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(1.04.0+6.0)ijk
  • B(6.04.0+1.0)ijk
  • C(1.0+4.0+6.0)ijk
  • D(4.01.0+6.0)ijk
  • E(1.06.0+4.0)ijk

Q3:

Two points in the Cartesian plane have the coordinates (2.00,4.00)mm and (3.00,3.00)mm. Find the distance between them.

Q4:

Two points in a plane have polar coordinates 𝑃2.500,𝜋6m and 𝑃3.800,2𝜋3m.

Determine the Cartesian coordinates of 𝑃.

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

Determine the Cartesian coordinates of 𝑃.

  • A(1.990 m, 3.095 m)
  • B(1.900 m, 3.290 m)
  • C(1.805 m, 3.140 m)
  • D(1.875 m, 3.050 m)
  • E(2.110 m, 3.000 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 𝑃1 that is directly over the North Pole to a point 𝑃2 that is at a latitude of 45.0. In determining the motion of the satellite, use a value of 6,371 km for the radius of Earth.

What is the magnitude of the displacement vector from 𝑃1 to 𝑃2?

At what angle below east is the direction of the displacement vector from 𝑃1 to 𝑃2?

Q6:

Under which conditions should coriolis acceleration be considered?

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

  • AThe floor should be a coordinate of zero, and the upward 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.
  • DHer hand should be a coordinate of zero, and the downward direction should be considered positive.

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