This document describes a simple pendulum with a string of negligible mass that does not stretch. It asks which statements are true if both the length of the string (L) and the arc length (S) increase by the same factor, while the mass (m) remains the same. The explanation shows that increasing L and S by the same factor does not change the angle θ, tension force magnitude, or tangential acceleration. Therefore, none of the given statements A through E are true.

1. Consider a simple pendulum (shown below) with a string of negligible mass that
does not stretch.
Question: If the length of the string (denoted as “L” in figure), and arc length (S)
increase by the same factor, which of the following statements are true right upon
release? Assume the mass (m) remains the same.
Tangential acceleration= -gsin(Θ)
S= LΘ
Note: These choices are comparing initial release quantities of the “before” scenario
and the “after” scenario (where S and L increased)
A) Tension force magnitude (denoted T) increases because the attachment point
is now further away from the mass
B) Tangential acceleration magnitude will increase
C) Tangential acceleration magnitude will decrease
D) A and B
E) A and C
F) None of these statements are true
ANSWER AND EXPLANATION ON BACK PAGE

2. Explanation: Consider the radial axis to be situated along “L” denoted in the figure.
The two forces acting along the axis would be Tension force (T) as well as the radial
axis component of gravity (mgcos(Θ)). When this mass is released, there is no
motion along the radial axis, its position is constant on this axis. Therefore, T and
mgcos(Θ) are equal and opposite. Furthermore, although the length of the string has
increased, Θ has not changed. Θ is equal to S/L and both S and L increased by the
same factor. Therefore, increasing the length did not change mgcos(Θ) meaning that
the magnitude of T did not change either. Consequently, choices “A”, “D”, and “E” are
eliminated. Using the tangential acceleration formula from above, we once again
note that Θ is still the same. Tangential acceleration has not changed after increasing
the length of the string. This eliminates choices “B” and “C”.
ANSWER: F, none of these statements are true.
References
"The Simple Pendulum - Boundless Open Textbook." Boundless. N.p., n.d. Web. 01 Feb.
2015.