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Math Help - Simple typo question: Radial & circumferential components of velocity

  1. #1
    s3a
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    Simple typo question: Radial & circumferential components of velocity

    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 (which is exactly what it written). The solution to part (a) is also attached as TheSolution.jpeg, should it prove useful.

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

    Basically, 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 (final) 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.

    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?


    Despite this being just a small typo, I would still VERY much appreciate any confirmation/contradiction!
    Attached Thumbnails Attached Thumbnails Simple typo question: Radial & circumferential components of velocity-theproblem.jpeg   Simple typo question: Radial & circumferential components of velocity-thesolution.jpeg  
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  2. #2
    MHF Contributor ebaines's Avatar
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    Re: Simple typo question: Radial & circumferential components of velocity

    The "rate of change of theta" should indeed be written as "radians per second," not "radius" nor "radii" per second. Theta is an angular measurement, whereas "radius" is a length, and it makes no sense to talk about the change of theta per unit time as a length. So bottom line is- it's a typo. As for the circumfrential velocity - you calculate that as rate of change of theta times the distance from the origin at that point in time:

     V_c= R \frac {d \theta}{dt}
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  3. #3
    s3a
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    Re: Simple typo question: Radial & circumferential components of velocity

    Thank you very much for your confirmation!
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