The kinetic energy of a spring

In summary, the answer key for this problem recommends using the formula for kinetic energy, but when attempting to do so, it was apparently the wrong answer. This seems to be due to the fact that an unstretched spring has a velocity, and when using the kinetic energy equation, the velocity is not accounted for.
  • #1
Differentiate it
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2
Homework Statement
Could someone explain the situation here?
Relevant Equations
(1/2)mv^2
Screenshot_2022-12-03-18-00-35-87_1ab726e7599468b75300a3cdb0d53113.jpg

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?
 
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  • #4
Differentiate it said:
Yes, I tried
I think @BvU's point is that we need to see it!
 
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  • #5
Differentiate it said:
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?
 
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  • #7
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?
 
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  • #8
bob012345 said:
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?
 
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  • #9
bob012345 said:
The free end is suddenly pulled at a velocity v.
Not suddenly, I think. That would get you into complexities of internal oscillations.
 
  • #10
haruspex said:
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.
 
  • #11
bob012345 said:
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.
 
  • #12
haruspex said:
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.
 
  • #13
bob012345 said:
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.
 
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  • #14
Differentiate it said:
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?
 

1. What is the formula for calculating the kinetic energy of a spring?

The formula for calculating the kinetic energy of a spring is:
KE = 1/2 * k * x^2, where KE is the kinetic energy, k is the spring constant, and x is the displacement from the equilibrium position.

2. How does the mass of an object affect the kinetic energy of a spring?

The mass of an object does not directly affect the kinetic energy of a spring. The kinetic energy of a spring is determined by the spring constant and the displacement from the equilibrium position.

3. What is the relationship between the potential energy and kinetic energy of a spring?

The potential energy and kinetic energy of a spring are interrelated. As the potential energy of a spring increases, the kinetic energy decreases, and vice versa. This is because the total energy of the system (spring and object) remains constant.

4. Can the kinetic energy of a spring be negative?

Yes, the kinetic energy of a spring can be negative. This means that the object attached to the spring is moving in the opposite direction of the spring's displacement. However, the total energy of the system (spring and object) will still remain constant.

5. How does the spring constant affect the kinetic energy of a spring?

The spring constant has a direct effect on the kinetic energy of a spring. A higher spring constant means that the spring is stiffer, and therefore, the object attached to it will have a greater kinetic energy when displaced. On the other hand, a lower spring constant will result in a lower kinetic energy for the object.

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