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The earth’s radius is 6.37×106 m; it rotates once every 24 hours.

  • Calculate the angular speed of the earth.
  • Calculate the direction (positive or negative) of the angular velocity. Assume you are viewing from a point exactly above the north pole.
  • Calculate the tangential speed of a point on the earth’s surface located on the equator.
  • Calculate the tangential speed of a point on the earth’s surface located halfway between the pole and the equator.

The aim of the question is to understand the concept of angular and tangential speeds of a rotating body and the points on its surface, respectively.

If $\omega$ is the angular speed and $T$ is the time period of rotation, the angular speed is defined by the following formula:

\[\omega = \frac{2\pi}{T}\]

If the radius $r$ of the rotation of a point around the axis of rotation, then the tangential speed $v$ is defined by the following formula:

\[v = r \omega\]

Expert Answer

Part (a): Calculate the angular speed of the earth.

If $\omega$ is the angular speed and $T$ is the time period of rotation, then:

\[\omega = \frac{2\pi}{T}\]

For our case:

\[T = 24 \times 60 \times 60 \ s\]

So:

\[\omega = \frac{2\pi}{24\times 60 \times 60 \ s} = 7.27 \times 10^{-5} \ rad/s\]

Part (b): Calculate the direction (positive or negative) of the angular velocity. Assume you are viewing from a point exactly above the north pole.

When viewed from a point exactly above the north pole, the earth rotates counter-clockwise, so the angular velocity is positive (following the right-hand convention).

Part (c): Calculate the tangential speed of a point on the earth’s surface located on the equator.

If the radius $r$ of the rigid body is known, then the tangential speed $v$ can be calculated using the formula:

\[v = r \omega\]

For our case:

\[ r = 6.37 \times 10^{6} m\]

And:

\[ \omega = 7.27 \times 10^{-5} rad/s\]

So:

\[v = ( 6.37 \times 10^{6} m)(7.27 \times 10^{-5} rad/s)\]

\[v = 463.1 m/s\]

Part(d): Calculate the tangential speed of a point on the earth’s surface located halfway between the pole and the equator.

A point on the earth’s surface located halfway between the pole and the equator rotates in a circle of radius given by the following formula:

\[\boldsymbol{r’ = \sqrt{3} r }\]

\[r’ = \sqrt{3} (6.37 \times 10^{6} m) \]

Where $r$ is the radius of the earth. Using the tangential speed formula:

\[v = \sqrt{3} ( 6.37 \times 10^{6} m)(7.27 \times 10^{-5} rad/s)\]

\[v = 802.11 m/s\]

Numerical Result

Part (a): $\omega = 7.27 \times 10^{-5} \ rad/s$

Part (b): Positive

Part (c): $v = 463.1 m/s$

Part (d): $v = 802.11 m/s$

Example

The radius of the Moon is $1.73 \times 10^{6} m$

– Calculate the angular speed of the moon.
– Calculate the tangential speed of a point on the moon’s surface located midway between the poles.

Part (a): One day on Moon is equal to:

\[T = 27.3 \times  24 \times  60 \times  60 \ s\]

So:

\[\omega = \frac{2\pi}{T} = \frac{2\pi}{27.3 \times  24 \times  60 \times  60 \ s}\]

\[\boldsymbol{\omega = 2.7 \times  10^{-6} \ rad/s}\]

Part (b): Tangential speed on the given point is:

\[v = r \omega\]

\[v = ( 1.73 \times  10^{6} m)(2.7 \times  10^{-6} \ rad/s)\]

\[ \boldsymbol{v = 4.67 m/s}\]

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