Dear Hammock Physics 102 class,

A number of you have come to my office hours asking about Professor TeeDee's cryptic solution to midterm question number 42. Recall that the problem asks for you to give the tension on a ridgeline, given inclination angle alpha on the suspension ropes above the ridgeline, inclination angle beta on the suspension ropes below the ridgeline, and weight w. Professor TeeDee's solution is illustrated below

The key to understanding this solution is to think about the forces on the join point J where the ridgeline attaches to the suspension ropes.

Slightly abusing notation by reversing direction, the force F2 is the restorative force due to stretching, the force F1 is a force due to gravity on the hammock/user along the rope, and force P is the tension on the ridgeline. We know how to compute F2; we know that the force due to gravity is equal in magnitude to the restorative force on that line. Importantly, we know that in vector arithmetic P + F2 + F1 = 0; J is stationary, so the forces cancel.

Remember the geometry of vector addition : to compute F2+F1 we can draw vector F2, reposition a vector with the same length and angle as F1 at the head of F2, and the sum is the vector extending from F2's base to the head of the repositioned F1. This is illustrated below.

The first thing to note is that the direction of the vector F2+F1 has to be opposite to the direction of P. This is handy, because it means we can use right triangles in the analysis.

Now you need to recall some geometric identities, also illustrated in this picture, of angles equal to alpha and beta. You also need to recall a little bit of trigonometry, that says the length of the adjacent side is the cosine of the angle times the length of the hypotenuse. Knowing the length F2 and the angle alpha, we can compute the length of line segment ac; knowling length F1 and the angle beta, we can compute the length of line segment bc. The difference between lengths of ac and bc gives the length of ab, which is the magnitude of P, and is equal to Professor's TeeDee's solution.

Professor TeeDee reminds you all that the final exam will ask this same question, without giving you angles alpha and beta, only the spring constants and lengths of the suspension ropes and ridgeline.

Please note that June 15 is the last day you can drop this class without penalty. There are still plenty of spaces available in Professor JustJeff's popular survey course on Hammock Camping.

Hammock Physics 102 TA

## Bookmarks