Solving Dynamics Homework Problems

In summary, I was trying to solve for the velocity of a particle that is moving in a spiral path described by r=1m and θ=2z rad, and moving along the z axis at a constant speed of |v | = 1000 m/s, and I was stuck. But after solving for the velocity in cylindrical coordinates, I was able to find that the velocity of the particle in terms of cylindrical coordinates is v= dr/dt êr + ω êθ + dz/dt êz.
  • #1
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For some reason I just don't get my dynamics homework. Here are the few problems I am having.

Problem 1:
A river is flowing north at 3 m/s (uniform current). If you want to travel in a straight line from point C to point D in a boat that moves at a constant speed of 10 m/s relative to the water, in what direction should you point the boat? How long does it take to make the crossing?

Point C is on the west bank of the river and the river is 500 m wide. Point D is 400 m north of point C on the east side of the river. Basicaly points C and D and the east side of the river make a right triangle with a base of 500 and height of 400 and the hypontenuse CD.

I found the angle of CD to be 38.66 degrees.

I set my coordinate system so that north is positive y or j direction and that east is positive x or i direction.

My first step was to find the the velocity of the the boat and to do that i used the following equation.

Velocity of the boat relative to river (10 m/s) = Vcos(38.66)i + Vsin(38.66)j - 3j

V = velocity of the boat

I found the magnitude by taking the square root of all of that and solving for V (i had to use the quadratic formula to get the roots for V). Therefore:
V= 11.62 m/s

Now I'm stuck. I have no idea where to go from here to find the direction. I substituted 11.62 back into my original velocity equation to get:

V wrt river = 9.07i + 4.26j

I'm thinking that that is only the componets of the velocity not the actual direction the boat is pointing maybe I'm wrong I don't know. Please help.

**Forgot to add the other problem I was stuck on

Problem 2:

A particle moves in a spiral path described by r = 1m and θ = 2z rad (where z is in meters) and moves along the z axis at a constant speed of |v | = 1000 m/s. What is the velocity of the particle in terms of cylindrical coordinates?

I have an equatoin that shows the velocity in cylindrical coordinates which is:
v = dr/dt êr + ω êθ + dz/dt êz

And there I am stuck. I think I can find ω by multiplying the the velocity by 2z because that would be ds/dt * dθ/ds. But after that how would i go about finding dr/dt and dz/dt? The ê are the unit vecots in the r, θ, and z directions respectively.
 
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  • #2
In problem 1 you say "Velocity of the boat relative to river (10 m/s) = Vcos(38.66)i + Vsin(38.66)j - 3j" That, since it take into account the motion of the river, is relative to the shore if the boat were aimed directly at point D (and if the river were flowing south!).

Assume you aim your boat at an angle θ to the shore. Then the velocity of your both, relative to the river, is 10cos(θ)i+ 10sin(θ)j. The velocity of the river, relative to the shore is +3j (NOT -3j) so the velocity of the boat relative to the shore is 10cos(θ)i+ (10sin(θ)+ 3)j and you want that to take you directly to D: you want (10sin(θ)+3)/10cos(θ)= 400/500. Solve that for θ.

2. Well, since r= 1, a constant, dr/dt= 0, doesn't it? ω IS dθ/dt so, no, you don't multiply by 2z : if θ= 2z, then dθ/dt= 2 dz/dt.

Your formula becomes 0 er+ 2 dz/dt eθ+ dz/dt ez. Set the length of that to 1000 and solve for dz/dt. Then plug that value back into your formula
 
  • #3
Awesome thanks for the help. Everything worked out nicely.
 

1. How do I approach solving dynamics homework problems?

To approach solving dynamics homework problems, you should first carefully read and understand the problem statement. Then, identify the given information and what is being asked for. Next, draw a diagram to visualize the problem and label all known and unknown variables. Finally, use the appropriate equations and principles to solve for the unknown variable.

2. What are the key principles and equations used in dynamics problems?

The key principles used in dynamics problems are Newton's laws of motion and the principle of conservation of energy and momentum. The key equations used are force = mass x acceleration, work = force x distance, and kinetic energy = 1/2 x mass x velocity squared.

3. How do I handle problems involving multiple objects or systems?

To solve problems involving multiple objects or systems, you should first draw separate free-body diagrams for each object or system, and then apply Newton's laws of motion and the principle of conservation of energy and momentum to each individual diagram. Finally, combine the results to solve for the overall problem.

4. What are common mistakes to avoid when solving dynamics homework problems?

Some common mistakes to avoid when solving dynamics homework problems include not carefully reading the problem statement, not identifying all relevant information, and using incorrect equations or principles. It is also important to double check your calculations and units to ensure accuracy.

5. How can I check my work and ensure I have the correct solution?

You can check your work and ensure the correct solution by plugging your calculated values back into the original equations to see if they satisfy the given conditions. You can also compare your solution to similar problems or ask a classmate or teacher to review your work.

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