Vibrator, Pulleys an d Waves Physics Problem! Wha...? How can I solve this?

Question

In the arrangement shown in Figure P14.40, an object of mass, m = 1.0 kg, hangs from a cord around a light pulley. The length of the cord between point P and the pulley is L = 2.0m
http://www.webassign.net/sf/p14_40.gif

(a) When the vibrator is set to a frequency of 175 Hz, a standing wave with six loops is formed. What must be the linear mass density of the cord?
_______kg/m

(b) How many loops (if any) will result if m is changed to 2.25 kg?
________ loops

(c) How many loops (if any) will result if m is changed to 9.00 kg?
________ loops

Answer 1

For transverse vibration of a tensioned wire or string,
fundamental frequency F1 = sqrt(T/(m/L))/L (see ref.) where T=tension, L=length, m=mass. With 6 loops we have the 6th harmonic, 6*F1, so F1 = 175/6. The m/L term is linear mass density (mass of a unit length of string).
To solve (a), we rearrange the F1 equation to
m/L = T/(F1*L)^2. (Units are kg-m/sec^2 / (m/sec)^2 = kg/m.)
With T = 1*g and L = 2, we have m/L = g/(175/6 * 2)^2.
For (b), since F1 is proportional to sqrt(T), increasing T by a factor of 2.25 = 9/4 increases F1 by 3/2 so L would sustain the 4th harmonic resulting in 4 loops.
For (c) increasing T by 9/1 = 9 increases F1 by 3, so L would sustain the 2nd harmonic resulting in 2 loops.

Answer 2

Vibrator Set