In the figure, a string, tied to a sinusoidal oscillator at P and running over a
ID: 3278309 • Letter: I
Question
In the figure, a string, tied to a sinusoidal oscillator at P and running over a support at Q, is stretched by a block of mass m. Separation L = 1.4 m, linear density mu = 1.4 g/m, and the oscillator frequency f = 110 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. (a) What mass m allows the oscillator to set up the fourth harmonic on the string? (b) What standing wave mode, if any, can be set up if m = 1 kg (Give 0 if the mass cannot set up a standing wave? (a) Number 0.22 Units (b) Number 3.679 UnitsExplanation / Answer
as we can see
length of the wire under tension, l = 1.4 m
linear density of the wire, mu = 1.4 g/m
osscilator frequency f = 110 Hz
a) for the fourth harmonic, as seen in the picture
lambda = L/2
now speed of wave on a string v = sqroot(T/mu)
but v = lambda*f
so, lambda*f = sqroot(T/mu)
L*f/2 = sqroot(T/mu)
1.4*110/2 = sqroot(T/0.0014)
T = 8.3006
but from the diagram T = mg
so m = 0.846 kg
b) let mass m = 1 kg
then T = mg = 9.81 N
so v = sqroot(T/mu) = sqroot(9.81/0.0014) = 83.708 m/s = lambda*f
noiw f = 110 Hz
so, lambda = 0.7609
for closed closed case, lambda = 2L/n
so for no integral valuer of n this is true
hence no atanding wave will form at m = 1 kg
so 0