Friday, May 6, 2016

Hints for Quiz 15

Formulae:    v= λ f;    μ=m/l;    v²=Fₜ/μ; Pμω²A²v;   T=1/f;    ω=2πf; v²=v₀²+2ad; d=vt+at²; L, m=1,2,... (fixed at both ends);    Lλ   m=0,1,...(fixed at one end),

v= λ f          (v Wave speed, λ wavelength, f frequency)
μ=m/l          (μ linear density, m mass, l cord length)
v²=Fₜ/μ        (v speed of transverse waves in a cord,  Fₜ tension force)
Pμω²A²v (P power transported by transverse waves in a string, cord, rope)
v²=v₀²+2ad; d=v₀t+at² (a acceleration, d displacement, t time)
Standing waves with the wavelength λ in a string with the length L:  L, m=1,2,... (strings are fixed at both ends)
Lλ,   m=0,1,...(strings are fixed at only one end)

1. A wire of uniform linear mass density hangs from the ceiling. It takes the time t for a pulse to travel the length of the wire. How long is the wire?
Use: μ=m/l;  F=mg; v²=v₀²+2ad; d=v₀t+at²

2. A stretched string with a mass m and a length l is placed under a tension F. How much power must be supplied to the string to generate traveling waves that have a frequency f and an amplitude A?
Use: μ=m/l;  v²=Fₜ/μ; P=½μω²A²v; v= λ f.

3.  A string of mass m and length l can carry a wave with the power P.  The amplitude of the wave is A.  What is the period of oscillation if the tension applied to the string is F?
Use: μ=m/l;  v²=Fₜ/μ; P=½μω²A²v; T=1/f; v= λ f.

4. A string of mass m and length l can carry a wave with the power P. When a tension of F is applied to the string it oscillates with a period T. What is the amplitude of the wave?
Use: μ=m/l;  v²=Fₜ/μ; P=½μω²A²v; T=1/f.

5. In the figure, point A is below the ceiling on the distance h. Determine how much longer it will take for a pulse to travel along wire 1 than it would along wire 2. Both wires are made of the same material and have identical cross sections.
Use:   μ=m/l,    v²=Fₜ/μ; F=mg.

6. A guitar string with a mass m and a length l is attached to a guitar at two points separated by the distance x. What tension must the guitar string have so that its fundamental note is f?
Use L =½mλ, m=1,2,...; v= λ f; μ=m/l;  v²=Fₜ/μ;

7. Two adjacent frequencies of transverse standing waves on a string held fixed at both ends are f1 and f2. What is the fundamental frequency of the transverse standing waves on this string?
Use: L =½mλ, m=1,2,...;

8.  A wave of amplitude A, wavelength λ, and frequency f.  What is its speed?
Use: v= λ f
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