Total centripetal acceleration ringworld

[Maiia]

Homework Statement

A) In Larry Niven’s science fiction novel Ringworld, a ring of material or radius 1.54 × 1011 m rotates about a star with a rotational speed of 1.4 × 106 m/s. The inhabitants of this ring world experience a normal contact force ~n. Acting alone, this normal force would produce an inward acceleration of 9.44 m/s2. Additionally, the star at the center of the ring exerts a gravitational force(also pointing inwards) on the ring and its inhabitants. What is the total centripetal acceleration of the inhabitants? The universal gravitational constant is 6.67259 × 10^ −11 N · m2/kg2 . Answer in units of m/s2.
B) The difference btw the total acceleration and the acceleration provided by the normal force is due to the gravitational attraction of the central star. Find the approximate mass of the star. Answer in units of kg.

I wasn't quite sure how to approach this...
I tried first setting the acc of Fn to 4pir^2/T2 to find the period, which I got to be 802,516.9827s and then substituting that into R^3orbit/T^2= GMplanet/4pir^2 to find the mass of the planet...which is more for the 2nd part I guess..

 

[LowlyPion]

Wouldn't you first calculate the centripetal acceleration for the rotational speed at that radius? Then the difference between that and the actual actual given normal force would be the gravitational acceleration toward the sun?

Knowing that force and radius then ...

Picture of Ringworld
http://www.fantasticfiction.co.uk/images/n0/n2565.jpg

 

[thomate1]

Hello, it seems like you have made some progress in approaching this problem. Let's break it down step by step to make it clearer.

First, let's analyze the situation. We have a ringworld with a radius of 1.54 × 10^11 m, rotating around a star with a speed of 1.4 × 10^6 m/s. The inhabitants on the ringworld experience a normal force, which alone produces an inward acceleration of 9.44 m/s^2. However, we also have to consider the gravitational force from the star, which also acts inwards.

Now, we can use Newton's second law, F=ma, to find the total acceleration of the inhabitants. The total force acting on them is the sum of the normal force and the gravitational force: F_total = F_normal + F_gravity. We can write this as ma_total = ma_normal + ma_gravity.

For the normal force, we can use the given information to find its magnitude. F_normal = ma_normal = m(9.44 m/s^2). Since we are looking for the total acceleration, we can rearrange this equation to get a_normal = 9.44 m/s^2.

For the gravitational force, we can use Newton's law of universal gravitation, F_gravity = GmM/r^2, where G is the universal gravitational constant, m is the mass of the inhabitants, M is the mass of the star, and r is the distance between the star and the ringworld. We know the value of G and r, and we are looking for the total acceleration, so we can rearrange this equation to get a_gravity = G(M/r^2).

Now, we can substitute these values into our original equation and solve for the total acceleration, a_total = a_normal + a_gravity. This will give us the total centripetal acceleration experienced by the inhabitants on the ringworld.

For the second part, we can use the same equation, F_gravity = GmM/r^2, but this time we are looking for the mass of the star, M. We already know the value of G, and we can find the value of m from the first part of the problem. We also know the value of r, so we can rearrange the equation and solve for M to find the approximate mass of the star.

I hope this helps clarify the problem and guide