Question

When tall buildings are constructed on earth, the duration of day night
(A) Slightly Increases
(B) Slightly decreases
(C) Has no change
(D) None of these

Answer 1

Hint: Consider earth to be a spherical object of mass m and radius R. Now, initially the moment of inertia about its geographical rotational axis is the same as the spherical object. When tall buildings are constructed, the overall radius of the earth expands laterally, thus causing changes. Using this, conclude our answer.

Complete Step by Step Solution:
It is a known fact that earth is spherical in shape and rotates about its geographical axis. The moment of inertia of earth in simple terms is defined as the body’s resistance to torque in changing its rotational speed and angular velocity. The moment of inertia heavily depends on the shape of the object. Since, earth is spherical, the mathematical representation of moment of inertia of earth is given as a product of mass and the square of the radius of the spherical component.
Now, tall buildings will have greater mass due to heavy usage of steel and cement to stabilize the structure with the base. This adds more mass to the earth’s surface and hence increases the overall mass. Along with that, tall buildings also increase the lateral radius when considering earth’s radius as a whole.
This results in an increase in moment of inertia of the earth. We know that angular momentum is a product of moment of inertia and angular velocity, given as
\[ \Rightarrow L = I\omega’\]
Now, when tall buildings are constructed , Moment of inertia increases, which means,
\[\Rightarrow L = {I}{\omega’ }\](Since, net acceleration of earth is same)
Equating both, we get,
\[\Rightarrow I\omega = {I}{\omega’ }\]
\[\Rightarrow \dfrac{{I\omega }}{{{I}}} = {\omega}\]
This means that\[{\omega } < \omega’ \], which concludes that angular velocity decreases. As angular velocity decreases , time period increases.

Hence, option (a) is the right answer.

Note: Time period of oscillation is defined as the ratio of angular displacement of the moving body and the angular velocity of the moving/rotating body. In this case, as angular velocity decreases, time period increases.