I need to plot a graph for this physics problem and I don't know how to do it.

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I have a fish bowl problem that is a circle with the top "cut-off". The height is 4*r/3 which is formed by removing the top third of a sphere (of radius r). The fishbowl is fixed in the sand so thats its rim is parallel to the ground. A small marble of mass m rests at the bottom of the fish bowl. Assuming all surfaces are frictionless and ignoring air resistance, find the maximum inital velocity that could be given to the marble for it to land back in the fish bowl.
This is my problem. I need help plotting the trajectory of the marble for its predicted projection. I also need help figuring out a code to plot the fish bowl so that I could also plot the projection with it.

Respuestas (2)

Walter Roberson
Walter Roberson el 29 de Nov. de 2023
By knowing the cutoff height you can calculate the radius of the circular opening.
By knowing the cutoff height you can calcuate the tan() of the angle that the sphere makes at the opening.
You now have the situation of launching a particle at a known angle, and needing it to land at the other rim of the circlular opening.
As gravity is symmetric in an environment with no air friction, you know that the peak of the arc must be at the center of the circle.
You should be able to construct a relatively simple differential equation for the path.

Image Analyst
Image Analyst el 29 de Nov. de 2023
It's a bit tricky. So the marble is at the bottom of a sphere with no water in it and you flick it. It sails upward along the glass until it gets to the rim, then it's just a regular projectile. Now first you need to know the distance along the opening at the top of the fishbowl because that is how far the marble can travel and still fall into the fishbowl. Once you know that distance, and the angle it's heading on as it leaves the glass, you can get Vx and Vy. In other words, if you know the velocity component vectors and the angle, you can determine how far it will travel, which should be to the other side of the fishbowl opening at the top.
Now that you know the velocities, you can use the formulas for angular velocity. You can get the angular velocity, however it's a bit tricky since the vertical velocity is affected by gravity, unlike it would be if, say, you flicked the marble along a circle parallel with the earth. So you need to take into account that gravity is pulling down on the marble slowing down it's vertical velocity component.
So it looks like it can be solved by splitting it up into two parts like I said. Part 1 above the glass, and part 2 in the glass bowl. It might take you a few hours wrangling equations to get the solution but once you have it you can just code it up easily in MATLAB.
My attached projectile demo might help, as will asking your instructor.

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