Introduction:
To determine the speed at which the Earth would have to rotate about its axis so that a person on the equator weighs 3/5th of its present value, we need to consider the gravitational force acting on the person. The weight of an object is determined by the force of gravity pulling on it. It is given by the equation:

Weight = mass × gravitational acceleration

where the gravitational acceleration is denoted by 'g'. The gravitational acceleration depends on the mass of the Earth and the distance from the center of the Earth to the object, which can be approximated as the radius of the Earth, denoted by 'r'.

Calculating the Present Weight:
To determine the present weight of a person on the equator, we need to know the gravitational acceleration at the equator. The Earth is not a perfect sphere, but rather an oblate spheroid, which means the equatorial radius is slightly larger than the polar radius. The average value for the radius of the Earth is approximately 6,371 kilometers.

Present Weight = mass × gravitational acceleration
Present Weight = mass × g

Determining the Required Speed:
To find the required speed at which the Earth would have to rotate so that a person on the equator weighs 3/5th of its present value, we can equate the centrifugal force with the force of gravity.

Centrifugal Force = Force of Gravity

The centrifugal force is given by the equation:

Centrifugal Force = mass × (angular velocity)^2 × radius

where the angular velocity is denoted by 'ω' and represents the rotational speed.

Force of Gravity = mass × gravitational acceleration
mg = mass × g

Setting up the Equations:
To proceed further, we need to relate the angular velocity (ω) to the rotational speed of the Earth. The rotational speed is measured in terms of the number of rotations per unit time, usually seconds.

Rotational Speed = 2π × (Number of Rotations per Unit Time)

Since we want the answer in terms of 'g' and 'r', we can substitute the value of rotational speed in terms of angular velocity and radius.

Rotational Speed = 2π × (angular velocity × radius)

By substituting this value into the equation for centrifugal force, we get:

Centrifugal Force = mass × (2π × (angular velocity × radius))^2 × radius
Centrifugal Force = mass × (4π^2 × (angular velocity × radius)^3)

Solving for Required Speed:
Now, we can equate the centrifugal force to the force of gravity and solve for the required rotational speed.

Centrifugal Force = Force of Gravity
mass × (4π^2 × (angular velocity × radius)^3) = mass × g

Simplifying the equation, we can cancel out the mass on both sides:

4π^2 × (angular velocity × radius)^3 = g

Now, solving for angular velocity (ω):

(angular velocity × radius)^3 = g / (4π^2