A guitar string vibrates in its fundamental mode, with nodes at its ends. The length of the rope segment that vibrates freely is 0.386 m. The maximum transverse acceleration of a point at the midpoint of the segment is 8.40x103 m / s2, and the maximum transverse velocity is 3.80 m / s. a) Calculate the amplitude of this standing wave. b) How fast are transverse traveling waves in this rope?
a |
a) 1.72 m b) 543 m/s |
b |
a) 1.72x10-3 m b) 272 m/s |
c |
a)6.45x10-3 m b) 234 m/s |
d |
a) 5.77x10-9 m b) 275 m/s |
the Displacement of the string is
Y(x,t)= A sin kx sin wt
Distance between the adj nods = lambda/2
length of the string lamba = 2 L
Amplitude of the wave ASinkx
then sin kx = sin(2pi/2l) * L/2 = 1
y(x,t) = A sin Wt
using L = 0.386 m
diff above eqn with respect to t,
Vy = dy/dy = W A Coswt
Max travservse velocity Vmax = W A = 3.8 m/s^2
Diff Vy w r t t to get accleration
a= dV/dt = -W^2 A sin wt
for max , a = -w^2 A
angular velocity W = a/V
W = (8.4 e 3)/3.8 = 2.21 e3 rad/s
Amplitude A =V/w = 3.8/2.21 e 3 = 1.72 *10^-3 m --(ANSWER)
----------------------
as V = Lambda*f
V = 2L * W/2pi
V = wL/pi
V = (0.386 * 2.21 e 3)/(3.14)
V = 272 m/s ------(ANSWER)
option B it is
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