for conversion to KE)?

 

[Mr Smailes]

No, I asked

I.e. when PE and KE are equal.
Perhaps I should start with a more basic question: what is the relationship between max PE and max KE (assuming we are defining PE as that which is available for conversion to KE)?

Max P.E is equal to minimum K.E and vice versa

 

[haruspex]

Max P.E is equal to minimum K.E and vice versa

Max PE > min PE = max KE > min KE = max PE? That makes no sense.

 

[Mr Smailes]

Max PE > min PE = max KE > min KE = max PE? That makes no sense.

Maximum potential energy is at either end where K.E is zero. In the middle would the K.E not be maximum and the potential energy be 0?

 

[haruspex]

Maximum potential energy is at either end where K.E is zero. In the middle would the K.E not be maximum and the potential energy be 0?

Right. But what is the relationship between max PE and max KE?

 

[Mr Smailes]

Right. But what is the relationship between max PE and max KE?

The maximum kinetic energy must be equal to the maximum potential energy presuming all of the potential energy is converted to kinetic

 

[haruspex]

The maximum kinetic energy must be equal to the maximum potential energy presuming all of the potential energy is converted to kinetic

Ok, so when the PE and KE are equal, how do they compare to their maxima?

 

[Mr Smailes]

Ok, so when the PE and KE are equal, how do they compare to their maxima?

They will be lower then the maxima when they are equal, it will be at some point between the max amplitude and the equilibrium position

 

[haruspex]

They will be lower then the maxima when they are equal, it will be at some point between the max amplitude and the equilibrium position

Don't worry about amplitude for now, concentrate on energy.
Suppose the block is moving away from the equilibrium position. As it moves some particular distance, say it gains an amount $\Delta E$ of PE. How much KE does it lose in the process?

 

[Mr Smailes]

Don't worry about amplitude for now, concentrate on energy.
Suppose the block is moving away from the equilibrium position. As it moves some particular distance, say it gains an amount $\Delta E$ of PE. How much KE does it lose in the process?

The same amount?

 

[haruspex]

The same amount?

Right. It is an example of conservation of work.
So I ask again, if $max KE=max PE=E$, then what are the KE and PE equal to when they equal each other?

 

[Mr Smailes]

So at the halfway point?

 

[haruspex]

So at the halfway point?

At the moment, we're just discussing energy. Please try to answer my question in post #26.

 

[Mr Smailes]

KE and PE will add together to equal the total energy

 

[haruspex]

KE and PE will add together to equal the total energy

Right, so each will be E/2.
The next step is to answer my second question in post #2. If the displacement is x and the effective spring constant is k, what is the PE?

 

[Mr Smailes]

E/2 = 1/2kx^2

 

[BvU]

E/2 = 1/2kx^2

It looks as if you are trying to take two steps at once. How does this compare to what I learned, that $E = {1\over 2} kx^2$ ?

 

[haruspex]

It looks as if you are trying to take two steps at once. How does this compare to what I learned, that $E = {1\over 2} kx^2$ ?

I previously defined E in this thread as the max KE.

 

[haruspex]

E/2 = 1/2kx^2

Right.
To be clear, let's call that displacement $x_{E/2}$.
So how does that displacement compare with the maximum displacement, $x_E$?

 

[Mr Smailes]

E/2 = 1/2k(x/2)^2 and E = 1/2kx^2