Help with Mechanics: Towing a Barge with Flexible Cable

[teru]

mechanics is hard...need help!

hello physics forums,
this is lengthy but pretty much challenging..really appreciate any help ^o^

A tugboat is towing a barge with a flexible cable. After towing steadily at constant speed for sometime, the tug propeller thrust T(t) is decreased from 100kN at a steady rate (dT/dt = constant).

given: mass of tug, m1 = 10,000 kg
mass of barge, m2 = 100,000 kg
tug drag characteristic, D1 = 2000.(v1)^2 N
barge drag charateristic, D2 = 6700.(v2)^2 N
cable stiffness 800 kN/m
vb = 0.4904 m/s

1) Determine the maximum magnitude of the 'steady rate' that ensures the tow cable is always in tension during the deceleration from the constant speed to a barge velocity, vb m/s (ignore the cable mass and its sag).

2) Can the non-linear differential equations be solved numerically. how will v1 and v2 behave with respect to time, t?

 

[HallsofIvy]

1) I'm not sure how "cable stiffness" affects this but:
Let F be the tension in the cable. There are three forces on the tugboat: the thrust from its propellor, T, which is positive, the drag, which is negative, and the pull from the barge, F, which is negative: $10000\frac{dv_1}{dt}= T- 2000v_1^2-F$.
There are two forces on the barge, the pull from the tugboat, F, which is positive, and the drag, 6700 v₂², which is negative: $100000\frac{dv_2}{dt}= F- 6700v_2^2$.

Question 1 asks, what is the maximum of dT/dt so that F never becomes 0.

 

[M98Ranger]

Hi there,

I understand that mechanics can be challenging, but with some guidance and practice, I believe you can master it! Let's break down the problem and see how we can approach it.

First, let's draw a free body diagram of the system. We have the tugboat and barge connected by a flexible cable, and we also have the forces acting on each object (thrust, drag, and tension in the cable).

Next, we can apply Newton's second law to each object to set up the equations of motion. We know that the tugboat and barge are moving at a constant speed, so the sum of the forces in the x-direction must be equal to zero. This allows us to solve for the tension in the cable, which should always be in tension to keep the objects connected.

Now, we can use the given drag characteristics to solve for the drag forces on each object. From there, we can set up an equation for the acceleration of the system, which will depend on the rate at which the thrust is decreased.

To ensure that the cable is always in tension during the deceleration, we need to find the maximum magnitude of the steady rate (dT/dt) that will not cause the tension to become zero. We can set up an inequality using our equation for acceleration and the maximum tension we found earlier.

As for the second part of the problem, yes, the non-linear differential equations can be solved numerically. You can use numerical methods such as Euler's method or Runge-Kutta method to solve for the velocities of the tugboat and barge with respect to time. The behavior of v1 and v2 will depend on the initial conditions and the rate at which the thrust is decreased.

I hope this helps you get started on the problem. Remember to always draw a free body diagram and apply Newton's laws to solve mechanics problems. Good luck!