The earth has a radius of 6380 km and turns around once on its axis in 24 h. (b) If arad at the equator is greater than g, objects will fly off the earth's surface and into space. (We will see the reason for this in Chapter 5.) What would the period of the earth's rotation have to be for this to occur?

Verified step by step guidance

1. First, understand that the problem involves calculating the period of Earth's rotation where the centripetal acceleration at the equator equals the gravitational acceleration. This is the condition for objects to 'fly off' the Earth's surface.
2. The centripetal acceleration \( a_{rad} \) at the equator is given by \( a_{rad} = \frac{v^2}{r} \), where \( v \) is the tangential velocity and \( r \) is the radius of the Earth.
3. The tangential velocity \( v \) can be expressed in terms of the period \( T \) of rotation as \( v = \frac{2\pi r}{T} \). Substitute this expression for \( v \) into the formula for \( a_{rad} \).
4. Set the centripetal acceleration equal to the gravitational acceleration \( g \), which is approximately \( 9.8 \text{ m/s}^2 \). This gives the equation \( \frac{(2\pi r)^2}{T^2 r} = g \).
5. Solve the equation \( \frac{4\pi^2 r}{T^2} = g \) for \( T \) to find the period of rotation where objects would fly off the Earth's surface. Rearrange to find \( T = \sqrt{\frac{4\pi^2 r}{g}} \).

Key Concepts

Centripetal Acceleration

Centripetal acceleration is the acceleration experienced by an object moving in a circular path, directed towards the center of the circle. It is given by the formula arad = v^2/r, where v is the tangential velocity and r is the radius of the circular path. At the Earth's equator, this acceleration is due to the Earth's rotation.

Gravitational Acceleration

Gravitational acceleration, denoted as g, is the acceleration due to Earth's gravity experienced by objects near its surface. It has an approximate value of 9.81 m/s². This force acts downward towards the center of the Earth and is responsible for keeping objects grounded unless counteracted by another force.

Rotational Period

The rotational period is the time it takes for an object to complete one full rotation around its axis. For Earth, this period is currently 24 hours. To determine when objects would fly off the surface, we need to find the period at which the centripetal acceleration at the equator equals or exceeds gravitational acceleration.