Equation of motion with variable acceleration

[mitochondria]

I have recently been working on a project regarding black holes and the spaghettification aspect of it interests me quite a bit. So, I have set out to try to derive some mathematical descriptions of the geometry of the object being spaghettify.

I have spent a few hours (uncessfully) trying to get an expression of acceleration in terms of time, which will eventually (hoepfully) lead me to an equation that desribes the changing distance (being stretched by differential force) of 2 furtherst points in the axis perpendicular to the gravitational field of the object (spherical) being spaghettified.

My problem is that I don't know the equations of motion in which the acceleration is not constant *frown*.

This is what I have so far for the two points described above:

$$F_{diff} =\frac{4GMmr}{x^3}$$

(for x >> r, where x is the distance between the two center of masses and r is the radius of the sphere)Divide both sides by m:
$$a =\frac{2GMr}{d^3}$$

(I think...) If I want to get an expression of how r changes over time I need to integrate the expression with respect to t. In this case I need to find an expression for d in terms of t - which I can't because I don't know the equation of motions with variable acceleration...

Thanks in advance!

 

[anathema]

Thank you for sharing your project on black holes and the spaghettification phenomenon. It is indeed a fascinating topic and I commend you for delving into the mathematical aspect of it.

From your post, it seems like you have a good understanding of the concept and have made some progress in deriving the equations for the acceleration of the two points in the perpendicular axis. However, as you have rightly pointed out, the challenge lies in dealing with variable acceleration.

In this case, you may want to consider using the equations of motion for a non-uniformly accelerated object, such as the one derived by Galileo and later expanded upon by Newton. These equations take into account the changing acceleration of an object and can be used to describe the motion of the two points in your system.

Another approach could be to use the concept of differential equations, which are commonly used in physics to describe systems with variable quantities. By setting up appropriate differential equations for the motion of the two points, you may be able to solve for the changing distance between them over time.

I would also recommend consulting with other scientists or researchers who have studied black holes and their behavior, as they may have insights or resources that could aid in your project. Additionally, you could also consider looking into existing literature or studies on spaghettification to see if there are any relevant equations or models that could be applied to your project.

Overall, I encourage you to continue your research and exploration of this interesting topic. I am confident that with determination and the right resources, you will be able to derive the mathematical descriptions you are seeking. Best of luck in your project!

 

[kyle8921]

I am excited to see your interest in studying the spaghettification aspect of black holes. It is a complex phenomenon that requires advanced mathematical descriptions to fully understand. I can offer some insights and suggestions to help you in your project.

Firstly, the equation you have derived for the differential force (F_diff) is correct. It takes into account the gravitational force exerted by the black hole on the object being spaghettified. However, in order to fully describe the motion of the object, we need to consider the net force acting on it. This includes not only the differential force, but also the object's own inertial forces.

Secondly, the equation you have derived for acceleration (a) is also correct, but it is only valid for a specific scenario where the distance between the two points (d) is much larger than the radius of the black hole (r). This is known as the "far-field" approximation and it simplifies the calculations. However, when studying the spaghettification process, we need to consider the entire range of distances from the black hole, not just the far-field.

To take into account the changing acceleration, we need to use the general equation of motion:

F_net = ma

where F_net is the net force acting on the object and m is its mass. This equation can be rearranged to solve for acceleration (a):

a = F_net/m

Now, the challenge lies in finding the net force acting on the object. This involves considering all the forces (gravitational, inertial, etc.) and their varying strengths at different distances from the black hole.

One approach to solving this problem is to use numerical methods, such as computer simulations, to track the motion of the object and calculate the net force at each point in time. This will give you a more accurate representation of the spaghettification process compared to analytical equations.

In summary, to fully describe the motion of an object being spaghettified by a black hole, we need to consider the changing acceleration and the net force acting on the object. This can be achieved through numerical methods or by using more advanced mathematical techniques, such as differential equations. I hope this helps in your project and I wish you all the best in your research.