What Forces Determine a Submarine's Diving Trajectory?

[yoyo]

A submarine of mass m is diving at a 45 degree angle which means that the submarine's propulsion system is generating constant thrust T in the direction of the given angle. Assume that the density of the submarine is d1 and the density of water is d2. Further, assume that there is no resistance to motion.

a) List the forces acting on the submarine and draw an appropriate diagram.
-I think the forces that are acting on the submarine is gravity, force normal,buoyancy, but I'm not sure of what other forces there may be.

b) Examine the forces that you listed and draw a conclusion about the trajectory of the submarine.
- Have no idea of what the trajectory due to the forces are. Can someone please explain?

c) Set up a coordinate system to describe the motion (the x-axis can be the surface of the ocean, the submarine's initial location can be put at the origin) Write down Newton's equations and solve them assuming that the submarine did not have any initial velocity.
- I understand how to setup the coordinate system, but having trouble with setting up Newton's equation and solving it.

d) Plot three trajectories of the submarine for the cases d1>d2, d1=d2, and d1<d2, Label each trajectory accordingly.
- i have very little experience with plotting trajectories...can somebody help?

 

[HallsofIvy]

a) List the forces acting on the submarine and draw an appropriate diagram.
-I think the forces that are acting on the submarine is gravity, force normal,buoyancy, but I'm not sure of what other forces there may be.

Well, since you were told there is a thrust, T, don't you think you should include that? And what is "force normal"? Normal to what?
I see thrust, bouyancy, and gravity.

b) Examine the forces that you listed and draw a conclusion about the trajectory of the submarine.
- Have no idea of what the trajectory due to the forces are. Can someone please explain?

The submarine will accelerate in the direction of the net force (F= ma). Which direction do you think the net force is in?

c) Set up a coordinate system to describe the motion (the x-axis can be the surface of the ocean, the submarine's initial location can be put at the origin) Write down Newton's equations and solve them assuming that the submarine did not have any initial velocity.
- I understand how to setup the coordinate system, but having trouble with setting up Newton's equation and solving it.

Newton's equation IS F= ma. The reference to Newton's equationS means you want to use that for each of the x, y, z directions.

d) Plot three trajectories of the submarine for the cases d1>d2, d1=d2, and d1<d2, Label each trajectory accordingly.
- i have very little experience with plotting trajectories...can somebody help?

You will find that the trajectories are straight lines. What happens to bouyancy in each of the cases given?

 

[thepatientmental]

a) The forces acting on the submarine are gravity, thrust, buoyancy, and drag (assuming no resistance to motion). A diagram of the forces can be drawn with the submarine at the origin and the forces acting in different directions: gravity acting downwards, thrust acting at a 45 degree angle, buoyancy acting upwards, and drag acting opposite to the direction of motion.

b) Based on the forces acting on the submarine, we can conclude that the submarine will continue to dive deeper into the water due to the constant thrust and gravity acting downwards. The buoyancy force will counteract the weight of the submarine, but it may not be enough to stop the submarine from diving deeper. The lack of resistance to motion also means that there will be no force opposing the submarine's motion, allowing it to continue its trajectory.

c) Setting up a coordinate system with the x-axis as the surface of the ocean and the origin at the submarine's initial location, Newton's equations can be written as follows:

ΣF = ma, where ΣF is the sum of all forces acting on the submarine, m is the mass of the submarine, and a is the acceleration.

In the x-direction: ΣFx = Tcos45° - Drag = ma
In the y-direction: ΣFy = Tsin45° - mg + Buoyancy = ma

Solving these equations with the given information, we can determine the acceleration of the submarine in both the x and y directions, and therefore, its trajectory.

d) To plot the trajectories, we can use the equations we derived in part c and input different values for d1 and d2. The three cases are:
- d1>d2: In this case, the submarine will sink deeper into the water due to the higher density of the submarine compared to water. This will result in a steeper trajectory.
- d1=d2: If the densities are equal, the submarine will maintain a constant depth as the buoyancy force will perfectly balance the weight of the submarine. The trajectory will be horizontal.
- d1<d2: In this case, the submarine will float upwards due to the higher density of water compared to the submarine. The trajectory will be upwards, but not as steep as the first case.

Labeling each trajectory accordingly, we can plot them on a graph with depth (y-axis) and time (x-axis). The resulting graph will show the different trajectories for each case.