A 12-kg object hangs in equilibrium from a string of total...

A $12-\mathrm{kg}$ object hangs in equilibrium from a string of total length $L=5.0 \mathrm{~m}$ and linear mass density $\mu=$ $0.0010 \mathrm{~kg} / \mathrm{m}$. The string is wrapped around two light, frictionless pulleys that are separated by the distance $d=$ $2.0 \mathrm{~m}$ (Fig. P14.45a). (a) Determine the tension in the string. (b) At what frequency must the string between the pulleys vibrate in order to form the standing-wave pattern shown in Figure $\mathrm{P} 14.45 \mathrm{~b}$ ?

Key Concepts

Mechanical Equilibrium in Heavy Ropes

When a rope (or string) has mass, its weight causes the tension to vary from point to point along its length. In a rope freely hanging under gravity in equilibrium, the tension at a given point is equal to the weight of all the rope hanging below that point. When additional masses are attached to the rope, the tension at a given point is determined by both the weight of the attached mass (if it hangs below that point) and the weight of the rope below that point. In systems in which the rope is arranged over pulleys and portions are oriented in different directions (vertical or horizontal), the equilibrium conditions must be applied separately to each segment; however, the rope may be divided by supports in such a way that different segments can have different tensions even though the rope is continuous.

Wave Propagation on a Stretched String

Small?amplitude transverse waves travel along a stretched string with a speed given by v = ?(T/?), where T is the tension in the string and ? is the linear mass density. This relation shows that for a given string, an increase in the tension produces a higher wave speed. This concept is used when one must determine the frequency of standing waves on a given segment, since the wave speed enters into the relation between frequency, wavelength, and mode number.

Standing Waves on a String Fixed at Both Ends

When a section of string is fixed (or effectively pinned) at both ends, only certain wavelengths can produce standing waves. For the fundamental mode, the distance between the fixed ends equals one?half of a wavelength (?/2 = L, so ? = 2L). Higher harmonics occur when the string length is an integer multiple of half wavelengths. The relation f = v/? then yields the frequencies of the allowed standing wave patterns. In problems of this type, once the appropriate tension (and hence wave speed) is found for the vibrating portion, the standing?wave condition together with v = ?(T/?) allows one to compute the required frequency.

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