Given information:
- The planet has the same mass and diameter as the Earth.
- Objects become weightless at the equator.

To find:
The time period of rotation of the planet in minutes (as defined on Earth).

Explanation:
The weightlessness observed at the equator suggests that the centrifugal force due to the rotation of the planet cancels out the gravitational force at that location. This implies that the centripetal force acting on objects at the equator is equal to the gravitational force.

1. Centripetal force:
The centripetal force is given by the equation:

Fc = m * ω² * r

Where,
- Fc is the centripetal force
- m is the mass of the object
- ω is the angular velocity of the rotation
- r is the distance from the center of rotation

Since objects become weightless at the equator, the centripetal force is equal to the gravitational force:

Fc = Fg

2. Gravitational force:
The gravitational force is given by the equation:

Fg = (G * m * M) / r²

Where,
- G is the gravitational constant
- m is the mass of the object
- M is the mass of the planet
- r is the distance from the center of the planet

3. Equating the forces:
Setting Fc equal to Fg, we have:

m * ω² * r = (G * m * M) / r²

Simplifying the equation, we get:

ω² = (G * M) / r³

4. Time period:
The time period of rotation (T) is the time taken for one complete rotation. It is related to the angular velocity (ω) by the equation:

T = 2π / ω

5. Substituting values:
- The mass and diameter of the planet are the same as the Earth, so M and r can be replaced with the mass and radius of the Earth.
- The gravitational constant G is a constant value.

Substituting these values into the equation for ω², we get:

ω² = (G * M) / r³ = (G * M) / (R^3)

Where,
- R is the radius of the Earth

6. Calculating the time period:
Substituting the value of ω² into the equation for the time period T, we have:

T = 2π / ω = 2π * sqrt(r³ / (G * M))

Since the radius of the planet is the same as the Earth, we have:

T = 2π * sqrt((R^3) / (G * M))

7. Converting to minutes:
To convert the time period into minutes, we need to multiply by the conversion factor of 1 minute / 60 seconds. Since the time period is given in seconds, we divide by 60 to obtain the time period in minutes:

T = (2π * sqrt((R^3) / (G * M))) / 60

8. Evaluating the expression:
Using the known values for the radius of the Earth (R) and the gravitational constant (G), we