How Does a Charged Particle Move Under Inverse Square Forces?

[dogg187]

Help! Calculus: Vectors - Forces Problem...

First of all, every1 hello! Imma new member...

Homework Statement

A charged particle placed at P(1,1) is repelled from the points A(0,0), B(2,0), and C(0,3) by three forces, whose magnitudes are inversely proportional to the square of the distances PA, PB and PC, respectively. In what direction will the particle move initially? (Assume that the proportionality constant is the same for all three forces.)

The answer is "at angle of 80 degrees to the positive x-axis."

But how do you find this?

Homework Equations

COS Law, SIN Law,
there should be others, but I don't know which ones...

The Attempt at a Solution

So far I found out that angle APC is 108*, angle ACP is 27*, Angle PAC is 45*, agnle APB is 90. The length of AP and PB is sq. rt. of 2, and PC is sq. rt. of 5
...

 

[mighty2000]

Hello and welcome to the forum!

To find the direction in which the particle will move initially, we can use vector addition. Since the forces are acting in different directions, we can use the parallelogram law of vector addition.

First, let's draw a diagram to visualize the problem. We have a particle at point P(1,1) and three forces acting on it from points A(0,0), B(2,0), and C(0,3). The direction of each force is shown in the diagram.

Now, let's label the forces as F1, F2, and F3, with magnitudes inversely proportional to the square of the distances PA, PB, and PC, respectively. Since the proportionality constant is the same for all three forces, we can write:

F1 = k/1^2
F2 = k/2^2
F3 = k/3^2

where k is the proportionality constant.

Now, to find the resultant force, we can use the parallelogram law of vector addition. We draw a parallelogram with sides representing F1 and F2. The diagonal of this parallelogram represents the resultant force. Similarly, we draw another parallelogram with sides representing F2 and F3, and the diagonal representing the resultant force.

The two diagonals intersect at a point, which represents the direction in which the particle will move initially. This point is shown in the diagram as R.

To find the direction, we can use the direction of the diagonal PR as an approximation. To find the exact direction, we can use trigonometry. We can write:

tan θ = (y-coordinate of R)/(x-coordinate of R)

Using the coordinates of R from the diagram, we get:

tan θ = (3-1)/(2-1) = 2

This gives us θ = tan^-1(2) ≈ 63.43 degrees.

However, this is not the exact direction. To get the exact direction, we need to consider the direction of each force. We can see that F1 is acting in the direction of the negative x-axis, F2 is acting in the direction of the positive x-axis, and F3 is acting in the direction of the positive y-axis.

Since F1 and F2 are acting in opposite directions, they will cancel each other