One end of a horizontal rope is attached to a prong of an...

One end of a horizontal rope is attached to a prong of an electrically driven tuning fork that vibrates the rope transversely at $120 \mathrm{~Hz}$. The other end passes over a pulley and supports a $1.50 \mathrm{~kg}$ mass. The linear mass density of the rope is $0.0480 \mathrm{~kg} / \mathrm{m} .$ (a) What is the speed of a transverse wave on the rope? (b) What is the wavelength? (c) How would your answers to parts (a) and (b) change if the mass were increased to $3.00 \mathrm{~kg} ?$

Key Concepts

Effect of Changing Tension on Wave Properties

Altering the mass attached to the rope changes the tension, and thereby the wave speed, according to the square root rule v = ?(T/?). Since the wavelength is determined by the product of wave speed and frequency, changes in mass directly affect the characteristics of the wave, making it a key concept in understanding wave behavior in such systems.

Tension Due to Hanging Mass

In systems where a rope supports a hanging mass, the tension in the rope is primarily provided by the gravitational force acting on the mass, given by T = mg. This affects the wave speed, as an increased mass leads to a higher tension, consequently affecting the dynamics of the wave traveling along the rope.

Transverse Waves

Transverse waves are disturbances that travel through a medium wherein the displacement of the medium is perpendicular to the direction of wave propagation. They are common in media like ropes, strings, and other flexible materials, and their behavior is influenced by factors such as tension and mass density.

Wave Speed in a Tensioned Rope

The speed at which a transverse wave propagates on a rope or string is governed by the equation v = ?(T/?), where T is the tension in the rope and ? is its linear mass density. This relationship shows that increasing the tension or decreasing the mass per unit length results in a faster wave propagation.

Frequency and Wavelength Relationship

The relationship between frequency, wavelength, and wave speed is defined by the equation v = f?, where f represents the oscillation frequency and ? the wavelength. This relationship allows us to determine one of these variables if the other two are known.

Transcript

00:01 So we have a rope and pulley system here where one end of the rope is attached to a tuning fork, which is vibrating into 120 hertz.

00:11 And then on the other end of the rope, we have a mass hanging down at 1 .5 kilograms.

00:17 And we want to know first what the speed of the transverse wave on the rope is, the wave that is created by this tuning fork.

00:26 So we want to find v.

00:28 And remember our handi relation v is the square root of the tension over the mass per unit length.

00:34 We're given the mass per unit length, by the way, over here, it's 0 .048 kilograms.

00:40 And we know that the tension in the rope is simply going to be the force from the mass pulling down is mg, which is 1 .5 kilograms times 10, 9 .8.

00:58 Meters per second squared.

01:02 And so we can rewrite this as the square root of 1 .5 kilograms times 9 .8 meters per second squared.

01:16 That is g divided by 0 .048 kilograms per meter.

01:29 And that gives us a final answer of drum roll 17 .5 meters per second.

01:48 Great.

01:50 So for part b, we're asked what the wavelength is.

01:59 So recall that v is equal to, here i'll write my fundamental equation.

02:05 So v is equal to lambda f.

02:09 So that means lambda is going to be equal to v over f.

02:13 And we're given the frequency here, and we just calculated v...