Work-Energy Theorum: Spring potential energy vs Kinetic Energy

In summary, the conversation discusses the application of the Work-Energy Theorem to determine the spring constant of a horizontal coiled spring. The law of conservation of energy is used and the equation for spring potential energy is identified. The participant notes an algebra error and confirms that the spring constant should always be positive. They also mention the importance of considering the negative sign in the equation for the force of a spring. The conversation ends with a clarification on the calculation of work done by a spring against external forces and the resulting negative change in kinetic energy.
  • #1
Senjai
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[SOLVED] Work-Energy Theorum: Spring potential energy vs Kinetic Energy

Homework Statement



A 1350-kg car rolling on a horizontal surface has a speed v = 40 km/h when it strikes a horizontal coiled spring and is brought to rest in a distance of 2.5 m. What is the spring constant of the spring? Ignore Friction and assume spring is mass-less.

Homework Equations


[tex] W = \Delta E[/tex]
[tex] E_{pspring} = \frac{1}{2}(kx^2) [/tex]
[tex] E_k = \frac{1}{2}(mv^2) [/tex]


The Attempt at a Solution



First right off the bat, i converted 40 km/h to its m/s equivalent of aprox. 11.11 m/s

i state the law of conservation of energy: Energy before = Energy after

Therefore:

[tex]
E_k = E_{pspring}
\frac{1}{2}(mv^2) = \frac{1}{2}(kx^2)
[/tex]

then i isolate k

[tex] k = \frac{-mv^2}{x^2} [/tex]

now here's the issue, is x negative? because the displacement is against the direction of motion?
and 2.5m = x, (-2.5)^2 gives me a answer of 4266 Nm
but -(2.5)^2 is entirely different.. This has been a long lasting math issue for me.

And what if x is positive?

i know k MUST be positive right?
 
Last edited:
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  • #2
(2.5)^2 is correct. There is no negative energy in the nature.
 
  • #3
There is no minus sign in mv^2 = kx^2
or in k = mv^2/x^2. No way you can get k negative!
The minus sign in F = -kx is supposed to help keep track of the fact that the force of the spring is opposite to the direction of stretch but it does seem to have a habit of getting in the way. k is ALWAYS positive.
 
  • #4
Thanks, the negative sign on mv^2 was an algebra error... Thanks for the clarification guys!
 
  • #5
Your attempt is correct but you missed somethingthat for a spring if you take natural length as the datum, the force on change in length is given as:
[tex] \vec{F}= -k \vec{x} [/tex]

and hence work done by a spring against external forces
[tex] W_{s}=\int\vec{F}\vec{.dx} [/tex]
over the required limits

in our case the answer is
[tex]W_{s}=-\frac{kx^{2}}{2}[/tex]
as
[tex]W_{s}=\Delta E[/tex]
[tex]\Delta E=-\frac{mv^{2}}{2}[/tex]
the change part was where you lost it all...the KE FELL TO ZERO. HENCE A NEGATIVE CHANGE.
 
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1. What is the Work-Energy Theorem?

The Work-Energy Theorem is a fundamental principle in physics that states that the net work done on an object is equal to the change in its kinetic energy.

2. How is the Work-Energy Theorem related to spring potential energy?

The Work-Energy Theorem can be applied to objects connected to a spring, where the work done by the spring force is equal to the change in the object's potential energy stored in the spring.

3. What is spring potential energy?

Spring potential energy is the potential energy stored in a spring when it is stretched or compressed from its equilibrium position. It is directly proportional to the amount the spring is stretched or compressed.

4. How is spring potential energy different from kinetic energy?

Spring potential energy is a type of potential energy, meaning it is energy that an object possesses due to its position or state. Kinetic energy, on the other hand, is the energy an object possesses due to its motion. They are different types of energy and are not interchangeable.

5. Can the Work-Energy Theorem be applied to systems with both spring potential energy and kinetic energy?

Yes, the Work-Energy Theorem can be applied to systems with both types of energy. The total work done on the system will be equal to the change in the total energy, which is the sum of the spring potential energy and kinetic energy.

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