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, calculate the centripetal acceleration (a_rad) at the equator due to the Earth's rotation. Use the formula a_rad = \frac{v^2}{r}, where v is the linear speed at the equator and r is the radius of the Earth.
2. To find the linear speed v at the equator, use the formula v = \frac{2\pi r}{T}, where T is the period of rotation (24 hours initially) and r is the radius of the Earth. Convert T into seconds for the calculation.
3. Substitute the expression for v from step 2 into the formula for a_rad in step 1. This will give you a_rad in terms of r and T.
4. Set the centripetal acceleration a_rad equal to the acceleration due to gravity g (approximately 9.8 m/s^2) to find the critical condition where objects would start to fly off if a_rad exceeds g.
5. Solve the equation from step 4 for T to find the period of rotation where a_rad equals g. This will give you the minimum period T for which objects at the equator will start to fly off into space.

Key Concepts

Centripetal Acceleration

Centripetal acceleration is the acceleration directed towards the center of a circular path that keeps an object moving in that path. It is calculated using the formula a_rad = v^2 / r, where v is the tangential velocity and r is the radius of the circular path. At the equator, this acceleration must be greater than the gravitational acceleration (g) for objects to fly off the Earth's surface.

Tangential Velocity

Tangential velocity is the linear speed of an object moving along a circular path, measured at any point along the circumference. For the Earth, this velocity can be calculated by dividing the circumference of the Earth by the rotation period. If the period of rotation decreases, the tangential velocity increases, which affects the centripetal acceleration experienced by objects at the equator.

Gravitational Acceleration (g)

Gravitational acceleration, denoted as g, is the acceleration experienced by an object due to the gravitational force exerted by the Earth. Near the Earth's surface, this value is approximately 9.81 m/s². For an object to remain on the surface, the centripetal acceleration must be less than or equal to g; if it exceeds g, the object will no longer be able to stay on the surface and will 'fly off' into space.