Finding Spring Constant & Energy w/ Doubt in Exercise

In summary: Is this what you meant by your "position where it is ##E_{\text{k}}=U##" question?Yes, I meant the position where ##E_{\text{k}}=U##.
  • #1
Dunkodx
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Summary:: Doubt in a spring exercise

Text of the exercise "a mass of ##m = 0.4 \ \text{kg} ## is attached to a spring and it oscillates horizontally with period ##T = 1.57 \text{s}##; the amplitude of the oscillation is ##d = 0.4 \text{m}##. Determine the spring constant, the total energy of the system and the position where the kinetic energy is equal to the potential energy."

I have found the spring constant with the relation ##T = \frac{2\pi}{\omega}## and I've used the conservation of energy to say that ##E=\frac{1}{2} k \left(\frac{d}{2}\right)^2=\frac{1}{2}mv^2=\frac{1}{2} m\omega^2 \left(\frac{d}{2}\right)^2##; I have a doubt for the last request, that is the position where it is ##E_{\text{k}}=U##; my reasoning is that in a generic position it is, again for the conservation of energy, that ##\frac{1}{2}kx^2+\frac{1}{2}mv^2=\frac{1}{2}k \left(\frac{d}{2}\right)^2##, but since we want the position where ##E_{\text{k}}=U## and it is ##E=E_{\text{k}}+U## substituting ##E_{\text{k}}=U## leads to ##E_m=2U \implies E_{\text{k}}+U= 2\cdot \frac{1}{2} k \left(\frac{d}{2}\right)^2\implies \frac{1}{2}kx^2+\frac{1}{2}mv^2=k \left(\frac{d}{2}\right)^2##. Again, since ##\frac{1}{2}kx^2=\frac{1}{2}mv^2## because I am interested when kinetical and potential energy are the same, I get ##kx^2=k\left(\frac{d}{2}\right)^2 \implies x=\frac{d}{2}##.

However the solution says that ##x=\frac{d}{2\sqrt{2}}##, where is my mistake? Thanks.
 
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  • #2
What is ##E_m## in your equation soup above?
 
  • #3
@hutchphd: It was a typo, that ##E_m## is the same as all the other ##E## in the post. Sorry for the confusion.
 
  • #4
O.K. Why do you say ##E=\frac1 2 k (\frac d 2 )^2##
 
  • #5
Because there are only conservative forces, so when the mass reaches the maximum amplitude (that is, when it is at position ##\frac{d}{2}##) the kinetical energy is ##0## because the mass has no velocity and there is only potential energy ##\frac{1}{2} k \left(\frac{d}{2}\right)^2##; that means that ##E=\frac{1}{2}k \left(\frac{d}{2}\right)^2##.
 
  • #6
The amplitude is the size of the excursion from equilibrium in my vernacular. Check this.
 
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  • #7
The question explicitly states that the amplitude is d. And there is no "maximum amplitude". Amplitude is the maximum displacement from equilibrium. Unless otherwise stated (like in peak-to-peak amplitude).
 
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  • #8
Dunkodx said:
Because there are only conservative forces, so when the mass reaches the maximum amplitude (that is, when it is at position ##\frac{d}{2}##) the kinetical energy is ##0## because the mass has no velocity and there is only potential energy ##\frac{1}{2} k \left(\frac{d}{2}\right)^2##; that means that ##E=\frac{1}{2}k \left(\frac{d}{2}\right)^2##.
If the equilibrium position is x=0, then displacement (x) is in the range ## -d≤x≤d##.
Extremum-to-extremum distance is 2d (twice the amplitude).
That means maximum potential energy is ##\frac 1 2 kd^2##, not ##\frac 1 2 k(\frac d 2)^2##.
 
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Related to Finding Spring Constant & Energy w/ Doubt in Exercise

1. How do you find the spring constant in an exercise?

The spring constant, also known as the force constant, can be found by dividing the force applied to the spring by the displacement of the spring. This can be represented by the equation k = F/x, where k is the spring constant, F is the applied force, and x is the displacement of the spring.

2. What is the significance of finding the spring constant in an exercise?

The spring constant is a measure of the stiffness of a spring and is an important factor in understanding the behavior of the spring. It is used in various equations to calculate the potential energy, kinetic energy, and work done by the spring.

3. How can doubt affect the accuracy of finding the spring constant and energy in an exercise?

Doubt can affect the accuracy of the results when finding the spring constant and energy in an exercise by introducing errors in the measurements or calculations. It is important to minimize doubt by using precise and accurate equipment and techniques.

4. Can the spring constant and energy be determined without doubt in an exercise?

While it is impossible to completely eliminate doubt, it is possible to minimize it by using precise and accurate measurements and techniques. This will result in more accurate and reliable results for the spring constant and energy in an exercise.

5. What are some examples of exercises where finding the spring constant and energy is important?

Finding the spring constant and energy is important in various exercises, such as determining the force required to stretch a spring to a certain length, calculating the energy stored in a spring, and understanding the behavior of springs in machines and structures.

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