Vector Calculus to derive Newton's second law

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SUMMARY

The discussion centers on deriving Newton's second law using vector calculus in the context of a particle moving in a circle with angular velocity ω. The user correctly identifies the relationship between radius (r) and angular velocity (ω) to find linear velocity (v) but struggles with differentiating to find torque (τ) and angular acceleration (α). The product rule application leads to confusion regarding the terms a=(v×ω)+(r×α), indicating a need for clarification on vector relationships and their physical interpretations.

PREREQUISITES
  • Understanding of vector calculus, particularly cross products and differentiation.
  • Familiarity with Newton's laws of motion and their applications in rotational dynamics.
  • Knowledge of angular velocity (ω) and angular acceleration (α) in circular motion.
  • Basic principles of torque (τ) and moment of inertia (I).
NEXT STEPS
  • Study the relationship between linear and angular quantities in circular motion.
  • Learn about the product rule in vector calculus and its applications in physics.
  • Explore the derivation of torque (τ) and its relation to angular acceleration (α) in rotational dynamics.
  • Review examples of applying Newton's second law in both linear and rotational contexts.
USEFUL FOR

Students studying physics, particularly those focusing on mechanics and rotational dynamics, as well as educators looking for examples of vector calculus applications in deriving physical laws.

pestydestroyer
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THIS THREAD HAS BEEN MOVED FROM ANOTHER FORUM BECAUSE IT IS HOMEWORK. BUT THERE IS NO TEMPLATE.

Hi everyone,
The problem gives that a particle moves in a circle with angular velocity ω. I know that r×ω=v, which is the velocity of the particle. However, I am told to differentiate and find τ=Iα but, using the product rule, I found a=(v×ω)+(r×α). I am not sure where to go from here as these quantities don't seem familiar to me. Was there a step I missed or is there something I am missing in this step?
Thanks so much for any help.
 
Last edited by a moderator:
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Do you see any vectors that are parallel to one another?

Also this looks like a homework problem and so you should edit your post to use the homework template otherwise someone will delete it and warn you.
 

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