Understanding Hooke's Law Integration: Exploring Vectors and Derivatives

In summary, the conversation discusses the equation m(dv/dt)v = (d/dt)(.5mv^2) and its integration. The participants debate whether the equation is an integration or not, with one person arguing that it follows the chain rule and the other mentioning the importance of multiplying by v before integrating. They also clarify the meaning of the spring constant 'k' and its relation to the strength of the spring. It is mentioned that the conversation may pertain to simple harmonic motion or Hooke's law.
  • #1
g.lemaitre
267
2
Screenshot2012-08-06at42820AM.png


Do you see where it says

m(dv/dt)v = (d/dt)(.5mv^2)

If it's an integration which I don't think it is then I would think it should be

(d/dt)((mv^3)/3) because you're taking the two v's and adding an additional power. I don't think it is an integration because it says right there in the book that you can't integrate, so if it's now then where does the 1/2 come from?
 
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  • #2
Hi g.lemaitre
It is indeed an integration and it is correct
just derive (1/2mv²) and you will see that you get back to mvv' or mv dv/dt
Cheers...
 
  • #3
It's a case of the chain rule:

[itex]\frac{d}{dt}\left(\frac{1}{2}mv^2\right)\:=\:\frac{d}{dv}\left(\frac{1}{2}mv^2\right)\:\times\frac{dv}{dt}\:=\:mv\frac{dv}{dt}[/itex].

Your textbook isn't very clear. You could integrate the left hand side of the equation wrt t, before multiplying by v. It's the right hand side that you can't integrate wrt to t until you've multiplied by v. That's why you need to multiply through by v. Then the left hand side integrates (wrt t) to give [itex]\frac{1}{2}mv^2[/itex]. The right hand integrates to give [itex]-\frac{1}{2}kx^2[/itex] + constant. Try differentiating this wrt t using the chain rule!
 
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  • #4
The spring constant 'k' is not a measure of the STRENGTH of the spring.
It is a measure of the STIFFNESS... units are N/m
The strength is measured by ultimate tensile stress
edit...is this question about simple harmonic motion?
or hookes law (elasticity)
 
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  • #5


Hooke's Law is a fundamental principle in physics that describes the relationship between the force applied to a spring and the resulting displacement of the spring. It states that the force applied to a spring is directly proportional to the displacement of the spring from its equilibrium position.

In the context of this integration, we are looking at the relationship between velocity and acceleration, which are both vector quantities. The equation m(dv/dt)v = (d/dt)(.5mv^2) is a representation of the mathematical relationship between these two quantities.

The left side of the equation, m(dv/dt)v, represents the vector quantity of force (F) multiplied by the vector quantity of velocity (v). This is equivalent to the work done on an object, which is the product of force and displacement. The right side of the equation, (d/dt)(.5mv^2), represents the derivative of the kinetic energy of the object, which is equal to the work done on the object.

The 1/2 in the equation comes from the fact that the kinetic energy of an object is equal to 1/2 times the mass of the object multiplied by the square of its velocity (KE = 1/2mv^2). This is a fundamental concept in physics and is not related to integration.

In conclusion, the equation m(dv/dt)v = (d/dt)(.5mv^2) is not an integration, but rather a representation of the relationship between force, velocity, and kinetic energy. The 1/2 in the equation comes from the definition of kinetic energy and is not related to integration.
 

What is Hooke's Law integration?

Hooke's Law integration is a mathematical method used to calculate the displacement or the change in length of a spring or a material when subjected to a force.

What is the formula for Hooke's Law integration?

The formula for Hooke's Law integration is F = -kx, where F is the force applied, k is the spring constant, and x is the displacement.

What is the difference between Hooke's Law integration and differentiation?

Hooke's Law integration calculates the displacement or the change in length of a spring or a material when a force is applied, while differentiation calculates the slope or rate of change of a function.

How is Hooke's Law integration used in real life?

Hooke's Law integration is used in various fields such as engineering, physics, and biology to predict the behavior of materials under different forces and to design structures and devices.

What are some limitations of Hooke's Law integration?

Hooke's Law integration assumes that the material or spring follows a linear relationship between force and displacement, which may not always be the case. Additionally, it does not account for factors such as friction, temperature, and elasticity of the material.

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