0.13 Phy1100: force -- vector solutions for coplanar forces concurrent  (Page 10/13)

Add vector C to the diagram

Go to the tail of vector A and draw vector C with an unspecified length at an angle of 120 degrees. The tip of vector C will touch the tail ofvector A and the tail of vector C will be somewhere down and to the right of the tail of vector A.

The lines should cross

If you did everything right, the lines that describe vector B and vector C should cross. Use a pushpin to mark the place where they cross. This point willbe the tip of vector B and will be the tail of vector C.

Measure the lengths of vector B and vector C

At this point, you can measure the lengths of the lines that describe vector B and vector C. Using the same plotting scale that you used for the 10-newtonvector A, you can determine the magnitudes of each of those vectors.

Since this is the same problem as before, the magnitudes for those two vectors should be the same asbefore. Vector B should be about 5.3 newtons and vector C should be about 7.3 newtons.

Was this easier than the parallelogram?

I suspect that you probably found the physical construction of the triangle solution to be easier than the physical construction of the parallelogramsolution. I further suspect that the trigonometric solution, which I will explain in the next section, will be even easier for blind students.

Trigonometric solution to the triangle of forces

Let's pretend that you don't have a graphic solution for this problem. However, you do have a rough sketch of what the triangle looks like onyour graph board. Use your Braille labeler to label the three vectors in the sketch A, B, and C. Then label the angles opposite each side as a, b, and c respectively.

Here are the facts as we know them at this point:

• Vector A = 10 newtons.
• Angle a = 105 degrees.
• Sine 105 degrees = 0.966
• Angle b = 30 degrees
• Sine 30 degrees = 0..5
• Angle c = 45 degrees
• Sine c = 0.707

The law of sines

We will solve this problem using the law of sines that I explained earlier. Given the names that we have applied to the sides and the angles for this triangle, the law ofsines can be written as follows:

(A/sin a) = (B/sin b) = (C/sin c)

(Note that upper and lower-case letters were switched relative to the law of sines given near the beginning of this module.)

Solve for the length of side B

Using the first two ratios and rearranging terms gives

B = A*sin b/sin a = 5.18

Solve for the length of side C

Using the first and third ratio and rearranging terms gives

C = A*sin c/sin a = 7.32

The answers

These are the answers that we are looking for:

• B = 5.18 newtons
• C = 7.32 newtons

Using components to analyze for equilibrium

Tent pole exercise

Please sketch this scenario on your graph board. Then we will solve the problem using a script and components.

A bird's eye view of the top of a tent pole shows guy wires extending outward and exerting radial10-newton forces on a tent pole at the following angles :

• 0 degrees
• 45 degrees
• 90 degrees
• 135 degrees
• 180 degrees
• 225 degrees
• 270 degrees

Tactile graphics

The image in the file named Phy1100e1.svg contains seven vectors of equal length emanating out from a common point at the angles given above.