4- Motion in plane using Cartesian and Polar coordinates

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16 Terms

1
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How can any vector A be represented in the Cartesian coordinate system?

Ax​ and Ay​ = the projections of the vector A onto the x- and y-axes.

<p>A<sub>x​</sub> and A<sub>y​</sub> = the projections of the vector <strong>A</strong> onto the x- and y-axes.</p>
2
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What are i^ and j^ in the Cartesian coordinate system?

Unit vectors along the x- and y-axes, respectively.

3
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How is the velocity vector v expressed in Cartesian coordinates?

x˙ and y˙ = the time derivatives of the position components along the x- and y-axes.

<p>x˙ and y˙ = the time derivatives of the position components along the x- and y-axes.</p>
4
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How are the components of velocity expressed in Cartesian coordinates?

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How are the components of acceleration expressed in Cartesian coordinates?

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6
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How is Newton's second law written in Cartesian coordinates?

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7
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How is the position of a point defined in polar coordinates?

Its defined by a distance r from a fixed origin and an angle θ from a fixed reference line.

<p>Its defined by a distance r from a fixed origin and an angle θ from a fixed reference line. </p>
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What are the unit vectors in the polar coordinate system?

  • r^ pointing in the direction of increasing r

  • θ^ pointing in the direction of increasing θ

9
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How do the unit vectors in polar coordinates differ from those in Cartesian coordinates?

  • In Cartesian coordinates, i^ and j^​ are constant

  • In polar coordinates, unit vectors r^ and θ^ change with the position of the object

<ul><li><p>In Cartesian coordinates, i^ and j^​ are constant</p><p></p></li><li><p>In polar coordinates, unit vectors r^ and θ^ change with the position of the object</p></li></ul><p></p>
10
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What is the expression for the position vector in polar coordinates?

  • r is the distance

  • r^ is the unit vector pointing in the direction of increasing r.

<ul><li><p>r is the distance</p><p></p></li><li><p>r^ is the unit vector pointing in the direction of increasing r.</p></li></ul><p></p>
11
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How is the velocity vector expressed in polar coordinates?

r˙=the radial velocity

ω = the angular velocity.

<p>r˙=the radial velocity</p><p>ω = the angular velocity.</p>
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What are the time derivatives of the unit vectors r^ and θ^ in polar coordinates?

θ˙= the angular velocity.

<p>θ˙= the angular velocity.</p>
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What is the angular velocity in polar coordinates?

Its the rate of change of the angle θ

<p>Its the rate of change of the angle θ</p>
14
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How is the acceleration vector derived in polar coordinates?

r¨ = the radial acceleration

<p>r¨ = the radial acceleration</p>
15
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<p>How is the force vector expressed in polar coordinates?</p>

How is the force vector expressed in polar coordinates?

Fr​ and Fθ are the radial and tangential components of the force.

<p>F<sub>r</sub>​ and F<sub>θ</sub> are the radial and tangential components of the force.</p>
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What is the form of Newton's second law in polar coordinates?

Fr​ and Fθ are the radial and tangential components of the force.

<p>F<sub>r</sub>​ and F<sub>θ</sub> are the radial and tangential components of the force.</p>