How fast should the Earth rotate about its Axis so that the apparent weight of the body at equator be zero and how long would a day be then?

In order for the apparent weight of a body at the equator to be zero, the centripetal force due to the Earth's rotation must exactly cancel out the gravitational force pulling the body towards the center of the Earth.

The centripetal force is given by the equation Fc = mω^2r, where m is the mass of the body, ω is the angular velocity of the Earth's rotation, and r is the radius of the Earth.

The gravitational force is given by the equation Fg = mg, where g is the acceleration due to gravity.

Setting these two forces equal to each other, we get:

mω^2r = mg

Solving for ω, we get:

ω = √(g/r)

Substituting in the values for g (9.81 m/s^2) and the radius of the Earth (6,371 km or 6,371,000 m), we get:

ω = √(9.81/6,371,000) = 7.29 x 10^-5 rad/s

To find the corresponding rotational period (the length of a day), we use the formula T = 2π/ω:

T = 2π/(7.29 x 10^-5) = 86,164 seconds

Therefore, the Earth would need to rotate at a speed of approximately 7.29 x 10^-5 rad/s in order for the apparent weight of a body at the equator to be zero, and a day would be approximately 86,164 seconds long.

Part 1: Calculating the Equatorial Rotation Speed

For the apparent weight of a body at the equator to be zero, the centrifugal force due to Earth's rotation must cancel out the force of gravity. The centrifugal force is given by:

```
F_c = m - v^2 / r
```

where:

- m is the mass of the body
- v is the linear velocity of the body
- r is the radius of the Earth's rotation

The force of gravity is given by:

```
F_g = m - g
```

where g is the acceleration due to gravity (approximately 9.81 m/s^2 at Earth's surface).

Setting F_c = F_g and solving for v, we get:

```
v = sqrt(r - g)
```

Substituting r with Earth's equatorial radius of 6378.137 km (6,378,137 m), we get:

```
v = sqrt(6,378,137 m - 9.81 m/s^2) ≈ 7,900 m/s
```

Part 2: Calculating the Duration of a Day

The speed of Earth's rotation is related to the length of a day. The period of rotation (T) is the time it takes for Earth to complete one rotation about its axis:

```
T = 2πr / v
```

Substituting v with the value calculated in Part 1, we get:

```
T = 2π - 6,378,137 m / 7,900 m/s ≈ 5,061 seconds
```

Therefore, the Earth would need to rotate about its axis at a speed of approximately 7,900 m/s at the equator to make the apparent weight of bodies at the equator zero. At this speed, the duration of a day would be approximately 5,061 seconds, which is about 1 hour and 24 minutes.