Suppose the people standing at rest with respect to the Earth on the Earth's equator are weightless.The radius of the earth is 6.37 * 10⁶ m.The length of the day under this condition would be A)21 hours. B)6 hours. C)2.1 hours. D)84.4 min. E)41.5 min.

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Quizplus · Accepted Answer

For people standing at rest with respect to the Earth on the Earth's equator to be weightless, the centripetal force required to keep them moving in a circle must equal the gravitational force. The formula for centripetal force is $F_c = \frac{mv^2}{r}$, and the gravitational force is given by $F_g = \frac{GmM}{r^2}$, where $m$ is the mass of the object (person), $v$ is the tangential velocity, $r$ is the radius of the Earth, $G$ is the gravitational constant, and $M$ is the mass of the Earth.Setting these two forces equal to each other and solving for $v$, we get $v = \sqrt{\frac{GM}{r}}$. The circumference of the Earth is $2\pi r$, so the time $T$ to make one rotation (the length of the day) is $T = \frac{2\pi r}{v} = 2\pi \sqrt{\frac{r}{G M}}$.Plugging in the values for $G$, $M$, and $r$ (with $r = 6.37 \times 10^6 m$), and solving for $T$, we find that $T$ is approximately 84.4 minutes. This calculation assumes that the only force acting on the people is gravity (which is being counteracted by the centripetal force due to their motion), making them feel weightless.

Suppose the People Standing at Rest with Respect to the Earth