Optimal Coordinate System for Applying Newton's 2nd Law on an Inclined Plane

[ysk1]

(Please look at the diagram attached at the bottom)

Question:

Newton's 2nd law, F=ma, is a vector equation. To add or subtract vectors it is often easiest to decompose the vector into components. Whereas a particular set of vector components is only valid in a particular coordinate system, the vector equality holds in any coordinate system, giving you freedom to pick a coordinate system that most simplifies the equations that result from the component equations.
It's generally best to pick a coordinate system with as many unknowns as possible along the coordinate axes. Vectors that lie along the axes appear in only one of the equations for each component, rather than in two equations with trigonometric prefactors. Note that it is sometimes advantageous to use different coordinate systems for each body in the problem.

In this problem, you should use Cartesian coordinates and your axes should be stationary with respect to the inclined plane.

Given the criteria just described, what orientation of the coordinate axes should you use in this problem?
In the answer options, "tilted" means with the x-axis oriented parallel to the plane (i.e., at angle theta to the horizontal), and "level" means with the x-axis horizontal.
1.
A) tilted for both block 1 and block 2
B) tilted for block 1 and level for block 2
C) level for block 1 and tilted for block 2
D) level for both block 1 and block 2


2. What is the sum of the x components of the forces acting on block 2? Take forces acting up the incline to be positive.
Express your answer in terms of some or all of the variables tension T, m_2, the magnitude of the acceleration of gravity g, and theta.




I don't get what these questions are asking, especially question 1.
Because I'm stuck on question 1, I can't proceed to question 2.
The question doesn't even explain what Cartesian coordinates are. :(
Could anyone please tell me how I should do these problems.
Thank you.

 

[Doc Al]

Cartesian coordinates are just your usual x-y coordinates. Hint: Choose your axes so that the motion of the object is parallel to one axis. That makes your equations much simpler.

Which way do the blocks move?

 

[edavey8205]

Yes I would define your origin first and to make it a little easier i would use block 2 to be level - place your x-y origin at the base of block 2 so that there are no angle for that block simply because it has one more force then block one. Hope that can help start it. If you need more help let me know.

 

[ysk1]

I got #1, but I can't get the right answer for #2.

2. What is the sum of the x components of the forces acting on block 2? Take forces acting up the incline to be positive.
Express your answer in terms of some or all of the variables tension T, m_2, the magnitude of the acceleration of gravity g, and theta.

As an answer, I got the following but it's wrong. I think that's right, but why is it wrong?:

sum of F_2x = T - mgsin(theta)

 

[Doc Al]

sum of F_2x = T - mgsin(theta)

Make sure you specify which mass you are using in your equation.

 

[ysk1]

Now I have problem with the question below:

Write equations for the constraints and other given information

In this problem, the fact that the length of the string does not change imposes a constraint on relative accelerations of the two blocks. Find a relationship between the x component of the acceleration of block 2, a_2x, and the acceleration of block 1. Pay careful attention to signs.
Express a_2x in terms of a_1x and/or a_1y, the components of the acceleration vector of block 1.



I don't know what the question is asking.
Please tell me how I should solve this.
Thank you.

 

[Doc Al]

Take a look at this thread https://www.physicsforums.com/showthread.php?p=1094861. See my comments on finding the "acceleration constraint" of the system.