Centripetal Acceleration of object orbiting earth

[Fusilli_Jerry89]

Homework Statement

An object orbits the Earth at a constant speed in a circle of radius 6.38x10^6m, very close to but not touching the Earth's surface. What is its centripetal acceleraion?

Homework Equations

The Attempt at a Solution

Quick question, when would GM/r^2 equal to the centripetal acceleration, and when wouldn't it? I thought it always did but some calculations don't have the same numbers if you do it both ways. For this question would it be?

 

[Dick]

Well, force is GMm/r^2 (Newton et al) and acceleration is force/m (Newton et al). So acceleration=GM/r^2. If you have evidence otherwise, cough it up.

 

[m2003]

The centripetal acceleration of an object orbiting the Earth can be calculated using the formula a = v^2/r, where v is the velocity of the object and r is the radius of its orbit. In this case, the object's velocity is constant and equal to the speed of its orbit, so we can use the formula a = (2πr)/T^2, where T is the period of the orbit. This formula can also be written as a = (4π^2r)/T^2.

To find the period, we can use the formula T = 2πr/v, where v is the speed of the orbit. Plugging in the given values, we get T = 2π(6.38x10^6m)/v. Since the speed is constant, we can use the given radius to calculate the period.

Once we have the period, we can plug it into the formula for centripetal acceleration to get a = (4π^2r)/T^2 = (4π^2(6.38x10^6m))/(T^2) = 9.8m/s^2. This is the centripetal acceleration of the object orbiting the Earth at a constant speed.

To address your question about when GM/r^2 equals the centripetal acceleration, it is important to note that GM/r^2 represents the gravitational force between two objects, while the centripetal acceleration is the acceleration towards the center of a circular motion. These two quantities are not always equal, as the centripetal acceleration also takes into account the velocity and radius of the orbit. However, in the case of a circular orbit, the centripetal acceleration can be calculated using the gravitational force as a = GM/r^2. This is because the force of gravity is the only force acting on the object, causing it to accelerate towards the center of the orbit.

I hope this helps clarify the concept of centripetal acceleration and how it relates to the gravitational force in orbital motion. Let me know if you have any further questions.