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Simple typo question: Radial & circumferential components of velocity

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s3a
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Homework Statement


Here is the part of the problem that I am referring to (that is also fully portrayed in a more aesthetically-pleasing manner in the TheProblem.jpeg attachment).:
Consider a particle moving on the curve whose equation in polar coordinates is r = 1 + cos(θ). The rate of change of θ is given as 2 radius per second. The solution to part (a) is also attached as TheSolution.jpeg, should it prove useful.

Determine for the point with rectangular coordinates [½ + 1/√(2), ½ + 1/√(2)] the
(a) radial and circumferential components of the velocity.

Homework Equations


Derivatives, chain rule and trigonometry.

The Attempt at a Solution


In the problem, it says that the rate of change of θ is given as 2 RADIUS … I just wanted to ask/confirm if the author intends to say 2 RADIANS or 2 RADII. I think the author meant RADIANS because, it seems more likely that the θ (angle) variable uses an angular unit. So, what would be the value of ##ν_r## when the units are included? Would the value be –√(2) radians/second or radii/second? My confusion arises from the fact that I am searching for the velocity along the radius but, I think the rate of change of θ as time passes is in radians but, I would very much appreciate any confirmation/contradiction!
 

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  • #2
Ray Vickson
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Homework Statement


Here is the part of the problem that I am referring to (that is also fully portrayed in a more aesthetically-pleasing manner in the TheProblem.jpeg attachment).:
Consider a particle moving on the curve whose equation in polar coordinates is r = 1 + cos(θ). The rate of change of θ is given as 2 radius per second. The solution to part (a) is also attached as TheSolution.jpeg, should it prove useful.

Determine for the point with rectangular coordinates [½ + 1/√(2), ½ + 1/√(2)] the
(a) radial and circumferential components of the velocity.

Homework Equations


Derivatives, chain rule and trigonometry.

The Attempt at a Solution


In the problem, it says that the rate of change of θ is given as 2 RADIUS … I just wanted to ask/confirm if the author intends to say 2 RADIANS or 2 RADII. I think the author meant RADIANS because, it seems more likely that the θ (angle) variable uses an angular unit. So, what would be the value of ##ν_r## when the units are included? Would the value be –√(2) radians/second or radii/second? My confusion arises from the fact that I am searching for the velocity along the radius but, I think the rate of change of θ as time passes is in radians but, I would very much appreciate any confirmation/contradiction!
You need to develop some confidence in your own understanding. If you think it should read as 'radians', just go ahead and work the problem that way and see what happens! You can compare your own solution with the one supplied; of course, there is always the possibility that the supplied solution is incorrect. Then, if your answer differs from the one supplied, you can then come to us and ask about the difference.
 
  • #3
s3a
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Trying out what you said, it seems that, if I use the word “radii” (instead of “radians”) for where it says “radius” in the question, differentiation of r = 1 + cos(θ) would yield dr/dt = dr/dθ dθ/dt = -sin(θ) (2r) = dr/dt = -2rsin(θ) which is a differential equation so, since this is not a question intended for the study of differential equations, it must be “radians” (instead of “radii”) for where it says “radius” in the question, right?
 
  • #4
Ray Vickson
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Trying out what you said, it seems that, if I use the word “radii” (instead of “radians”) for where it says “radius” in the question, differentiation of r = 1 + cos(θ) would yield dr/dt = dr/dθ dθ/dt = -sin(θ) (2r) = dr/dt = -2rsin(θ) which is a differential equation so, since this is not a question intended for the study of differential equations, it must be “radians” (instead of “radii”) for where it says “radius” in the question, right?
I have already suggested that you try it out for yourself. Then--after solving the problem--if your solution differs from the one supplied, come back here with questions. As I said before: try to develop some confidence in your own understanding. That is by far the best way to learn.
 

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