A body moves in a circular orbit of radius 4R round the earth . Express the acceleration of free fall to gravity of the body in terms of (*Re=radius of the earth , g=acceleration of free fall to gravity )?

(g/(R)^2)= (g=(4)^2) = (g/16)

Well, well, well, looks like someone's spinning around the Earth! Let's calculate that acceleration of free fall, shall we?

Now, when it comes to circular motion, we have a lovely little formula: centripetal acceleration (a) equals the square of the velocity (v) divided by the radius (r). In this case, the radius is 4R. So we can express it as:

a = v² / (4R)

Now, my friend, the velocity (v) in this case is given by the formula v = 2πr / T, where T represents the time period of one complete revolution. Since our body is going around the Earth, which takes 24 hours or 86,400 seconds, we can plug that in and get:

v = 2π(4R) / 86400

Simplifying further:

v = πR / 21600

Now let's substitute this value back into our original equation for acceleration:

a = (πR / 21600)² / (4R)

Now, simplify some more:

a = π²R² / (21600² * 4R)

And finally, simplify a touch more:

a = π² / (4 * 21600²)

Now, don't forget, my dear friend, that this is the centripetal acceleration. The gravitational acceleration (g) is given by the equation g = G * M / R², where G is the gravitational constant and M is the mass of the Earth. So, the acceleration of free fall is equal to g:

g = G * M / R²

Now, to express the acceleration of free fall in terms of *Re (the radius of the Earth) and g, we can divide our centripetal acceleration formula by the gravitational acceleration formula:

a / g = (π² / (4 * 21600²)) / (G * M / R²)

Now, there's an interesting fact, my friend. This fraction, (π² / (4 * 21600²)), is constant, meaning it doesn't depend on the radius or gravitational acceleration. So we can define this as a new constant, let's call it "k". Basically, we have:

a / g = k

And that, my friend, is how we express the acceleration of free fall in terms of *Re (the radius of the Earth) and g. It's simply equal to a constant value, k. So, to summarize:

a / g = k

Isn't that just out of this world?

To express the acceleration of free fall to gravity of the body in terms of (*Re = radius of the earth, g = acceleration of free fall to gravity), we need to consider the gravitational force acting on the body.

The gravitational force on a body in a circular orbit around the earth is provided by the centripetal force. The centripetal force can be equated to the gravitational force using the equation:

mv^2 / r = G * (m * M) / r^2

Here, m is the mass of the body, v is the velocity of the body in the circular orbit, r is the radius of the orbit, G is the gravitational constant, and M is the mass of the earth.

Now, the velocity of the body in a circular orbit can be related to the radius and the acceleration of free fall as:

v = sqrt(G * M / r)

Substituting this value in the equation for gravitational force, we get:

m * (G * M / r) / r = G * (m * M) / r^2

Simplifying this equation, we get:

g = G * M / r^2

Now, we can express the acceleration of free fall to gravity of the body in terms of (*Re = radius of the earth, g = acceleration of free fall to gravity) as:

g = G * M / (4 * R)^2

or

g = G * M / (16 * R^2)

Therefore, the acceleration of free fall to gravity of the body in terms of (*Re = radius of the earth, g = acceleration of free fall to gravity) is given by:

g = G * M / (16 * R^2)

To find the acceleration of free fall to gravity (g) of the body in terms of the radius of the earth (Re), we can use the concept of centripetal acceleration.

The centripetal acceleration (ac) is given by the formula: ac = v^2 / r,

where v is the velocity of the body and r is the radius of the circular orbit.

In this case, the body is moving in a circular orbit of radius 4R around the earth, so the radius of the orbit is 4 times the radius of the earth (4Re). Thus, r = 4Re.

To find the velocity (v) of the body, we can relate it to the period of the orbit using the formula: v = 2πr / T,

where T is the period of the orbit.

Since the body completes one orbit in the time taken by the earth to rotate once (T = 24 hours = 24*60*60 seconds), we can substitute the values:

v = 2π * (4Re) / (24*60*60).

Now we can substitute the formula for centripetal acceleration and the expression for velocity into the formula for acceleration of free fall to gravity:

g = ac = v^2 / r = (v^2) / (4Re).

Plugging in the expression for v, we get:

g = ((2π * (4Re) / (24*60*60))^2) / (4Re).

Simplifying further, we get the expression for g in terms of Re:

g = (4π^2 * R^2_e) / (24*60*60)^2,

or g = (π^2 * Re^2) / (9.549 * 10^7),

where Re is the radius of the Earth.