Question

Two waves traveling in opposite directions on a stretched rope interfere to give the standing wave described by the following wave function:

y(x,t) = 4 sin⁡(2πx) cos⁡(120πt),

where, y is in centimetres, x is in meters, and t is in seconds. The rope is two meters long, L = 2 m, and is fixed at both ends.

A)Which of the following represents the two individual waves, y1 and y2, which produce the above standing waves:

1)y1 = 2 sin⁡(2πx ‒ 120πt), and y2 = 2 sin⁡(2πx ‒ 120πt)

2)y1 = 4 sin⁡(2πx ‒ 120πt), and y2 = 4 sin⁡(2πx + 120πt)

3)y1 = 2 sin⁡(2πx ‒ 120πt), and y2 = 2 sin⁡(2πx + 120πt)

4)y1 = 1 cos⁡(2πx ‒ 120πt), and y2 = 3 cos⁡(120πx + 2πt)

5)y1 = 1 sin⁡(2πx ‒ 120πt), and y2 = 3 sin⁡(2πx + 120πt + π)

Other:

B)The distance between two successive antinodes is:

1) d_AA = 0.25 m

2) d_AA = 0.15 m

3) d_AA = 1 m

4) d_AA = 0.5m

5) d_AA = 2 m

Other:

C) The fundamental resonance frequency on this rope is:

1) f1 = 60 Hz

2) f1 = 30 Hz

3) f1 = 25 Hz

4) f1 = 20 Hz

5) f1 = 15 Hz

Other:

C)The maximum transverse speed of an element on the rope located at the position x = 1.5 m is:

1) v_(y,max) = 480 π cm/s

2) v_(y,max) = 240 cm/s

3) v_(y,max) = 0 cm/s

4) v_(y,max) = 120 π cm/s

5) v_(y,max) = 60 cm/s

Other:

D)In terms of the oscillation period, T, at which of the following times would all elements on the string have a zero vertical displacement, y(x,t) = 0, for the first time:

1) t = T/8

2) t = T/4

3) t = T/2

4) t = 3T/4

5) t = T

Other:

e) If the oscillation frequency is decreased by a factor of four, f_new = f/4, while keeping the tension force, the length of the rope, and the linear mass density constants, then how many loops would appear on the rope?

1) One loop

2) Two loops

3) Three loops

4) Four loops

5) No loops appear, because the conditions are not satisfied for the standing waves to exist.

Answer 1