Question

A string of length L, mass per unit length u, and tension Tis vibrating at it's fundamental frequency. What effect will the following have on the fundamental frequency? (a) The length of the string is doubled, with all other factors held constant. Of will be reduced by a factor of 2. 1 Of will be increased by a factor of 2. f will be reduced by a factor of 4. Of will be increased by a factor of 4. (b) The mass per unit length is doubled, with all other factors held constant. Of will be reduced by a factor of V3. O f will be increased by a factor of V2. f will be reduced by a factor of V2. Of will be increased by a factor of V3. (c) The tension is doubled, with all other factors held constant. Of will be reduced by a factor of V2, f will be reduced by a factor of V3. f will be increased by a factor of V2. Of will be increased by a factor of V3.

Answer 1

The expression for the fundamental frequency of a wave is as follows: f = 27 Vue Here, f is the fundamental frequency, 1 is the length of string, T is the tension in the string and u is the mass per unit length. The expression for frequency as the length changes is as follow: J'a 271ū Here, f' is the new frequency and l' is the new length. Substitute 21 for 1'. f'= 1 2(21) Vu 1/1 T 21 21 Vu 1 T Substitute 21 Vu for f in above equation. The fundamental frequency will be increased by factor of -

(b) The expression for the fundamental frequency of a wave is as follows: 21 Vu The expression for frequency as the mass per unit length changes is as follow: Here, fi is the new frequency and 4, is the new mass per unit length. Substitute 2 u for 4. IT Substitute - for f in above equation. 21 Yu The fundamental frequency will be increased by factor of

© The expression for the fundamental frequency of a wave is as follows: 1 IT 21 Vu The expression for frequency as the tension changes is as follow: 1221 Vu Here, f, is the new frequency and T, is the new tension. Substitute 2T for T2 fizI 21 f'=27VH Substitute 1 T 21 Vu for f in above equation. fl=/25 The fundamental frequency will be increased by factor of V2.