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125.

Problem 17.35P (HRW)

A transverse sinusoidal wave is generated at one end of a long, horizontal string by a bar that moves up and down through a distance of 1.00 cm. The motion is continuous and is repeated regularly 120 times per second. The string has linear density 120 g/m and is kept under a tension of 90.0 N.

We have to find (a) the maximum value of the transverse speed u and (b) the maximum value of the transverse component of the tension. (c) We have to show that the two maximum values calculated above occur at the same phase values for the wave. We have to find transverse displacement y of the string at these phases. (d) We have to find the maximum rate of energy transfer along the string. (e) We have to find the transverse displacement y when the maximum transfer occurs. (f) We have to find the minimum rate of energy transfer along the string. (g) We have to find the transverse displacement y when this minimum transfer occurs.

Solution:             Click For PDF Version

As the transverse sinusoidal wave is being generated by moving one end of the sting up and down 120 times per second, the frequency, f, of the wave is

and the angular frequency is

The linear mass density,, of the string is

As the string is kept under a tension F = 90.0 N, the speed of transverse wave motion is

Wave number k of the sinusoidal wave is

As the wave is generated by moving one end of the string up and down through a distance of 1.00 cm, the amplitude of transverse oscillation of the string, a, is

We now have all the parameters that determine the equation for the sinusoidal wave motion

Instantaneous transverse speed of oscillation of the string at position x is given by

(a)

We note from the function giving speed of transverse motion of the string that its maximum value is

(b)

The transverse component of the tension in the string at position x and at time t is given by , which for small displacements can be approximated by . We thus have

We note that the maximum value of the transverse component of the tension in the string is

(c)

We note that these two maximum values occur for the same phase value, .

The transverse displacement of the string at this phase is zero.

(d)

The power transferred along the string is given by

Therefore, the maximum power transferred along the string is

(e)

At this phase the transverse displacement of the string is zero.

(f)

Minimum power transferred is zero. It occurs when phase .

(g)

The transverse displacement of the string at these phase is maximum and is