Understanding the Chain Rule in Mechanics: Solving for Acceleration and Force

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SUMMARY

The discussion focuses on applying the chain rule in mechanics to derive acceleration and force for a particle moving with a speed defined by v(x) = α / x. The participant initially attempted to differentiate directly with respect to time, leading to confusion. The correct approach involves using the chain rule to express acceleration as a = (dv/dx)(dx/dt), ultimately yielding the force equation F(x) = -mα²/x³. This highlights the necessity of understanding the relationship between position, velocity, and time in mechanics.

PREREQUISITES
  • Understanding of basic mechanics, specifically Newton's second law (F = ma).
  • Familiarity with calculus, particularly differentiation and the chain rule.
  • Knowledge of kinematics, including the relationship between velocity, acceleration, and position.
  • Experience with particle motion in a frictionless environment.
NEXT STEPS
  • Study the application of the chain rule in calculus, particularly in physics contexts.
  • Learn how to derive equations of motion for particles under various forces.
  • Explore advanced topics in mechanics, such as Lagrangian mechanics and energy conservation.
  • Investigate the implications of frictionless motion in real-world scenarios.
USEFUL FOR

Students of physics, educators teaching mechanics, and anyone interested in understanding the mathematical foundations of motion and forces in a frictionless environment.

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Homework Statement




6. A particle of mass m moves along a frictionless, horizontal plane with a speed given by

v(x) = α / x. Where x is the distance of the object from the origin and α is a constant.

Working with F = ma, we want to get the acceleration. You have v = v(x). You want a = dv/dt. Find (dv/dx)(dx/dt). Find the force F(x) to which the particle is subjected to.




The Attempt at a Solution



I guess my problem is I don't understand why I need to use chain rule since v = dx/dt. I thought I could take the derivative in respect to t on both sides, and get dv/dt = - α / x^2, then multiply both sides by m to get the force equation.

the answer is given, -m α^2/ x^3. So can someone explain what I'm missing here...
 
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You need to use the chain rule because x is some function of t. What you have done above is find dv/dx. Now you have correctly identified dx/dt as v and you know v = a/x, so what is (dv/dx)*(dx/dt)?
 
wow awesome thanks that clears up a lot. I almost gave up on anyone answering me. I read that there was a way to close a thread or say that the problem is solved but I'm not seeing it on here so I guess'll have to leave it as is.
 
The forum software was upgraded recently and I think only mentors can mark it solved at the minute. Just leave it as it is for now. :smile:
 

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