(II) A wire is composed of aluminum with length ℓ1=0.600 m and mass per unit length μ1=2.70 g /

(II) A wire is composed of aluminum with length ℓ1=0.600 m...

(II) A wire is composed of aluminum with length $\ell_{1}=0.600 \mathrm{m}$ and mass per unit length $\mu_{1}=2.70 \mathrm{g} / \mathrm{m}$ joined to a steel section with length $\ell_{2}=0.882 \mathrm{m}$ and mass per unit length $\mu_{2}=7.80 \mathrm{g} / \mathrm{m} .$ This composite wire is fixed at both ends and held at a uniform tension of 135 $\mathrm{N}$ . Find the lowest frequency standing wave that can exist on this wire, assuming there is a node at the joint between aluminum and steel. How many nodes (including the two at the ends) does this standing wave possess?

Key Concepts

Composite Systems in Wave Mechanics

Composite systems involve segments made of different materials, each with distinct physical properties such as mass per unit length. When analyzing such systems, it is crucial to account for the variation in wave speed across different sections while maintaining continuity conditions at the interface between the materials.

Harmonics and Frequency Quantization

Harmonics refer to the integer multiples of the fundamental frequency that a system can support when the wave is constrained by its boundary conditions. The quantization of frequencies is a key aspect when analyzing standing waves on strings, as only specific wavelengths (and thus frequencies) satisfy the node conditions at the boundaries.

Boundary Conditions

Boundary conditions specify the constraints at the ends or junctions of a medium where waves are established. For instance, fixed ends enforce nodes, and in composite systems, additional enforced nodes (such as at the junction between different materials) further dictate the allowed vibration patterns and frequencies.

Standing Waves

Standing waves are the result of the interference between two waves traveling in opposite directions in a medium. They are characterized by fixed nodes where the amplitude remains zero and antinodes where the amplitude fluctuates, and are a fundamental concept in the study of waves on strings and other oscillatory systems.

Wave Speed in a Medium

The speed at which a wave propagates along a string or wire is governed by the tension in the medium and its mass per unit length, encapsulated in the formula v = ?(T/?). This relationship is essential for determining how quickly disturbances travel through different sections of a composite wire.

Transcript

00:01 We have a wire that is composed of aluminum with lens l1 equals 0 .6 meter and mass density of mu1 equals 2 .7 gram per meter.

00:15 And it is joined to a different section, the steel section, that has a different lens and a different mass.

00:27 So this wire is fixed at both ends and it is held at a uniform tension of ft equals 135 neuter.

00:49 So we want to find the lowest frequency standing wave that can exist on this wire, assuming the joint is a knot.

01:01 And we ask how many standing wave will we have? okay.

01:09 So in order to do that, so frequency for each string has to be the same for a standing wave to exist because the string is continuous at its junction.

01:21 So each string will obey the relationship that v equals lambda times f and lambda is 2l over an n.

01:33 F is f as a yeah okay because n and this v equals f equals f x equals f and this v equals f t over as the definition of f.

01:50 So lambda f equals v.

01:54 So we can combine these to find the notes.

01:59 Because if we rearrange this, we get f equals n over 2l times square root of ft over mu.

02:08 So we can say f to the left equals f to the right.

02:13 So n equals left 2 times l.

02:18 2 times l to the left and f to the left over mu equals n to the right to l to the right, square root of f to the right over mu.

02:37 And f is the same, it's uniform, so we can cancel that.

02:43 So this gives us n to the left over n to the right equals l to the left over n to the right equals l to the left over l to the right times square root of mew to the left and mew to the right.

03:01 Now we use all the parameters that we were told, square root of 2 .7 gram per meter and 7 .8 gram per meter...

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