P17_053.PDF

53. We refer to the points where the rope is attached as A and B , respectively. When A and B are not

displaced horizontal, the rope is in its initial state (neither stretched (under tension) nor slack). If they are

displaced away from each other, the rope is clearly stretched. When A and B are displaced in the same

direction, by amounts (in absolute value)

ξ A |

and

ξ B |

,thenif

ξ A |

ξ B |

then the rope is stretched,

the rope is slack. We must be careful about the case where one is displaced but the

other is not, as will be seen below.

ξ A |

ξ B |

(a) The standingwave solution for the shorter cable, appropriate for the initial condition ξ =0at

t = 0, and the boundary conditions ξ =0at x =0and x = L (the x axis runs vertically here),

is ξ A = ξ m sin( k A x ) sin( ω A t ). The angular frequency is ω A =2 π/T A ,andthewavenumberis

k A =2 π/λ A where λ A =2 L (it begins oscillating in its fundamental mode) where the point of

attachment is x = L/ 2. The displacement of what we are callingpoint A at time t = ηT A (where

η is a pure number) is

T A ηT A = ξ m sin(2 πη ) .

The fundamental mode for the longer cable has wavelength λ B =2 λ A =2(2 L )=4 L , which implies

(by v = fλ and the fact that both cables support the same wave speed v )that f B = 2 f A or

ω B = 2 ω A . Thus, the displacement for point B is

ξ B = ξ m sin 2 π

4 L

ξ A = ξ m sin 2 π

2 L

sin 2 π

√ 2 sin( πη ) .

Runningthrough the possibilities ( η = 4 , 2 , 4 , 1 , 4 , 2 , 4 , and 2) we ﬁnd the rope is under tension

in the followingcases. The ﬁrst case is one we must be very careful in our reasoning, since A is not

displaced but B is displaced in the positive direction; we interpret that as the direction away from

A (rightwards in the ﬁgure) – thus making the rope stretch.

η = 1

sin 1

2 π

T A

ηT A = ξ m

ξ A =0

ξ B = ξ m

√ 2 > 0

η = 3

ξ A =

−

ξ m < 0 ξ B = ξ m

2 > 0

ξ m

2 < 0

where in the last case they are both displaced leftward but A more so than B so that the rope is

indeed stretched.

(b) The values of η (where we have deﬁned η = t/T A ) which reproduce the initial state are

η =1 ξ A =0 ξ B =0 and

η =2 ξ A =0 ξ B =0 .

η = 7

ξ A =

−

ξ m < 0 ξ B =

−

are to the right, but point A is farther to the right than B . In the second case, they are displaced

towards each other.

η = 1

ξ A = x m > 0 ξ B = ξ m

√ 2 > 0

η = 5

ξ A = ξ m > 0 ξ B =

−

ξ m

2 < 0

ξ m

√ 2 < 0

whereinthethirdcase B is displaced leftward toward the undisplaced point A .

(d) The ﬁrst design works eﬀectively to damp fundamental modes of vibration in the two cables (espe-

cially in the shorter one which would have an antinode at that point), whereas the second one only

damps the fundamental mode in the longer cable.

η = 3

ξ A =0

ξ B =

−

and if

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika: