With what angular speed the earth should rotate so that the person lies at the poles feels no weight

To understand the angular speed required for a person at the poles to feel no weight, we need to delve into some concepts of physics, particularly those related to gravity and circular motion. When we talk about feeling weight, we are referring to the normal force that acts on a person due to gravity. At the poles, if the Earth were to rotate at a certain speed, the centrifugal force could counteract the gravitational force, making a person feel weightless.

Understanding Forces at Play

Weight is the force exerted by gravity on an object, which can be calculated using the formula:

Weight (W) = mass (m) × gravitational acceleration (g)

At the Earth's surface, the average gravitational acceleration (g) is approximately 9.81 m/s². However, when the Earth rotates, a person experiences a centrifugal force that acts outward, which can be calculated using the formula:

Centrifugal Force (F_c) = m × ω² × r

Here, ω is the angular speed in radians per second, and r is the radius of the Earth at the poles, which is about 6,371 kilometers (or 6,371,000 meters).

Setting Up the Equation

For a person to feel no weight at the poles, the centrifugal force must equal the gravitational force. Therefore, we can set the two forces equal to each other:

m × g = m × ω² × r

Since mass (m) appears on both sides of the equation, we can simplify it to:

g = ω² × r

Calculating Angular Speed

Now, we can rearrange this equation to solve for the angular speed (ω):

ω² = g / r

ω = √(g / r)

Substituting the values for g and r:

• g ≈ 9.81 m/s²
• r ≈ 6,371,000 m

Now, plug in the values:

ω = √(9.81 / 6,371,000)

Calculating this gives:

ω ≈ √(0.00000154) ≈ 0.00124 rad/s

Converting to More Familiar Units

To make this more relatable, we can convert angular speed from radians per second to revolutions per day. Since there are 2π radians in a full circle and 86,400 seconds in a day, we can calculate:

Revolutions per day = (ω × 86,400) / (2π)

Substituting the value of ω:

Revolutions per day ≈ (0.00124 × 86,400) / (2π) ≈ 0.022 revolutions per day

Final Thoughts

In summary, for a person at the poles to feel no weight, the Earth would need to rotate at an angular speed of approximately 0.00124 radians per second, which translates to about 0.022 revolutions per day. This is significantly slower than the current rotation speed of the Earth, which is about 1 revolution per day. Thus, if the Earth rotated much slower, the centrifugal force would balance out the gravitational force, leading to a sensation of weightlessness at the poles.