Solve problems a and b below. a. In the figure below, a siring it tied to a sinu

Question

Solve problems a and b below. a. In the figure below, a siring it tied to a sinusoidal oscillator at P and rum over a rigid support at Q, and is stretched by a block of mass m. The separation L = 1.77 m, the linear density mu = 1.6 gm, and the oscillator frequency f = 125 Hz. The amplitude of the motion at P is small enough for that point to be considered a node A node also exists at Q. If m = 2 00 kg is needed for the fourth harmonic (with four anti nodes) as shown, then find the mass m needed to get the seventh harmonic (with seven antinodes). b. A stretched string has a mass per unit length of 5.78 g cm and a tension of 14.8 N. A sinusoidal wave on this string has an amplitude of 0.113 mm and a frequency of 147 Hz and is traveling in the negative direction of an x axis. If the wave equation is of the form y(x, t) = y_m sin (kx + omega t), what arc (i) y_m (in mm), (ii) k, and (iii) omega?

Explanation / Answer

a)L=1.77m, µ=1.6g/m= 1600kg/m,f=125hz

f=nv/2L

f = [n*sqrt(T/µ)]/(2L)

f = [n*sqrt(mg/µ)]/(2L)

Plugging given data,

125=[7*sqrt((m*9.8)/(0.0016)]/(2*1.77)

m= 0.65kg

b) µ=5.78g/cm= 0.578kg/m,T=14.8 N, ym=0.113mm, f=147Hz

i)

ym= 0.113mm

iii)

=2f = 2*3.14*147 = 923.16 rad/s

ii)

=v/f = sqrt(T/µ)/f = [sqrt(14.8/0.578)]/147 = 0.034m

k=2/ = (2*3.14)/0.034 = 184.7 Hz

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