Displacement and arc length problems

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Discussion Overview

The discussion revolves around the concepts of displacement, arc length, and their relationship in the context of physics equations. Participants explore the definitions and implications of these terms, particularly in relation to velocity and angular velocity.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants propose that the equation v=s/t should use arc length s instead of displacement for the reasoning to be valid.
  • Others argue that the distinction between displacement and arc length is crucial, as displacement is defined as the shortest distance between two points.
  • A participant clarifies that the equation should be interpreted as speed=distance/time rather than velocity=displacement/time.
  • One participant questions the relationship between speed, radius, and angular velocity, suggesting that velocity might be a more appropriate term than speed in certain contexts.
  • Another participant discusses the treatment of radius and angular velocity as scalars versus vectors, indicating that this affects the interpretation of the resulting product.
  • A separate question about the weight of a ship displacing 500,000 kg of water is raised, leading to a brief response about the weight being approximately 500 tonnes.

Areas of Agreement / Disagreement

Participants express differing views on the appropriate use of terms like speed and velocity, and whether arc length can be substituted for displacement in certain equations. The discussion remains unresolved regarding the implications of these distinctions.

Contextual Notes

There are limitations in the assumptions made about the definitions of displacement and arc length, as well as the treatment of angular velocity and radius as scalars or vectors. These factors contribute to the complexity of the discussion.

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1. v=s/t where s represents the displacement
2. s=rθ where s represents the arc length

v=rθ /t

Why can substitute here?
I guess that is not same things.

An arc length is not a straight line but displacement is which is shortest distance between initial and final point.
 
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Not quite sure I've grasped your difficulty, but for the reasoning to be valid, s in equation (1) must represent not a displacement but the arc length s. Thus v is not mean velocity over s, but speed.
 
Philip Wood said:
Not quite sure I've grasped your difficulty, but for the reasoning to be valid, s in equation (1) must represent not a displacement but the arc length s. Thus v is not mean velocity over s, but speed.

Thanks. I understand now.
The equation of (1) is not velocity=displacement/time, but speed=distance/time, right?
 
Right!
 
But I have some problems about it.
v=rθ /t

θ /t = angular velocity
Why speed = radius x angular velocity?
I think velocity = radius x angular velocity is suitable for it.
 
I assume you are not treating radius and angular velocity as vectors. If you're treating them as scalars, as in most introductory courses, then the multiplication yields a scalar, which is better called speed than velocity.

In more advanced work it is possible to assign a direction to angular velocity (making it a pseudo vector), and it is clearly possible to treat instantaneous radius as a vector. By assigning a special meaning to multiplication we can produce the instantaneous velocity vector.
 
what is the weight of a ship that displaces 500,000 kg of water?
 
500 tonnes, more or less.
 

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