Refer to the figure below
As you can see, the string can be broken into three parts:
- Section I: The portion of string from the object (represented as a square box) to the pulley. This portion is shown in blue. This portion starts at point A and ends at point B
- Section II: The portion of string that wraps around the upper side of the pulley. This portion is shown in red. This curved section starts at point B, wraps around the circle going through point C, and ends up at point D.
- Section III: The portion of string that stretches from the pulley down to the person's hand where the force is applied downward. The downward force is shown by the downward arrow. This force will lift the square box upward or simply keep it from falling to the ground. This section is shown in green. This portion starts at point D and ends up at point E.
I'm going to make the assumption that AB = DE. In other words, I'm assuming that the pulley is held such that the string falling on each side is the same length and that the end of one string is perfectly level with the other end.
If this assumption is held true, then we know that
AB = 13.7 and DE = 13.7 (since AB = DE)
So far AB+DE = 13.7+13.7 = 27.4 cm of string is used for the straight portions (section I and section III).
The string is 30.2cm long. Therefore that leaves 30.2-27.4 = 2.8 cm for the curved red section
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That curved red section (section II) is a semi-circle. It is exactly halfway around the distance of the full circle.
Double the value 2.8 to get 5.6
The circumference of the circle is 5.6 cm
So C = 5.6 will be used to find the diameter d
C = pi*d
pi*d = C
pi*d/pi = C/pi
d = C/pi
d = 5.6/pi
d = 1.78253536262922
The diameter of the pulley is roughly 1.7825 cm