A transverse sinusoidal wave is generated at one end of a...

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 $\mathrm{cm} .$ The motion is continuous and is repeated regularly 120 times per second. The string has linear density 120 $\mathrm{g} / \mathrm{m}$ and is kept under a tension of 90.0 $\mathrm{N} .$ Find the maximum value of (a) the transverse speed $u$ and $(\mathrm{b})$ the transverse component of the tension $\tau$ . (c) Show that the two maximum values calculated above occur at the same phase values for the wave. What is the transverse displacement $y$ of the string at these phases? (d) What is the maximum rate of energy transfer along the string? (e) What is the string? (g) What is the transverse displacement $y$ when this mini- mum transfer occurs? transverse displacement $y$ when this maximum transfer occurs? (f) What is the minimum rate of energy transfer along the string? (g) What is the transverse displacement $y$ when this minimum transfer occurs?

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 $\mathrm{cm} .$ The motion is continuous and is repeated
regularly 120 times per second. The string has linear density 120 $\mathrm{g} / \mathrm{m}$ and is kept under a tension of 90.0 $\mathrm{N} .$ Find the maximum
value of (a) the transverse speed $u$ and $(\mathrm{b})$ the transverse component of the tension $\tau$ . (c) Show that the two maximum values calculated above
occur at the same phase values for the wave. What is the transverse displacement $y$ of the string at these phases? (d) What is the maximum rate of energy transfer along the string? (e) What is the string? (g) What is the transverse displacement $y$ when this mini-
mum transfer occurs?
transverse displacement $y$ when this maximum transfer occurs?
(f) What is the minimum rate of energy transfer along the string? (g) What is the transverse displacement $y$ when this minimum transfer occurs?

Key Concepts

Transverse Components of Tension

In a string under tension, the transverse component of the tension results from the slope of the string at any point. This component, which is proportional to the spatial derivative of the displacement, is important for understanding the forces that act to restore the string to its equilibrium position, and it is related to the curvature of the wave.

Transverse Sinusoidal Waves

Transverse sinusoidal waves are disturbances that travel along a medium, such as a string, where the displacement of the medium is perpendicular to the direction of wave propagation. These waves are characterized by parameters including amplitude, frequency, wavelength, and phase, which determine the shape and evolution of the wave as it moves.

Wave Speed on a String

The speed at which a wave travels along a string is determined by the properties of the string, specifically its tension and linear density. The relationship is given by v = ?(T/?), where T is the tension in the string and ? is the mass per unit length. This speed plays a crucial role in linking the frequency and wavelength of the wave.

Transverse Velocity and Derivatives

The transverse velocity of points on the string is the time derivative of the transverse displacement. It indicates how fast a point on the string is moving up or down. For a sinusoidal wave, the maximum transverse speed is found when the rate of change of displacement with respect to time is greatest, typically when the displacement is zero.

Energy Transfer and Power in Waves

Waves carry energy along the medium, and the rate at which energy is transferred (power) depends on the amplitude, frequency, tension, and other properties of the wave. The instantaneous power can vary along the wave, and its average value over one period provides insight into energy propagation, with maximum power transfer occurring at the phases where the product of the transverse velocity and transverse force is highest.

Phase Relationships in Periodic Motion

Different physical quantities in a sinusoidal wave, such as displacement, velocity, and force, are not in phase with each other. Their phase relationships are key to understanding the dynamics of the wave, including when maximum values of velocity or force occur and how these relate to energy transfer. For example, maximum transverse velocity typically occurs at zero displacement, illustrating the phase difference between displacement and its time derivative.

Transcript

00:01 In this question, we have this wave that has a peak -to -peak amplitude of 1 cm.

00:06 It has a frequency of 100 per second.

00:08 It has linear mass density of 120 grams per meter.

00:13 And then finally, tension of 90 newton.

00:17 Part a asked us to find the maximum transfer speed of a point.

00:24 So we already did this equation in chapter 15.

00:28 And the equation is ym times omega.

00:34 But i think it will be helpful due to part b to actually write out, quickly write out the derivation again.

00:42 It's because we have this wave can be described by ym sine x minus omega -t.

00:53 So when we want to find u, u is the derivative of y with respect to t.

01:02 So it's ym, negative ym, omega, cosine, kx minus omega t.

01:12 So its maximum value will be ym times omega when cosine equals negative 1.

01:20 Okay, so in this case, we have peak to peak is 1 centimeter.

01:24 So the amplitude is half of that is 5mm.

01:31 We also need to find what omega is.

01:33 Omega is 2 pi times frequency, which is 120 hertz.

01:40 So this gives us a maximum transfer speed of 3 .7 meter per second.

01:46 More like 3 .8, 3 .77.

01:49 Second, we want to find the largest transverse component of tension.

01:56 What does that mean? so we have tension, and tension is always along the string.

02:04 So then there is this horizontal component of tension.

02:09 We want to find the maximum correspondental component of this tension.

02:16 How do we find that? so this can be found if we, let's, let me draw a zoom dot graph.

02:31 So for every single component, there is a y, there is an x, in this angle, theta.

02:48 This angle theta can be described by, actually, let's use a different angle.

02:56 We can use this bottom angle, theta.

03:01 This angle theta can be described by tangent theta equals delta y over delta x.

03:11 But then how is tension related to this theta? so tension is, this tension has a transverse component.

03:35 This is a transverse component.

03:42 Sorry, this is the transverse component of the tension.

03:45 And this is the direction of the tension.

03:52 So the component of tension in this direction will be tension in the transverse direction equals total tension times sine theta.

04:11 Okay, so we need to use d -y -d -x.

04:15 Like, we will take the derivative of this equation, this time with respect to x to find what tangent theta is.

04:23 And then we will use that tangent of theta, and we will take the maximum value of that.

04:29 We will take the maximum value of theta to find what tangent transverses, so the top of transverses.

04:40 Okay, so let's first take this derivative.

04:47 If we take this with respect to x, we get k times y m times sine cosine, k x minus omega t.

05:03 So the maximum value it could possibly obtain is k times y m.

05:10 So tangent theta maximum, which is also theta maximum since tangent is monotonously increasing, theta equals inverse tangent of k times while m.

05:32 So we need to find k first then we can find theta.

05:38 What is k? k is 2 pi over lambda but we also do not know what lambda is so we need to we can say k equals 2 pi over lambda but then lambda equals v over so because of that, k equals omega over v.

06:21 And that's probably the easiest expression we can use for now.

06:25 Okay, so omega is 2 pi f, v is square root of tau over mu, and then tau over mu, this whole thing again times y m.

06:45 So it is, in first tangent of 2 pi times 120 hertz over over square root of 90 newton over 0 .12 kilogram per meter times ym is 5mm.

07:08 Our calculator tells us this is around 7 .8 theta.

07:15 So that is the maximum theta we have.

07:18 And the larger the theta is, the larger the transfer's component.

07:24 Of the tension is.

07:27 So the largest transverse component of tension happens when theta equals 7 .83 degree and that is 90 newton times sine of 7 .83 degree and it is 12 .3 newton...

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