(b) the distance d.

Question

A lightweight string of length 56.6 cm is attached to the ceiling at two points separated by distance d, as shown in the diagram below. Three objects of equal mass are hung from the string, separating the string into four equal segments of length L. In the diagram, angle q = 69.0^O (i.e. this is "theta", the upper angle, in the picture.) Find:

  (a) the measure of angle f (i.e. phi) and

  (b) the distance d.

Enter your answer for Part (b), in cm, in the box below.

Answer 1

Assuming theta is the outer segment’s angle with the horizontal ceiling, and phi is the inner segment’s angle with the horizontal:

Each of the four segments has length:

L = 56.6 / 4 = 14.15 cm

Each inner segment supports half the middle object's weight. Each outer segment supports that plus one additional object's weight.

The vertical tension components therefore have a 3:1 ratio, while the horizontal components are equal:

tan(theta) = 3 * tan(phi)

(a) Find phi:

phi = arctan(tan(69 degrees) / 3)

phi = 40.97 degrees

phi ≈ 41.0 degrees

(b) Add the horizontal projections of all four segments:

d = 2L * cos(theta) + 2L * cos(phi)

d = 2(14.15) * [cos(69 degrees) + cos(40.97 degrees)]

d = 31.51 cm

Answer for part (b): 31.5 cm

The diagram was not attached, so these answers depend on the angle definitions stated above.