Can gravitational field strength equal the centripetal acceleration?

In summary, the conversation discusses the concept of weightlessness at the equator due to the Earth's rotation. It explains that for an object to be weightless, the centripetal force must equal the gravitational force, and this can be achieved if the normal force between the object and the ground is zero. This results in the equation Fg/m = v^2/R, where Fg is the gravitational field strength, m is the mass of the object, v is the velocity of the equator, and R is the radius of the Earth.
  • #1
TN17
47
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As a homework question, it asks, "...if the Earth were rotating so fast that the objects at the equator were apparently weightless?"

Somewhere, someone said that, quote:
In order for the rotation of the Earth to cancel weight, the gravitational field strength should equal the centripetal accel. (v^2/R)

Do they mean g=a(centripetal)?
I don't get how that makes sense.
 
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  • #2
The object on the equator moves along a circle of radius of the Earth (R) with the velocity of the equator (v). The centripetal force needed to this motion is provided by gravity Fg=GmM/R^2 (M is the mass of Earth) and the normal force N acting between the object and ground:

mv^2/R=GmM/R^2+N.

The object is weigthless if the ground does not push it upward, and the object does not push the ground, that is N=0. the If the normal force is 0 the centripetal force is equal to gravity at the equator.

mv^2/R=GmM/R^2

The gravitational field strength is Fg/m. Dividing the previous equation by m,

Fg/m = G M/R^2= v^2/R.

ehild
 

1. Can the gravitational field strength be greater than the centripetal acceleration?

No, the gravitational field strength is a measure of the force of gravity exerted on an object, while the centripetal acceleration is a measure of the acceleration towards the center of a circular path. These two quantities cannot be compared as they have different units.

2. How do gravitational field strength and centripetal acceleration relate to each other?

The gravitational field strength and centripetal acceleration are related through the equation F = ma, where F is the force of gravity, m is the mass of the object, and a is the centripetal acceleration. This equation shows that the gravitational field strength and centripetal acceleration are both dependent on the mass of the object.

3. Can gravitational field strength and centripetal acceleration have the same value?

Yes, it is possible for the gravitational field strength and centripetal acceleration to have the same value. This would occur when the mass of the object is such that the force of gravity and the centripetal acceleration are equal. This can happen in certain scenarios, such as when an object is in orbit around a central body.

4. How does the gravitational field strength affect centripetal acceleration?

The gravitational field strength does not directly affect the centripetal acceleration. However, it is a factor in determining the force of gravity, which in turn affects the centripetal acceleration through the equation F = ma.

5. Can the gravitational field strength and centripetal acceleration change over time?

Yes, both the gravitational field strength and centripetal acceleration can change over time. The gravitational field strength can change due to changes in the mass of objects or their distance from each other. The centripetal acceleration can change due to changes in the speed or direction of the object in circular motion.

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