13.2 Gravitation near earth's surface  (Page 3/6)

Apparent weight: accounting for earth’s rotation

As we saw in Applications of Newton’s Laws , objects moving at constant speed in a circle have a centripetal acceleration directed toward the center of the circle, which means that there must be a net force directed toward the center of that circle. Since all objects on the surface of Earth move through a circle every 24 hours, there must be a net centripetal force on each object directed toward the center of that circle.

Let’s first consider an object of mass m located at the equator, suspended from a scale ( [link] ). The scale exerts an upward force ${\overrightarrow{F}}_{\text{s}}$ away from Earth’s center. This is the reading on the scale, and hence it is the apparent weight    of the object. The weight ( mg ) points toward Earth’s center. If Earth were not rotating, the acceleration would be zero and, consequently, the net force would be zero, resulting in $F_{\text{s}}=mg$ . This would be the true reading of the weight.

With rotation, the sum of these forces must provide the centripetal acceleration, $a_{\text{c}}$ . Using Newton’s second law, we have

Note that $a_{\text{c}}$ points in the same direction as the weight; hence, it is negative. The tangential speed v is the speed at the equator and r is $R_{\text{E}}$ . We can calculate the speed simply by noting that objects on the equator travel the circumference of Earth in 24 hours. Instead, let’s use the alternative expression for $a_{\text{c}}$ from Motion in Two and Three Dimensions . Recall that the tangential speed is related to the angular speed $\left(\omega \right)$ by $v=r\omega$ . Hence, we have $a_c=\text{−}r{\omega }^2$ . By rearranging [link] and substituting $r=R_{\text{E}}$ , the apparent weight at the equator is

The angular speed of Earth everywhere is

Substituting for the values or $R_{\text{E}}$ and $\omega$ , we have $R_{\text{E}}{\omega }^2=0.0337{\text{m/s}}^2$ . This is only 0.34% of the value of gravity, so it is clearly a small correction.