How Is the Orbital Period of a Satellite Determined?

AI Thread Summary
The orbital period T of a satellite is derived using Newton's law of universal gravitation and centripetal force principles. The relationship shows that T is given by the formula T = 2π√(r³/GM), where G is the gravitational constant and M is the mass of the planet. The derivation starts with equating the centripetal force (mv²/r) to the gravitational force (GmM/r²). It concludes that both the orbital speed and period depend solely on the orbital radius r. This formula is essential for understanding satellite motion in gravitational fields.
mogley76
Messages
17
Reaction score
0

Homework Statement



by using Newtons law of universal gravitaion and the relationship for the centripetal force, show that the orbital period T for a satellite circling a planet of mass M in orbit of radius r is given by :
T= 2pi sqrt r^3/GM

Homework Equations



none given

The Attempt at a Solution



consider a satellite of mass m orbiting the Earth of mass M with a constant speed v along a circular path of radius r.

the centripetal force : mv^2/r is provided by the gravitational attraction of the earth

so F= mv^2/r = GmM/r^2

v of robit = sgrt GM/r

the orbital period T = 2pi/w
since v=wr T now eqauls : 2pi r /v

therefore T= 2 pi sqrt r^3/GM
as both G and M are constants it can be seen that both the orbital speed and period depend only on the orbital radius.


have i derived this correctly?
 
Physics news on Phys.org
Perfect!
 
yay!:cool:
 
When you start out with Fc = Fg as you did, you nearly always get the formula you want pretty quickly. It is useful to have the alternate form of Fc= 4π²mR/T² handy. Use it if you are interested in T, and mv²/R if you are interested in v.
 

Thread 'Struggling to understand the concept of this question (ice cube in water optics question)'

TL;DR Summary: I came across this question from a Sri Lankan A-level textbook. Question - An ice cube with a length of 10 cm is immersed in water at 0 °C. An observer observes the ice cube from the water, and it seems to be 7.75 cm long. If the refractive index of water is 4/3, find the height of the ice cube immersed in the water. I could not understand how the apparent height of the ice cube in the water depends on the height of the ice cube immersed in the water. Does anyone have an...
View full post »
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
View full post »

Similar threads

Time taken for a planet to collide with the Sun
Replies
10
Views
802
Calculate Mass of Satellite: Step-by-Step Guide
Replies
16
Views
3K
Planet moving in elliptical orbit around a star
Replies
13
Views
837
Potential Energy and raising a satellite from Earth into a Circular Orbit
Replies
5
Views
2K
Finding the period of the satellite's orbital motion
Replies
1
Views
2K
A satellite in orbit in a system with one planet and two moons
Replies
1
Views
1K
Find at what rate the orbit radius will grow
Replies
30
Views
2K
Doubling speed in circular orbit to attain elliptical orbit
Replies
3
Views
2K
Finding r of a Jovian-synchronous orbit and escape velocity
Replies
7
Views
2K
Finding Orbital Period of Unknown Planet
Replies
5
Views
2K

Hot Threads

Recent Insights

Back
Top