Understanding Velocity=r(omega)??

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SUMMARY

The equation for linear velocity, represented as v = r(ω), establishes that velocity is directly proportional to the radius (r) and the angular velocity (ω) measured in radians per second. This relationship arises from the definition of arc length (s = θr) and the derivative of angular displacement with respect to time (ω = dθ/dt). Consequently, when differentiating arc length with respect to time, the equation simplifies to v = rω, illustrating the connection between linear and angular motion.

PREREQUISITES
  • Understanding of angular velocity (ω) in radians per second
  • Familiarity with the concept of arc length (s = θr)
  • Basic knowledge of calculus, specifically derivatives
  • Concept of radians and their relationship to circular motion
NEXT STEPS
  • Study the relationship between linear and angular motion in physics
  • Learn about the derivation of the arc length formula (s = θr)
  • Explore the concept of radians and their applications in circular motion
  • Investigate the implications of angular velocity in rotational dynamics
USEFUL FOR

Students in physics, particularly those studying mechanics and circular motion, as well as educators seeking to clarify the relationship between linear and angular velocity.

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Homework Statement


I do not understand why velocity is equal to the radius times the angular velocity. Angular velocity is given in radians per second. How does this equal velocity when multiplied by the radius?

Thanks yall! I'm new here by the way.


Homework Equations


v=r(omega)


The Attempt at a Solution




??
 
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S=θr

where S is the Arc Length.

So think of it like r*θ/t
 
johnps said:

Homework Statement


I do not understand why velocity is equal to the radius times the angular velocity. Angular velocity is given in radians per second. How does this equal velocity when multiplied by the radius?

Thanks yall! I'm new here by the way.


Homework Equations


v=r(omega)


The Attempt at a Solution




??

What is the equation for the diameter of a circle, in terms of the radius? That should help it to make more sense.

Welcome to the PF, BTW.
 
By definition, s=θr. By definition, w=dθ/dt. That's why ds/dt=r*dθ/dt=r*w=v. Why the definition s=θr, you ask? That's how the radian was defined. It's simple, elegant, and leaves out a pesky constants that clutter up equations.
 
Aha, so s=(theta)(radius) because of the equation circumference = 2 (pi) (r) correct?

Sorry I'm not sure how to get the symbols yet. I'll figure it out.
 

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