Why is the Kinetic Energy of a Spring (1/6)mv²?

[Differentiate it]

Homework Statement

Could someone explain the situation here?

Relevant Equations

(1/2)mv^2

I tried just using the formula for kinetic energy but that was apparently the wrong answer. The answer key says it's (1/6)mv². I don't understand how they got that answer. Could someone explain?

 

[BvU]

Sure !
Make a sketch and propose something. Anything !
(do read: https://www.physicsforums.com/threads/homework-help-guidelines-for-students-and-helpers.686781/)

 

[Differentiate it]

Sure !
Make a sketch and propose something. Anything !
(do read: https://www.physicsforums.com/threads/homework-help-guidelines-for-students-and-helpers.686781/)

Yes, I tried

 

[Steve4Physics]

Yes, I tried

I think @BvU's point is that we need to see it!

 

[Herman Trivilino]

Homework Statement:: Could someone explain the situation here?
Relevant Equations:: (1/2)mv^2

I tried just using the formula for kinetic energy

So, that would be correct for a particle, why do you think it doesn't work for a spring?

 

[kuruman]

See
https://en.wikipedia.org/wiki/Effective_mass_(spring–mass_system)

 

[bob012345]

You have to think about what is going on physically with this spring. One end is fixed and the other is free. The free end is suddenly pulled at a velocity $v$. You are looking at it at time zero, the free end has a velocity but hasn't actually stretched yet, it is still length $l$. What's going on with the rest of the spring at that instant?

 

[erobz]

You have to think about what is going on physically with this spring. One end is fixed and the other is free. The free end is suddenly pulled at a velocity $v$. You are looking at it at time zero, the free end has a velocity but hasn't actually stretched yet, it is still length $l$. What's going on with the rest of the spring at that instant?

To add to what @bob012345 is hinting at, specifically think about the fixed end. What velocity is it moving at?

 

[haruspex]

The free end is suddenly pulled at a velocity v.

Not suddenly, I think. That would get you into complexities of internal oscillations.

 

[bob012345]

Not suddenly, I think. That would get you into complexities of internal oscillations.

I looked at this problem as idealized. The free end of an unstretched spring is found to be moving at velocity $v$. Suddenly was an unfortunate choice not because of what the actual dynamics of a real spring would be since this is an ideal spring, but because there is no need to involve time at all. This is a snapshot.

 

[haruspex]

since this is an ideal spring

An ideal spring is massless; this one isn't, which is why a sudden tug will likely induce internal oscillations.

 

[bob012345]

An ideal spring is massless; this one isn't, which is why a sudden tug will likely induce internal oscillations.

Sure, but for this problem there is only an agent pulling the free end with uniform velocity. It must be assumed that there is also a velocity distribution set up along the spring and we don't need to worry about the details of how that came about.

 

[haruspex]

Sure, but for this problem there is only an agent pulling the free end with uniform velocity. It must be assumed that there is also a velocity distribution set up along the spring and we don't need to worry about the details of how that came about.

Quite so, we have to assume some sort of steady state, whatever that means in this context. Hence, no sudden movements.

 

[bob012345]

Homework Statement:: Could someone explain the situation here?
Relevant Equations:: (1/2)mv^2I tried just using the formula for kinetic energy but that was apparently the wrong answer. The answer key says it's (1/6)mv². I don't understand how they got that answer. Could someone explain?

@Differentiate it, have you used the hints we gave you to make progress with this problem?