If the Earth has radius R and rotates with angular velocity $,$ show that the apparent weight of an object of mass m at latitude l is given by $()()22222$ m g $$ Rcos $+$ Rsin$$ cos$.$ The first space travellers to reach the planet Isobarios land in the polar regions and discover, to their surprise, that the apparent weight of objects, measured with a spring balance, does not vary with latitude even though the planet is rotating. The planet is spherically symmetric and the length of the $$day$$ is $2$ hours, $46$ minutes and $40$ seconds. Calculate the mean density of the planet. What would the explorers experience if they moved towards the planet$$s equator? $[$The gravitational constant is G $=6\mathrm{.}6710-11$ m$3$ kg$-1$ s $-2]$