In Fig. 16-43 an aluminum wire, of length L1=60.0 cm cross-sectional area 1.00 ×10^-2 cm^2, an

step 1 So this is a very interesting question. We have L1 and L2 and these two wires are made of differ

In Fig. 16-43 an aluminum wire, of length L1=60.0 cm...

In Fig. 16-43 an aluminum wire, of length $L_{1}=60.0 \quad \mathrm{cm}$ cross-sectional area 1.00 $\times 10^{-2} \mathrm{cm}^{2},$ and density 2.60 $\mathrm{g} / \mathrm{cm}^{3}$ , is joined to a steel wire, of density 7.80 $\mathrm{g} / \mathrm{cm}^{3}$ and the same cross-sectional area. The compound wire, loaded with a block of mass $m=10.0 \mathrm{kg},$ is arranged so that the distance $L_{2}$ from the joint to the supporting pulley is 86.6 $\mathrm{cm}$ . Transverse waves are set up on the wire by an ex- ternal source of variable frequency; a node is located at the pulley.

In Fig. 16-43 an aluminum wire, of
length $L_{1}=60.0 \quad \mathrm{cm}$
cross-sectional area 1.00
$\times 10^{-2} \mathrm{cm}^{2},$ and density
2.60 $\mathrm{g} / \mathrm{cm}^{3}$ , is joined to a
steel wire, of density 7.80 $\mathrm{g} / \mathrm{cm}^{3}$ and the same
cross-sectional area. The compound wire, loaded with a block of mass $m=10.0 \mathrm{kg},$ is
arranged so that the distance $L_{2}$ from the joint to the supporting
pulley is 86.6 $\mathrm{cm}$ . Transverse waves are set up on the wire by an ex-
ternal source of variable frequency; a node is located at the pulley.

Key Concepts

Wave Speed on a String

The speed at which transverse waves propagate along a stretched string or wire is determined by the tension in the string and its linear mass density. This relationship is given by the equation v = ?(T/?), where v is the wave speed, T is the tension, and ? is the mass per unit length. This concept is fundamental to understanding wave behavior in any medium that supports tension-induced wave propagation.

Linear Mass Density

Linear mass density is defined as the mass per unit length of a wire, string, or rope. It plays a crucial role in determining the dynamics of wave propagation because it factors directly into the wave speed. Understanding how to calculate and interpret linear mass density is essential when analyzing waves in different materials or geometrical configurations.

Tension due to Gravity

In many practical scenarios, particularly those involving hanging masses or suspended objects, the tension in the string arises from the weight of the load. The gravitational force acting on an object of mass m results in tension T = mg. This tension directly affects the speed of wave propagation along the string and is a key concept when analyzing oscillations and vibrations in such systems.

Standing Waves and Nodes

When waves reflect and interfere within a bounded medium, they can form standing wave patterns characterized by fixed nodes and antinodes. Nodes are points where the medium does not move, and their positions can be used to determine wavelengths and frequencies of the standing waves. Mastery of the boundary conditions leading to standing wave formation is essential for applications in musical instruments, transmission lines, and more.

Composite Media and Wave Transmission

When a wave travels through a medium composed of different segments or materials, such as a composite or compound string, the properties like density and tension may vary along its length. Analyzing wave transmission in these cases involves applying continuity conditions at the interfaces and understanding how changes in medium properties affect the propagation speed, reflection, and transmission of waves.

Transcript

00:01 So this is a very interesting question.

00:03 We have l1 and l2 and these two wires are made of different material.

00:08 One is aluminum and one is steel.

00:11 And we hand a mass of 10 kilograms from like the end of these two jointed strings.

00:19 And we want to induce, want to have a standing wave on the string so that the joint is a knot.

00:31 So let's first think about what the standing wave would look like.

00:36 Apparently there will be some standing wave.

00:40 I'm just drawing for reference, like as an indication, don't like, don't take how many periods i draw is the answer.

00:51 Maybe we can have some up here and then we can maybe have some up here.

00:57 But then the thing is that the joint better still stay stable.

01:02 And the next thing is that because aluminum and steel have very different, they have different mass density.

01:10 They have the same tension because the same tension induced by this 10 kilogram mass, but they have different mass density.

01:17 So they will have different velocity and because of that the wavelengths may be different on these two sections of the wide.

01:29 So let's first calculate the ratio of the velocities.

01:44 So let's think about it.

01:47 We have a row.

01:50 We have area, this is area, cross -sectional area, and we have row, and these two will be enough to give us the mass density in similar here.

02:02 For a steel string has the same cross section.

02:06 So we can find the velocity difference, the velocity for the aluminum string and the steel string.

02:13 Then how does that velocity connect to the wavelengths? let's write out l.

02:21 Let's say l is like the lens of the string, it can be l1 or l2.

02:26 So l is lambda over 2.

02:33 Times n and lambda is v over f it's v over so in we have other words we have frequency equals um nv over 2 l so this is true for both aluminum string and for the steel string and the frequency has to be the same for both sections why? because, i mean, it's the same wave.

03:15 If they have the different frequency, if you think about it, if this green wave oscillates five times, while the blue wave only oscillates three times, it's not going to be a continuous string, it's a joint.

03:29 It has to be a very continuous point.

03:33 You can only have one motion at this joint.

03:37 So because of that, the left and right has to have the same periods.

03:41 Have to have the same oscillating frequency.

03:44 So what that means is n1 v1 over 2l1 has to be equal to m2 v2 over 2l2.

03:54 So that is a boundary condition we have between the aluminum string and the steel string.

04:03 Okay, so this now we need to really plug in what the velocity is, because we know what l is but we don't know what velocity is.

04:16 So velocity is tau over mu...