Worksheet: Coordinate Systems

In this worksheet, we will practice performing vector calculations in different coordinate systems such as polar, cylindrical, and spherical coordinates.

Q1:

Consider a coordinate system in which the positive 𝑥 -axis is directed vertically upward.

What is the displacement of a particle 8.5 m directly above the origin?

What is the displacement of a particle 1.3 m directly below the origin?

Q2:

The coordinates of a particle in a rectangular coordinate system are ( 1 . 0 , 4 . 0 , 6 . 0 ) . What is the position vector of the particle?

  • A ( 1 . 0 6 . 0 + 4 . 0 ) i j k
  • B ( 4 . 0 1 . 0 + 6 . 0 ) i j k
  • C ( 1 . 0 + 4 . 0 + 6 . 0 ) i j k
  • D ( 6 . 0 4 . 0 + 1 . 0 ) i j k
  • E ( 1 . 0 4 . 0 + 6 . 0 ) i j k

Q3:

Two points in the Cartesian plane have the coordinates ( 2 . 0 0 , 4 . 0 0 ) m m and ( 3 . 0 0 , 3 . 0 0 ) m m . Find the distance between them.

Q4:

Two points in a plane have polar coordinates 𝑃 2 . 5 0 0 , 𝜋 6 m and 𝑃 3 . 8 0 0 , 2 𝜋 3 m .

Determine the Cartesian coordinates of 𝑃 .

  • A (2.265 m, 1.130 m)
  • B (2.165 m, 1.250 m)
  • C (2.115 m, 1.210 m)
  • D (2.225 m, 1.150 m)
  • E (2.100 m, 1.050 m)

Determine the Cartesian coordinates of 𝑃 .

  • A ( 1 . 9 9 0 m, 3.095 m)
  • B ( 1 . 8 7 5 m, 3.050 m)
  • C ( 2 . 1 1 0 m, 3.000 m)
  • D ( 1 . 8 0 5 m, 3.140 m)
  • E ( 1 . 9 0 0 m, 3.290 m)

Determine the distance between the points, to the nearest centimeter.

Q5:

A circular polar orbit around Earth is an orbit that passes directly overhead at both the North Pole and the South Pole. A satellite in a circular polar orbit around Earth at an altitude of 400 km moves in its orbit form a point 𝑃 1 that is directly over the North Pole to a point 𝑃 2 that is at a latitude of 4 5 . 0 . In determining the motion of the satellite, use a value of 6 3 7 1 km for the radius of Earth.

What is the magnitude of the displacement vector from 𝑃 1 to 𝑃 2 ?

At what angle below east is the direction of the displacement vector from 𝑃 1 to 𝑃 2 ?

Q6:

Under which conditions should coriolis acceleration be considered?

  • Awhen linearly accelerating reference frames are used
  • Bwhen translating reference frames are used
  • Cwhen fixed reference frames are used
  • Dwhen co-moving reference frames are used
  • Ewhen rotating reference frames are used

Q7:

A student is trying to determine the acceleration of a feather as she drops it to the ground. If the student is looking to achieve a positive velocity and positive acceleration, what is the most sensible way to set up her coordinate system?

  • AHer hand should be a coordinate of zero, and the downward direction should be considered positive.
  • BHer hand should be a coordinate of zero, and the upward direction should be considered positive.
  • CThe floor should be a coordinate of zero, and the downward direction should be considered negative.
  • DThe floor should be a coordinate of zero, and the upward direction should be considered positive.

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