Orbital Velocity vs Mass and Orbital Velocity vs Radius

[ha9981]

Homework Statement

Graph the relationship between the speed of a satellite orbiting a large mass in uniform circular motion. Also I need to learn how to properly write proportionality constant and proportionality equation.

Homework Equations

in the equation below, m is the mass of the larger mass. G is gravitational constant 6.67x10^-11.

v = $$\sqrt{Gm/r}$$

The Attempt at a Solution

I think that:

V is directly proportional to sqrt of Gm (v $$\propto$$ $$\sqrt{Gm/r}$$]
and
V is inversely proportional to sqrt of r ( v $$\propto$$ 1/r )

Now how do i graph these, i have ideas:
Remove the square root by making the y-axis squared.
And first proportionality is linear and the second one is decreasing from value of 1 because its 1/r relationship and maximum y value is 1.

For writing proportionality statements is everything in the numerator directly proportional, and everything in the denominator inversely proportional?
Also how does the proportionality constants work? I know you have to introduce them with I believe every relationship. Do they go with the inverse relationships as well? And with equations with numerator and denominator is there a constant on both top and bottom or is it made into one value.

 

[Redbelly98]

Welcome to PF

Homework Statement

Graph the relationship between the speed of a satellite orbiting a large mass in uniform circular motion. Also I need to learn how to properly write proportionality constant and proportionality equation.

Homework Equations

in the equation below, m is the mass of the larger mass. G is gravitational constant 6.67x10^-11.

v = $$\sqrt{Gm/r}$$

The Attempt at a Solution

I think that:

V is directly proportional to sqrt of Gm (v $$\propto$$ $$\sqrt{Gm/r}$$]
and
V is inversely proportional to sqrt of r ( v $$\propto$$ 1/r )

Looks good. However it may be better to state the relation between V and m, without the G, since G is a constant and not a variable.

Now how do i graph these, i have ideas:
Remove the square root by making the y-axis squared.

I don't think that's necessary, you should be able to simply graph y and x (whatever they are). Has your teacher said anything about trying to make the graph be a straight line?

And first proportionality is linear and the second one is decreasing from value of 1 because its 1/r relationship and maximum y value is 1.

I don't understand this.

For writing proportionality statements is everything in the numerator directly proportional, and everything in the denominator inversely proportional?

Pretty much. But that really only applies to variables in the numerator or denominator. It shouldn't be necessary (most of the time) to talk about how something varies with respect to a constant like G.

Also how does the proportionality constants work? I know you have to introduce them with I believe every relationship. Do they go with the inverse relationships as well?

Yes, constants are involved in inverse relationships.

And with equations with numerator and denominator is there a constant on both top and bottom or is it made into one value.

One constant should suffice, in general.

Hope this helps.

 

[ha9981]

When I am asked to write the proportionality constant for these things would I just write G or the eqn with G?

Also I am thinking that after I remove the G from that eqn above I should put a K outside as a proportionality constant.

About making the graphs linear I would make y-axis v^2 and the x-axis for the first graph m and the second graph 1/r. Is that right? And I am guessing graph would look like one with a slope of 1 .